3.11.45 \(\int \frac {(A+B x) (b x+c x^2)^{5/2}}{(d+e x)^5} \, dx\)
Optimal. Leaf size=633 \[ -\frac {5 \sqrt {c} \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b x+c x^2}}\right ) \left (4 A c e (2 c d-b e)-B \left (3 b^2 e^2-20 b c d e+24 c^2 d^2\right )\right )}{4 e^7}-\frac {5 \left (b x+c x^2\right )^{3/2} \left (3 e x \left (A e \left (b^2 e^2-8 b c d e+8 c^2 d^2\right )-B d \left (9 b^2 e^2-32 b c d e+24 c^2 d^2\right )\right )+d \left (A e \left (-b^2 e^2-12 b c d e+16 c^2 d^2\right )-B d \left (7 b^2 e^2-52 b c d e+48 c^2 d^2\right )\right )\right )}{96 d e^4 (d+e x)^3 (c d-b e)}-\frac {5 \sqrt {b x+c x^2} \left (-2 c e x \left (A e \left (b^2 e^2-16 b c d e+16 c^2 d^2\right )-B d \left (17 b^2 e^2-64 b c d e+48 c^2 d^2\right )\right )-A e \left (b^3 e^3+16 b^2 c d e^2-80 b c^2 d^2 e+64 c^3 d^3\right )+B d \left (-7 b^3 e^3+120 b^2 c d e^2-304 b c^2 d^2 e+192 c^3 d^3\right )\right )}{64 d e^6 (d+e x) (c d-b e)}+\frac {5 \left (A e \left (-b^4 e^4-16 b^3 c d e^3+144 b^2 c^2 d^2 e^2-256 b c^3 d^3 e+128 c^4 d^4\right )-B d \left (7 b^4 e^4-168 b^3 c d e^3+672 b^2 c^2 d^2 e^2-896 b c^3 d^3 e+384 c^4 d^4\right )\right ) \tanh ^{-1}\left (\frac {x (2 c d-b e)+b d}{2 \sqrt {d} \sqrt {b x+c x^2} \sqrt {c d-b e}}\right )}{128 d^{3/2} e^7 (c d-b e)^{3/2}}+\frac {\left (b x+c x^2\right )^{5/2} (-A e+3 B d+2 B e x)}{4 e^2 (d+e x)^4} \]
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Rubi [A] time = 0.90, antiderivative size = 633, normalized size of antiderivative = 1.00,
number of steps used = 8, number of rules used = 6, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used =
{812, 810, 843, 620, 206, 724} \begin {gather*} -\frac {5 \left (b x+c x^2\right )^{3/2} \left (3 e x \left (A e \left (b^2 e^2-8 b c d e+8 c^2 d^2\right )-B d \left (9 b^2 e^2-32 b c d e+24 c^2 d^2\right )\right )+d \left (A e \left (-b^2 e^2-12 b c d e+16 c^2 d^2\right )-B d \left (7 b^2 e^2-52 b c d e+48 c^2 d^2\right )\right )\right )}{96 d e^4 (d+e x)^3 (c d-b e)}-\frac {5 \sqrt {b x+c x^2} \left (-2 c e x \left (A e \left (b^2 e^2-16 b c d e+16 c^2 d^2\right )-B d \left (17 b^2 e^2-64 b c d e+48 c^2 d^2\right )\right )-A e \left (16 b^2 c d e^2+b^3 e^3-80 b c^2 d^2 e+64 c^3 d^3\right )+B d \left (120 b^2 c d e^2-7 b^3 e^3-304 b c^2 d^2 e+192 c^3 d^3\right )\right )}{64 d e^6 (d+e x) (c d-b e)}-\frac {5 \sqrt {c} \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b x+c x^2}}\right ) \left (4 A c e (2 c d-b e)-B \left (3 b^2 e^2-20 b c d e+24 c^2 d^2\right )\right )}{4 e^7}+\frac {5 \left (A e \left (144 b^2 c^2 d^2 e^2-16 b^3 c d e^3-b^4 e^4-256 b c^3 d^3 e+128 c^4 d^4\right )-B d \left (672 b^2 c^2 d^2 e^2-168 b^3 c d e^3+7 b^4 e^4-896 b c^3 d^3 e+384 c^4 d^4\right )\right ) \tanh ^{-1}\left (\frac {x (2 c d-b e)+b d}{2 \sqrt {d} \sqrt {b x+c x^2} \sqrt {c d-b e}}\right )}{128 d^{3/2} e^7 (c d-b e)^{3/2}}+\frac {\left (b x+c x^2\right )^{5/2} (-A e+3 B d+2 B e x)}{4 e^2 (d+e x)^4} \end {gather*}
Antiderivative was successfully verified.
[In]
Int[((A + B*x)*(b*x + c*x^2)^(5/2))/(d + e*x)^5,x]
[Out]
(-5*(B*d*(192*c^3*d^3 - 304*b*c^2*d^2*e + 120*b^2*c*d*e^2 - 7*b^3*e^3) - A*e*(64*c^3*d^3 - 80*b*c^2*d^2*e + 16
*b^2*c*d*e^2 + b^3*e^3) - 2*c*e*(A*e*(16*c^2*d^2 - 16*b*c*d*e + b^2*e^2) - B*d*(48*c^2*d^2 - 64*b*c*d*e + 17*b
^2*e^2))*x)*Sqrt[b*x + c*x^2])/(64*d*e^6*(c*d - b*e)*(d + e*x)) - (5*(d*(A*e*(16*c^2*d^2 - 12*b*c*d*e - b^2*e^
2) - B*d*(48*c^2*d^2 - 52*b*c*d*e + 7*b^2*e^2)) + 3*e*(A*e*(8*c^2*d^2 - 8*b*c*d*e + b^2*e^2) - B*d*(24*c^2*d^2
- 32*b*c*d*e + 9*b^2*e^2))*x)*(b*x + c*x^2)^(3/2))/(96*d*e^4*(c*d - b*e)*(d + e*x)^3) + ((3*B*d - A*e + 2*B*e
*x)*(b*x + c*x^2)^(5/2))/(4*e^2*(d + e*x)^4) - (5*Sqrt[c]*(4*A*c*e*(2*c*d - b*e) - B*(24*c^2*d^2 - 20*b*c*d*e
+ 3*b^2*e^2))*ArcTanh[(Sqrt[c]*x)/Sqrt[b*x + c*x^2]])/(4*e^7) + (5*(A*e*(128*c^4*d^4 - 256*b*c^3*d^3*e + 144*b
^2*c^2*d^2*e^2 - 16*b^3*c*d*e^3 - b^4*e^4) - B*d*(384*c^4*d^4 - 896*b*c^3*d^3*e + 672*b^2*c^2*d^2*e^2 - 168*b^
3*c*d*e^3 + 7*b^4*e^4))*ArcTanh[(b*d + (2*c*d - b*e)*x)/(2*Sqrt[d]*Sqrt[c*d - b*e]*Sqrt[b*x + c*x^2])])/(128*d
^(3/2)*e^7*(c*d - b*e)^(3/2))
Rule 206
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])
Rule 620
Int[1/Sqrt[(b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Dist[2, Subst[Int[1/(1 - c*x^2), x], x, x/Sqrt[b*x + c*x^2
]], x] /; FreeQ[{b, c}, x]
Rule 724
Int[1/(((d_.) + (e_.)*(x_))*Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2]), x_Symbol] :> Dist[-2, Subst[Int[1/(4*c*d
^2 - 4*b*d*e + 4*a*e^2 - x^2), x], x, (2*a*e - b*d - (2*c*d - b*e)*x)/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a,
b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[2*c*d - b*e, 0]
Rule 810
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> -Si
mp[((d + e*x)^(m + 1)*(a + b*x + c*x^2)^p*((d*g - e*f*(m + 2))*(c*d^2 - b*d*e + a*e^2) - d*p*(2*c*d - b*e)*(e*
f - d*g) - e*(g*(m + 1)*(c*d^2 - b*d*e + a*e^2) + p*(2*c*d - b*e)*(e*f - d*g))*x))/(e^2*(m + 1)*(m + 2)*(c*d^2
- b*d*e + a*e^2)), x] - Dist[p/(e^2*(m + 1)*(m + 2)*(c*d^2 - b*d*e + a*e^2)), Int[(d + e*x)^(m + 2)*(a + b*x
+ c*x^2)^(p - 1)*Simp[2*a*c*e*(e*f - d*g)*(m + 2) + b^2*e*(d*g*(p + 1) - e*f*(m + p + 2)) + b*(a*e^2*g*(m + 1)
- c*d*(d*g*(2*p + 1) - e*f*(m + 2*p + 2))) - c*(2*c*d*(d*g*(2*p + 1) - e*f*(m + 2*p + 2)) - e*(2*a*e*g*(m + 1
) - b*(d*g*(m - 2*p) + e*f*(m + 2*p + 2))))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[b^2 - 4*a*
c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && GtQ[p, 0] && LtQ[m, -2] && LtQ[m + 2*p, 0] && !ILtQ[m + 2*p + 3, 0]
Rule 812
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Sim
p[((d + e*x)^(m + 1)*(e*f*(m + 2*p + 2) - d*g*(2*p + 1) + e*g*(m + 1)*x)*(a + b*x + c*x^2)^p)/(e^2*(m + 1)*(m
+ 2*p + 2)), x] + Dist[p/(e^2*(m + 1)*(m + 2*p + 2)), Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^(p - 1)*Simp[g*(
b*d + 2*a*e + 2*a*e*m + 2*b*d*p) - f*b*e*(m + 2*p + 2) + (g*(2*c*d + b*e + b*e*m + 4*c*d*p) - 2*c*e*f*(m + 2*p
+ 2))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2
, 0] && RationalQ[p] && p > 0 && (LtQ[m, -1] || EqQ[p, 1] || (IntegerQ[p] && !RationalQ[m])) && NeQ[m, -1] &&
!ILtQ[m + 2*p + 1, 0] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])
Rule 843
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Dis
t[g/e, Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p, x], x] + Dist[(e*f - d*g)/e, Int[(d + e*x)^m*(a + b*x + c*x^
2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0]
&& !IGtQ[m, 0]
Rubi steps
\begin {align*} \int \frac {(A+B x) \left (b x+c x^2\right )^{5/2}}{(d+e x)^5} \, dx &=\frac {(3 B d-A e+2 B e x) \left (b x+c x^2\right )^{5/2}}{4 e^2 (d+e x)^4}-\frac {5 \int \frac {(2 b (3 B d-A e)+4 (3 B c d-b B e-A c e) x) \left (b x+c x^2\right )^{3/2}}{(d+e x)^4} \, dx}{16 e^2}\\ &=-\frac {5 \left (d \left (A e \left (16 c^2 d^2-12 b c d e-b^2 e^2\right )-B d \left (48 c^2 d^2-52 b c d e+7 b^2 e^2\right )\right )+3 e \left (A e \left (8 c^2 d^2-8 b c d e+b^2 e^2\right )-B d \left (24 c^2 d^2-32 b c d e+9 b^2 e^2\right )\right ) x\right ) \left (b x+c x^2\right )^{3/2}}{96 d e^4 (c d-b e) (d+e x)^3}+\frac {(3 B d-A e+2 B e x) \left (b x+c x^2\right )^{5/2}}{4 e^2 (d+e x)^4}+\frac {5 \int \frac {\left (b \left (A e \left (16 c^2 d^2-12 b c d e-b^2 e^2\right )-B d \left (48 c^2 d^2-52 b c d e+7 b^2 e^2\right )\right )+2 c \left (A e \left (16 c^2 d^2-16 b c d e+b^2 e^2\right )-B d \left (48 c^2 d^2-64 b c d e+17 b^2 e^2\right )\right ) x\right ) \sqrt {b x+c x^2}}{(d+e x)^2} \, dx}{64 d e^4 (c d-b e)}\\ &=-\frac {5 \left (B d \left (192 c^3 d^3-304 b c^2 d^2 e+120 b^2 c d e^2-7 b^3 e^3\right )-A e \left (64 c^3 d^3-80 b c^2 d^2 e+16 b^2 c d e^2+b^3 e^3\right )-2 c e \left (A e \left (16 c^2 d^2-16 b c d e+b^2 e^2\right )-B d \left (48 c^2 d^2-64 b c d e+17 b^2 e^2\right )\right ) x\right ) \sqrt {b x+c x^2}}{64 d e^6 (c d-b e) (d+e x)}-\frac {5 \left (d \left (A e \left (16 c^2 d^2-12 b c d e-b^2 e^2\right )-B d \left (48 c^2 d^2-52 b c d e+7 b^2 e^2\right )\right )+3 e \left (A e \left (8 c^2 d^2-8 b c d e+b^2 e^2\right )-B d \left (24 c^2 d^2-32 b c d e+9 b^2 e^2\right )\right ) x\right ) \left (b x+c x^2\right )^{3/2}}{96 d e^4 (c d-b e) (d+e x)^3}+\frac {(3 B d-A e+2 B e x) \left (b x+c x^2\right )^{5/2}}{4 e^2 (d+e x)^4}-\frac {5 \int \frac {-b \left (B d \left (192 c^3 d^3-304 b c^2 d^2 e+120 b^2 c d e^2-7 b^3 e^3\right )-A e \left (64 c^3 d^3-80 b c^2 d^2 e+16 b^2 c d e^2+b^3 e^3\right )\right )+16 c d (c d-b e) \left (4 A c e (2 c d-b e)-B \left (24 c^2 d^2-20 b c d e+3 b^2 e^2\right )\right ) x}{(d+e x) \sqrt {b x+c x^2}} \, dx}{128 d e^6 (c d-b e)}\\ &=-\frac {5 \left (B d \left (192 c^3 d^3-304 b c^2 d^2 e+120 b^2 c d e^2-7 b^3 e^3\right )-A e \left (64 c^3 d^3-80 b c^2 d^2 e+16 b^2 c d e^2+b^3 e^3\right )-2 c e \left (A e \left (16 c^2 d^2-16 b c d e+b^2 e^2\right )-B d \left (48 c^2 d^2-64 b c d e+17 b^2 e^2\right )\right ) x\right ) \sqrt {b x+c x^2}}{64 d e^6 (c d-b e) (d+e x)}-\frac {5 \left (d \left (A e \left (16 c^2 d^2-12 b c d e-b^2 e^2\right )-B d \left (48 c^2 d^2-52 b c d e+7 b^2 e^2\right )\right )+3 e \left (A e \left (8 c^2 d^2-8 b c d e+b^2 e^2\right )-B d \left (24 c^2 d^2-32 b c d e+9 b^2 e^2\right )\right ) x\right ) \left (b x+c x^2\right )^{3/2}}{96 d e^4 (c d-b e) (d+e x)^3}+\frac {(3 B d-A e+2 B e x) \left (b x+c x^2\right )^{5/2}}{4 e^2 (d+e x)^4}-\frac {\left (5 c \left (4 A c e (2 c d-b e)-B \left (24 c^2 d^2-20 b c d e+3 b^2 e^2\right )\right )\right ) \int \frac {1}{\sqrt {b x+c x^2}} \, dx}{8 e^7}+\frac {\left (5 \left (A e \left (128 c^4 d^4-256 b c^3 d^3 e+144 b^2 c^2 d^2 e^2-16 b^3 c d e^3-b^4 e^4\right )-B d \left (384 c^4 d^4-896 b c^3 d^3 e+672 b^2 c^2 d^2 e^2-168 b^3 c d e^3+7 b^4 e^4\right )\right )\right ) \int \frac {1}{(d+e x) \sqrt {b x+c x^2}} \, dx}{128 d e^7 (c d-b e)}\\ &=-\frac {5 \left (B d \left (192 c^3 d^3-304 b c^2 d^2 e+120 b^2 c d e^2-7 b^3 e^3\right )-A e \left (64 c^3 d^3-80 b c^2 d^2 e+16 b^2 c d e^2+b^3 e^3\right )-2 c e \left (A e \left (16 c^2 d^2-16 b c d e+b^2 e^2\right )-B d \left (48 c^2 d^2-64 b c d e+17 b^2 e^2\right )\right ) x\right ) \sqrt {b x+c x^2}}{64 d e^6 (c d-b e) (d+e x)}-\frac {5 \left (d \left (A e \left (16 c^2 d^2-12 b c d e-b^2 e^2\right )-B d \left (48 c^2 d^2-52 b c d e+7 b^2 e^2\right )\right )+3 e \left (A e \left (8 c^2 d^2-8 b c d e+b^2 e^2\right )-B d \left (24 c^2 d^2-32 b c d e+9 b^2 e^2\right )\right ) x\right ) \left (b x+c x^2\right )^{3/2}}{96 d e^4 (c d-b e) (d+e x)^3}+\frac {(3 B d-A e+2 B e x) \left (b x+c x^2\right )^{5/2}}{4 e^2 (d+e x)^4}-\frac {\left (5 c \left (4 A c e (2 c d-b e)-B \left (24 c^2 d^2-20 b c d e+3 b^2 e^2\right )\right )\right ) \operatorname {Subst}\left (\int \frac {1}{1-c x^2} \, dx,x,\frac {x}{\sqrt {b x+c x^2}}\right )}{4 e^7}-\frac {\left (5 \left (A e \left (128 c^4 d^4-256 b c^3 d^3 e+144 b^2 c^2 d^2 e^2-16 b^3 c d e^3-b^4 e^4\right )-B d \left (384 c^4 d^4-896 b c^3 d^3 e+672 b^2 c^2 d^2 e^2-168 b^3 c d e^3+7 b^4 e^4\right )\right )\right ) \operatorname {Subst}\left (\int \frac {1}{4 c d^2-4 b d e-x^2} \, dx,x,\frac {-b d-(2 c d-b e) x}{\sqrt {b x+c x^2}}\right )}{64 d e^7 (c d-b e)}\\ &=-\frac {5 \left (B d \left (192 c^3 d^3-304 b c^2 d^2 e+120 b^2 c d e^2-7 b^3 e^3\right )-A e \left (64 c^3 d^3-80 b c^2 d^2 e+16 b^2 c d e^2+b^3 e^3\right )-2 c e \left (A e \left (16 c^2 d^2-16 b c d e+b^2 e^2\right )-B d \left (48 c^2 d^2-64 b c d e+17 b^2 e^2\right )\right ) x\right ) \sqrt {b x+c x^2}}{64 d e^6 (c d-b e) (d+e x)}-\frac {5 \left (d \left (A e \left (16 c^2 d^2-12 b c d e-b^2 e^2\right )-B d \left (48 c^2 d^2-52 b c d e+7 b^2 e^2\right )\right )+3 e \left (A e \left (8 c^2 d^2-8 b c d e+b^2 e^2\right )-B d \left (24 c^2 d^2-32 b c d e+9 b^2 e^2\right )\right ) x\right ) \left (b x+c x^2\right )^{3/2}}{96 d e^4 (c d-b e) (d+e x)^3}+\frac {(3 B d-A e+2 B e x) \left (b x+c x^2\right )^{5/2}}{4 e^2 (d+e x)^4}-\frac {5 \sqrt {c} \left (4 A c e (2 c d-b e)-B \left (24 c^2 d^2-20 b c d e+3 b^2 e^2\right )\right ) \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b x+c x^2}}\right )}{4 e^7}+\frac {5 \left (A e \left (128 c^4 d^4-256 b c^3 d^3 e+144 b^2 c^2 d^2 e^2-16 b^3 c d e^3-b^4 e^4\right )-B d \left (384 c^4 d^4-896 b c^3 d^3 e+672 b^2 c^2 d^2 e^2-168 b^3 c d e^3+7 b^4 e^4\right )\right ) \tanh ^{-1}\left (\frac {b d+(2 c d-b e) x}{2 \sqrt {d} \sqrt {c d-b e} \sqrt {b x+c x^2}}\right )}{128 d^{3/2} e^7 (c d-b e)^{3/2}}\\ \end {align*}
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Mathematica [B] time = 6.29, size = 2680, normalized size = 4.23 \begin {gather*} \text {Result too large to show} \end {gather*}
Antiderivative was successfully verified.
[In]
Integrate[((A + B*x)*(b*x + c*x^2)^(5/2))/(d + e*x)^5,x]
[Out]
((-(B*d) + A*e)*x*(b + c*x)*(x*(b + c*x))^(5/2))/(4*d*(-(c*d) + b*e)*(d + e*x)^4) + ((x*(b + c*x))^(5/2)*(((-3
*c*d*(B*d - A*e) + (e*(7*b*B*d - 8*A*c*d + A*b*e))/2)*x^(7/2)*(b + c*x)^(7/2))/(3*d*(-(c*d) + b*e)*(d + e*x)^3
) + (((-2*c*d*(B*d*(6*c*d - 7*b*e) + A*e*(2*c*d - b*e)) + (e*(-7*b^2*B*d*e + A*(48*c^2*d^2 - 40*b*c*d*e - b^2*
e^2)))/4)*x^(7/2)*(b + c*x)^(7/2))/(2*d*(-(c*d) + b*e)*(d + e*x)^2) + ((((e*(-192*A*c^3*d^3 - 4*b^2*c*d*e*(91*
B*d - 17*A*e) + 3*b^3*e^2*(7*B*d + A*e) + 16*b*c^2*d^2*(21*B*d + 8*A*e)))/8 + (5*c*d*(A*e*(32*c^2*d^2 - 32*b*c
*d*e - b^2*e^2) - B*d*(48*c^2*d^2 - 56*b*c*d*e + 7*b^2*e^2)))/4)*x^(7/2)*(b + c*x)^(7/2))/(d*(-(c*d) + b*e)*(d
+ e*x)) + ((-1/8*(c*d*(-192*A*c^3*d^3 - 4*b^2*c*d*e*(91*B*d - 17*A*e) + 3*b^3*e^2*(7*B*d + A*e) + 16*b*c^2*d^
2*(21*B*d + 8*A*e))) + (b*e*(-192*A*c^3*d^3 - 4*b^2*c*d*e*(91*B*d - 17*A*e) + 3*b^3*e^2*(7*B*d + A*e) + 16*b*c
^2*d^2*(21*B*d + 8*A*e)))/8 - (7*b*((e*(-192*A*c^3*d^3 - 4*b^2*c*d*e*(91*B*d - 17*A*e) + 3*b^3*e^2*(7*B*d + A*
e) + 16*b*c^2*d^2*(21*B*d + 8*A*e)))/8 + (5*c*d*(A*e*(32*c^2*d^2 - 32*b*c*d*e - b^2*e^2) - B*d*(48*c^2*d^2 - 5
6*b*c*d*e + 7*b^2*e^2)))/4))/2)*((2*b^2*x^(5/2)*Sqrt[b + c*x]*(1 + (c*x)/b)^3*((5/(16*(1 + (c*x)/b)^3) + 5/(8*
(1 + (c*x)/b)^2) + (1 + (c*x)/b)^(-1))/2 - (15*b^3*((2*c*x)/b - (4*c^2*x^2)/(3*b^2) - (2*Sqrt[c]*Sqrt[x]*ArcSi
nh[(Sqrt[c]*Sqrt[x])/Sqrt[b]])/(Sqrt[b]*Sqrt[1 + (c*x)/b])))/(512*c^3*x^3*(1 + (c*x)/b)^3)))/(5*e) - (d*((2*b^
2*x^(3/2)*Sqrt[b + c*x]*(1 + (c*x)/b)^3*((3*(5/(8*(1 + (c*x)/b)^3) + 5/(6*(1 + (c*x)/b)^2) + (1 + (c*x)/b)^(-1
)))/8 + (15*b^2*((2*c*x)/b - (2*Sqrt[c]*Sqrt[x]*ArcSinh[(Sqrt[c]*Sqrt[x])/Sqrt[b]])/(Sqrt[b]*Sqrt[1 + (c*x)/b]
)))/(256*c^2*x^2*(1 + (c*x)/b)^3)))/(3*e) - (d*((2*b^2*Sqrt[x]*Sqrt[b + c*x]*(1 + (c*x)/b)^3*((15/(8*(1 + (c*x
)/b)^3) + 5/(4*(1 + (c*x)/b)^2) + (1 + (c*x)/b)^(-1))/6 + (5*Sqrt[b]*ArcSinh[(Sqrt[c]*Sqrt[x])/Sqrt[b]])/(16*S
qrt[c]*Sqrt[x]*(1 + (c*x)/b)^(7/2))))/e - (d*((2*b*c*Sqrt[x]*Sqrt[b + c*x]*(1 + (c*x)/b)^2*((3/(2*(1 + (c*x)/b
)^2) + (1 + (c*x)/b)^(-1))/4 + (3*Sqrt[b]*ArcSinh[(Sqrt[c]*Sqrt[x])/Sqrt[b]])/(8*Sqrt[c]*Sqrt[x]*(1 + (c*x)/b)
^(5/2))))/e - ((c*d - b*e)*((2*c*Sqrt[x]*Sqrt[b + c*x]*(1 + (c*x)/b)*(1/(2*(1 + (c*x)/b)) + (Sqrt[b]*ArcSinh[(
Sqrt[c]*Sqrt[x])/Sqrt[b]])/(2*Sqrt[c]*Sqrt[x]*(1 + (c*x)/b)^(3/2))))/e - ((c*d - b*e)*((2*Sqrt[b]*Sqrt[c]*Sqrt
[1 + (c*x)/b]*ArcSinh[(Sqrt[c]*Sqrt[x])/Sqrt[b]])/(e*Sqrt[b + c*x]) - (2*Sqrt[c*d - b*e]*ArcTanh[(Sqrt[c*d - b
*e]*Sqrt[x])/(Sqrt[d]*Sqrt[b + c*x])])/(Sqrt[d]*e)))/e))/e))/e))/e))/e) - 6*c*((e*(-192*A*c^3*d^3 - 4*b^2*c*d*
e*(91*B*d - 17*A*e) + 3*b^3*e^2*(7*B*d + A*e) + 16*b*c^2*d^2*(21*B*d + 8*A*e)))/8 + (5*c*d*(A*e*(32*c^2*d^2 -
32*b*c*d*e - b^2*e^2) - B*d*(48*c^2*d^2 - 56*b*c*d*e + 7*b^2*e^2)))/4)*((2*b^2*x^(7/2)*Sqrt[b + c*x]*(1 + (c*x
)/b)^3*((7*(3/(16*(1 + (c*x)/b)^3) + 1/(2*(1 + (c*x)/b)^2) + (1 + (c*x)/b)^(-1)))/12 + (35*b^4*((2*c*x)/b - (4
*c^2*x^2)/(3*b^2) + (16*c^3*x^3)/(15*b^3) - (2*Sqrt[c]*Sqrt[x]*ArcSinh[(Sqrt[c]*Sqrt[x])/Sqrt[b]])/(Sqrt[b]*Sq
rt[1 + (c*x)/b])))/(2048*c^4*x^4*(1 + (c*x)/b)^3)))/(7*e) - (d*((2*b^2*x^(5/2)*Sqrt[b + c*x]*(1 + (c*x)/b)^3*(
(5/(16*(1 + (c*x)/b)^3) + 5/(8*(1 + (c*x)/b)^2) + (1 + (c*x)/b)^(-1))/2 - (15*b^3*((2*c*x)/b - (4*c^2*x^2)/(3*
b^2) - (2*Sqrt[c]*Sqrt[x]*ArcSinh[(Sqrt[c]*Sqrt[x])/Sqrt[b]])/(Sqrt[b]*Sqrt[1 + (c*x)/b])))/(512*c^3*x^3*(1 +
(c*x)/b)^3)))/(5*e) - (d*((2*b^2*x^(3/2)*Sqrt[b + c*x]*(1 + (c*x)/b)^3*((3*(5/(8*(1 + (c*x)/b)^3) + 5/(6*(1 +
(c*x)/b)^2) + (1 + (c*x)/b)^(-1)))/8 + (15*b^2*((2*c*x)/b - (2*Sqrt[c]*Sqrt[x]*ArcSinh[(Sqrt[c]*Sqrt[x])/Sqrt[
b]])/(Sqrt[b]*Sqrt[1 + (c*x)/b])))/(256*c^2*x^2*(1 + (c*x)/b)^3)))/(3*e) - (d*((2*b^2*Sqrt[x]*Sqrt[b + c*x]*(1
+ (c*x)/b)^3*((15/(8*(1 + (c*x)/b)^3) + 5/(4*(1 + (c*x)/b)^2) + (1 + (c*x)/b)^(-1))/6 + (5*Sqrt[b]*ArcSinh[(S
qrt[c]*Sqrt[x])/Sqrt[b]])/(16*Sqrt[c]*Sqrt[x]*(1 + (c*x)/b)^(7/2))))/e - (d*((2*b*c*Sqrt[x]*Sqrt[b + c*x]*(1 +
(c*x)/b)^2*((3/(2*(1 + (c*x)/b)^2) + (1 + (c*x)/b)^(-1))/4 + (3*Sqrt[b]*ArcSinh[(Sqrt[c]*Sqrt[x])/Sqrt[b]])/(
8*Sqrt[c]*Sqrt[x]*(1 + (c*x)/b)^(5/2))))/e - ((c*d - b*e)*((2*c*Sqrt[x]*Sqrt[b + c*x]*(1 + (c*x)/b)*(1/(2*(1 +
(c*x)/b)) + (Sqrt[b]*ArcSinh[(Sqrt[c]*Sqrt[x])/Sqrt[b]])/(2*Sqrt[c]*Sqrt[x]*(1 + (c*x)/b)^(3/2))))/e - ((c*d
- b*e)*((2*Sqrt[b]*Sqrt[c]*Sqrt[1 + (c*x)/b]*ArcSinh[(Sqrt[c]*Sqrt[x])/Sqrt[b]])/(e*Sqrt[b + c*x]) - (2*Sqrt[c
*d - b*e]*ArcTanh[(Sqrt[c*d - b*e]*Sqrt[x])/(Sqrt[d]*Sqrt[b + c*x])])/(Sqrt[d]*e)))/e))/e))/e))/e))/e))/e))/(d
*(-(c*d) + b*e)))/(2*d*(-(c*d) + b*e)))/(3*d*(-(c*d) + b*e))))/(4*d*(-(c*d) + b*e)*x^(5/2)*(b + c*x)^(5/2))
________________________________________________________________________________________
IntegrateAlgebraic [F] time = 180.05, size = 0, normalized size = 0.00 \begin {gather*} \text {\$Aborted} \end {gather*}
Verification is not applicable to the result.
[In]
IntegrateAlgebraic[((A + B*x)*(b*x + c*x^2)^(5/2))/(d + e*x)^5,x]
[Out]
$Aborted
________________________________________________________________________________________
fricas [B] time = 15.83, size = 8045, normalized size = 12.71
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((B*x+A)*(c*x^2+b*x)^(5/2)/(e*x+d)^5,x, algorithm="fricas")
[Out]
[1/384*(240*(24*B*c^4*d^10 - 4*(17*B*b*c^3 + 2*A*c^4)*d^9*e + (67*B*b^2*c^2 + 20*A*b*c^3)*d^8*e^2 - 2*(13*B*b^
3*c + 8*A*b^2*c^2)*d^7*e^3 + (3*B*b^4 + 4*A*b^3*c)*d^6*e^4 + (24*B*c^4*d^6*e^4 - 4*(17*B*b*c^3 + 2*A*c^4)*d^5*
e^5 + (67*B*b^2*c^2 + 20*A*b*c^3)*d^4*e^6 - 2*(13*B*b^3*c + 8*A*b^2*c^2)*d^3*e^7 + (3*B*b^4 + 4*A*b^3*c)*d^2*e
^8)*x^4 + 4*(24*B*c^4*d^7*e^3 - 4*(17*B*b*c^3 + 2*A*c^4)*d^6*e^4 + (67*B*b^2*c^2 + 20*A*b*c^3)*d^5*e^5 - 2*(13
*B*b^3*c + 8*A*b^2*c^2)*d^4*e^6 + (3*B*b^4 + 4*A*b^3*c)*d^3*e^7)*x^3 + 6*(24*B*c^4*d^8*e^2 - 4*(17*B*b*c^3 + 2
*A*c^4)*d^7*e^3 + (67*B*b^2*c^2 + 20*A*b*c^3)*d^6*e^4 - 2*(13*B*b^3*c + 8*A*b^2*c^2)*d^5*e^5 + (3*B*b^4 + 4*A*
b^3*c)*d^4*e^6)*x^2 + 4*(24*B*c^4*d^9*e - 4*(17*B*b*c^3 + 2*A*c^4)*d^8*e^2 + (67*B*b^2*c^2 + 20*A*b*c^3)*d^7*e
^3 - 2*(13*B*b^3*c + 8*A*b^2*c^2)*d^6*e^4 + (3*B*b^4 + 4*A*b^3*c)*d^5*e^5)*x)*sqrt(c)*log(2*c*x + b + 2*sqrt(c
*x^2 + b*x)*sqrt(c)) - 15*(384*B*c^4*d^9 + A*b^4*d^4*e^5 - 128*(7*B*b*c^3 + A*c^4)*d^8*e + 32*(21*B*b^2*c^2 +
8*A*b*c^3)*d^7*e^2 - 24*(7*B*b^3*c + 6*A*b^2*c^2)*d^6*e^3 + (7*B*b^4 + 16*A*b^3*c)*d^5*e^4 + (384*B*c^4*d^5*e^
4 + A*b^4*e^9 - 128*(7*B*b*c^3 + A*c^4)*d^4*e^5 + 32*(21*B*b^2*c^2 + 8*A*b*c^3)*d^3*e^6 - 24*(7*B*b^3*c + 6*A*
b^2*c^2)*d^2*e^7 + (7*B*b^4 + 16*A*b^3*c)*d*e^8)*x^4 + 4*(384*B*c^4*d^6*e^3 + A*b^4*d*e^8 - 128*(7*B*b*c^3 + A
*c^4)*d^5*e^4 + 32*(21*B*b^2*c^2 + 8*A*b*c^3)*d^4*e^5 - 24*(7*B*b^3*c + 6*A*b^2*c^2)*d^3*e^6 + (7*B*b^4 + 16*A
*b^3*c)*d^2*e^7)*x^3 + 6*(384*B*c^4*d^7*e^2 + A*b^4*d^2*e^7 - 128*(7*B*b*c^3 + A*c^4)*d^6*e^3 + 32*(21*B*b^2*c
^2 + 8*A*b*c^3)*d^5*e^4 - 24*(7*B*b^3*c + 6*A*b^2*c^2)*d^4*e^5 + (7*B*b^4 + 16*A*b^3*c)*d^3*e^6)*x^2 + 4*(384*
B*c^4*d^8*e + A*b^4*d^3*e^6 - 128*(7*B*b*c^3 + A*c^4)*d^7*e^2 + 32*(21*B*b^2*c^2 + 8*A*b*c^3)*d^6*e^3 - 24*(7*
B*b^3*c + 6*A*b^2*c^2)*d^5*e^4 + (7*B*b^4 + 16*A*b^3*c)*d^4*e^5)*x)*sqrt(c*d^2 - b*d*e)*log((b*d + (2*c*d - b*
e)*x + 2*sqrt(c*d^2 - b*d*e)*sqrt(c*x^2 + b*x))/(e*x + d)) - 2*(2880*B*c^4*d^9*e + 15*A*b^4*d^4*e^6 - 240*(31*
B*b*c^3 + 4*A*c^4)*d^8*e^2 + 120*(53*B*b^2*c^2 + 18*A*b*c^3)*d^7*e^3 - 15*(127*B*b^3*c + 96*A*b^2*c^2)*d^6*e^4
+ 15*(7*B*b^4 + 15*A*b^3*c)*d^5*e^5 - 96*(B*c^4*d^4*e^6 - 2*B*b*c^3*d^3*e^7 + B*b^2*c^2*d^2*e^8)*x^5 + 48*(12
*B*c^4*d^5*e^5 - (33*B*b*c^3 + 4*A*c^4)*d^4*e^6 + 2*(15*B*b^2*c^2 + 4*A*b*c^3)*d^3*e^7 - (9*B*b^3*c + 4*A*b^2*
c^2)*d^2*e^8)*x^4 + (6000*B*c^4*d^6*e^4 - 15*A*b^4*d*e^9 - 8*(1981*B*b*c^3 + 250*A*c^4)*d^5*e^5 + 2*(6995*B*b^
2*c^2 + 2308*A*b*c^3)*d^4*e^6 - (4421*B*b^3*c + 3262*A*b^2*c^2)*d^3*e^7 + (279*B*b^4 + 661*A*b^3*c)*d^2*e^8)*x
^3 + (12480*B*c^4*d^7*e^3 + 73*A*b^4*d^2*e^8 - 40*(815*B*b*c^3 + 104*A*c^4)*d^6*e^4 + 4*(7079*B*b^2*c^2 + 2370
*A*b*c^3)*d^5*e^5 - (8707*B*b^3*c + 6452*A*b^2*c^2)*d^4*e^6 + (511*B*b^4 + 1059*A*b^3*c)*d^3*e^7)*x^2 + 5*(201
6*B*c^4*d^8*e^2 + 11*A*b^4*d^3*e^7 - 48*(109*B*b*c^3 + 14*A*c^4)*d^7*e^3 + 2*(2251*B*b^2*c^2 + 760*A*b*c^3)*d^
6*e^4 - (1363*B*b^3*c + 1022*A*b^2*c^2)*d^5*e^5 + (77*B*b^4 + 163*A*b^3*c)*d^4*e^6)*x)*sqrt(c*x^2 + b*x))/(c^2
*d^8*e^7 - 2*b*c*d^7*e^8 + b^2*d^6*e^9 + (c^2*d^4*e^11 - 2*b*c*d^3*e^12 + b^2*d^2*e^13)*x^4 + 4*(c^2*d^5*e^10
- 2*b*c*d^4*e^11 + b^2*d^3*e^12)*x^3 + 6*(c^2*d^6*e^9 - 2*b*c*d^5*e^10 + b^2*d^4*e^11)*x^2 + 4*(c^2*d^7*e^8 -
2*b*c*d^6*e^9 + b^2*d^5*e^10)*x), -1/192*(15*(384*B*c^4*d^9 + A*b^4*d^4*e^5 - 128*(7*B*b*c^3 + A*c^4)*d^8*e +
32*(21*B*b^2*c^2 + 8*A*b*c^3)*d^7*e^2 - 24*(7*B*b^3*c + 6*A*b^2*c^2)*d^6*e^3 + (7*B*b^4 + 16*A*b^3*c)*d^5*e^4
+ (384*B*c^4*d^5*e^4 + A*b^4*e^9 - 128*(7*B*b*c^3 + A*c^4)*d^4*e^5 + 32*(21*B*b^2*c^2 + 8*A*b*c^3)*d^3*e^6 - 2
4*(7*B*b^3*c + 6*A*b^2*c^2)*d^2*e^7 + (7*B*b^4 + 16*A*b^3*c)*d*e^8)*x^4 + 4*(384*B*c^4*d^6*e^3 + A*b^4*d*e^8 -
128*(7*B*b*c^3 + A*c^4)*d^5*e^4 + 32*(21*B*b^2*c^2 + 8*A*b*c^3)*d^4*e^5 - 24*(7*B*b^3*c + 6*A*b^2*c^2)*d^3*e^
6 + (7*B*b^4 + 16*A*b^3*c)*d^2*e^7)*x^3 + 6*(384*B*c^4*d^7*e^2 + A*b^4*d^2*e^7 - 128*(7*B*b*c^3 + A*c^4)*d^6*e
^3 + 32*(21*B*b^2*c^2 + 8*A*b*c^3)*d^5*e^4 - 24*(7*B*b^3*c + 6*A*b^2*c^2)*d^4*e^5 + (7*B*b^4 + 16*A*b^3*c)*d^3
*e^6)*x^2 + 4*(384*B*c^4*d^8*e + A*b^4*d^3*e^6 - 128*(7*B*b*c^3 + A*c^4)*d^7*e^2 + 32*(21*B*b^2*c^2 + 8*A*b*c^
3)*d^6*e^3 - 24*(7*B*b^3*c + 6*A*b^2*c^2)*d^5*e^4 + (7*B*b^4 + 16*A*b^3*c)*d^4*e^5)*x)*sqrt(-c*d^2 + b*d*e)*ar
ctan(-sqrt(-c*d^2 + b*d*e)*sqrt(c*x^2 + b*x)/((c*d - b*e)*x)) - 120*(24*B*c^4*d^10 - 4*(17*B*b*c^3 + 2*A*c^4)*
d^9*e + (67*B*b^2*c^2 + 20*A*b*c^3)*d^8*e^2 - 2*(13*B*b^3*c + 8*A*b^2*c^2)*d^7*e^3 + (3*B*b^4 + 4*A*b^3*c)*d^6
*e^4 + (24*B*c^4*d^6*e^4 - 4*(17*B*b*c^3 + 2*A*c^4)*d^5*e^5 + (67*B*b^2*c^2 + 20*A*b*c^3)*d^4*e^6 - 2*(13*B*b^
3*c + 8*A*b^2*c^2)*d^3*e^7 + (3*B*b^4 + 4*A*b^3*c)*d^2*e^8)*x^4 + 4*(24*B*c^4*d^7*e^3 - 4*(17*B*b*c^3 + 2*A*c^
4)*d^6*e^4 + (67*B*b^2*c^2 + 20*A*b*c^3)*d^5*e^5 - 2*(13*B*b^3*c + 8*A*b^2*c^2)*d^4*e^6 + (3*B*b^4 + 4*A*b^3*c
)*d^3*e^7)*x^3 + 6*(24*B*c^4*d^8*e^2 - 4*(17*B*b*c^3 + 2*A*c^4)*d^7*e^3 + (67*B*b^2*c^2 + 20*A*b*c^3)*d^6*e^4
- 2*(13*B*b^3*c + 8*A*b^2*c^2)*d^5*e^5 + (3*B*b^4 + 4*A*b^3*c)*d^4*e^6)*x^2 + 4*(24*B*c^4*d^9*e - 4*(17*B*b*c^
3 + 2*A*c^4)*d^8*e^2 + (67*B*b^2*c^2 + 20*A*b*c^3)*d^7*e^3 - 2*(13*B*b^3*c + 8*A*b^2*c^2)*d^6*e^4 + (3*B*b^4 +
4*A*b^3*c)*d^5*e^5)*x)*sqrt(c)*log(2*c*x + b + 2*sqrt(c*x^2 + b*x)*sqrt(c)) + (2880*B*c^4*d^9*e + 15*A*b^4*d^
4*e^6 - 240*(31*B*b*c^3 + 4*A*c^4)*d^8*e^2 + 120*(53*B*b^2*c^2 + 18*A*b*c^3)*d^7*e^3 - 15*(127*B*b^3*c + 96*A*
b^2*c^2)*d^6*e^4 + 15*(7*B*b^4 + 15*A*b^3*c)*d^5*e^5 - 96*(B*c^4*d^4*e^6 - 2*B*b*c^3*d^3*e^7 + B*b^2*c^2*d^2*e
^8)*x^5 + 48*(12*B*c^4*d^5*e^5 - (33*B*b*c^3 + 4*A*c^4)*d^4*e^6 + 2*(15*B*b^2*c^2 + 4*A*b*c^3)*d^3*e^7 - (9*B*
b^3*c + 4*A*b^2*c^2)*d^2*e^8)*x^4 + (6000*B*c^4*d^6*e^4 - 15*A*b^4*d*e^9 - 8*(1981*B*b*c^3 + 250*A*c^4)*d^5*e^
5 + 2*(6995*B*b^2*c^2 + 2308*A*b*c^3)*d^4*e^6 - (4421*B*b^3*c + 3262*A*b^2*c^2)*d^3*e^7 + (279*B*b^4 + 661*A*b
^3*c)*d^2*e^8)*x^3 + (12480*B*c^4*d^7*e^3 + 73*A*b^4*d^2*e^8 - 40*(815*B*b*c^3 + 104*A*c^4)*d^6*e^4 + 4*(7079*
B*b^2*c^2 + 2370*A*b*c^3)*d^5*e^5 - (8707*B*b^3*c + 6452*A*b^2*c^2)*d^4*e^6 + (511*B*b^4 + 1059*A*b^3*c)*d^3*e
^7)*x^2 + 5*(2016*B*c^4*d^8*e^2 + 11*A*b^4*d^3*e^7 - 48*(109*B*b*c^3 + 14*A*c^4)*d^7*e^3 + 2*(2251*B*b^2*c^2 +
760*A*b*c^3)*d^6*e^4 - (1363*B*b^3*c + 1022*A*b^2*c^2)*d^5*e^5 + (77*B*b^4 + 163*A*b^3*c)*d^4*e^6)*x)*sqrt(c*
x^2 + b*x))/(c^2*d^8*e^7 - 2*b*c*d^7*e^8 + b^2*d^6*e^9 + (c^2*d^4*e^11 - 2*b*c*d^3*e^12 + b^2*d^2*e^13)*x^4 +
4*(c^2*d^5*e^10 - 2*b*c*d^4*e^11 + b^2*d^3*e^12)*x^3 + 6*(c^2*d^6*e^9 - 2*b*c*d^5*e^10 + b^2*d^4*e^11)*x^2 + 4
*(c^2*d^7*e^8 - 2*b*c*d^6*e^9 + b^2*d^5*e^10)*x), -1/384*(480*(24*B*c^4*d^10 - 4*(17*B*b*c^3 + 2*A*c^4)*d^9*e
+ (67*B*b^2*c^2 + 20*A*b*c^3)*d^8*e^2 - 2*(13*B*b^3*c + 8*A*b^2*c^2)*d^7*e^3 + (3*B*b^4 + 4*A*b^3*c)*d^6*e^4 +
(24*B*c^4*d^6*e^4 - 4*(17*B*b*c^3 + 2*A*c^4)*d^5*e^5 + (67*B*b^2*c^2 + 20*A*b*c^3)*d^4*e^6 - 2*(13*B*b^3*c +
8*A*b^2*c^2)*d^3*e^7 + (3*B*b^4 + 4*A*b^3*c)*d^2*e^8)*x^4 + 4*(24*B*c^4*d^7*e^3 - 4*(17*B*b*c^3 + 2*A*c^4)*d^6
*e^4 + (67*B*b^2*c^2 + 20*A*b*c^3)*d^5*e^5 - 2*(13*B*b^3*c + 8*A*b^2*c^2)*d^4*e^6 + (3*B*b^4 + 4*A*b^3*c)*d^3*
e^7)*x^3 + 6*(24*B*c^4*d^8*e^2 - 4*(17*B*b*c^3 + 2*A*c^4)*d^7*e^3 + (67*B*b^2*c^2 + 20*A*b*c^3)*d^6*e^4 - 2*(1
3*B*b^3*c + 8*A*b^2*c^2)*d^5*e^5 + (3*B*b^4 + 4*A*b^3*c)*d^4*e^6)*x^2 + 4*(24*B*c^4*d^9*e - 4*(17*B*b*c^3 + 2*
A*c^4)*d^8*e^2 + (67*B*b^2*c^2 + 20*A*b*c^3)*d^7*e^3 - 2*(13*B*b^3*c + 8*A*b^2*c^2)*d^6*e^4 + (3*B*b^4 + 4*A*b
^3*c)*d^5*e^5)*x)*sqrt(-c)*arctan(sqrt(c*x^2 + b*x)*sqrt(-c)/(c*x)) + 15*(384*B*c^4*d^9 + A*b^4*d^4*e^5 - 128*
(7*B*b*c^3 + A*c^4)*d^8*e + 32*(21*B*b^2*c^2 + 8*A*b*c^3)*d^7*e^2 - 24*(7*B*b^3*c + 6*A*b^2*c^2)*d^6*e^3 + (7*
B*b^4 + 16*A*b^3*c)*d^5*e^4 + (384*B*c^4*d^5*e^4 + A*b^4*e^9 - 128*(7*B*b*c^3 + A*c^4)*d^4*e^5 + 32*(21*B*b^2*
c^2 + 8*A*b*c^3)*d^3*e^6 - 24*(7*B*b^3*c + 6*A*b^2*c^2)*d^2*e^7 + (7*B*b^4 + 16*A*b^3*c)*d*e^8)*x^4 + 4*(384*B
*c^4*d^6*e^3 + A*b^4*d*e^8 - 128*(7*B*b*c^3 + A*c^4)*d^5*e^4 + 32*(21*B*b^2*c^2 + 8*A*b*c^3)*d^4*e^5 - 24*(7*B
*b^3*c + 6*A*b^2*c^2)*d^3*e^6 + (7*B*b^4 + 16*A*b^3*c)*d^2*e^7)*x^3 + 6*(384*B*c^4*d^7*e^2 + A*b^4*d^2*e^7 - 1
28*(7*B*b*c^3 + A*c^4)*d^6*e^3 + 32*(21*B*b^2*c^2 + 8*A*b*c^3)*d^5*e^4 - 24*(7*B*b^3*c + 6*A*b^2*c^2)*d^4*e^5
+ (7*B*b^4 + 16*A*b^3*c)*d^3*e^6)*x^2 + 4*(384*B*c^4*d^8*e + A*b^4*d^3*e^6 - 128*(7*B*b*c^3 + A*c^4)*d^7*e^2 +
32*(21*B*b^2*c^2 + 8*A*b*c^3)*d^6*e^3 - 24*(7*B*b^3*c + 6*A*b^2*c^2)*d^5*e^4 + (7*B*b^4 + 16*A*b^3*c)*d^4*e^5
)*x)*sqrt(c*d^2 - b*d*e)*log((b*d + (2*c*d - b*e)*x + 2*sqrt(c*d^2 - b*d*e)*sqrt(c*x^2 + b*x))/(e*x + d)) + 2*
(2880*B*c^4*d^9*e + 15*A*b^4*d^4*e^6 - 240*(31*B*b*c^3 + 4*A*c^4)*d^8*e^2 + 120*(53*B*b^2*c^2 + 18*A*b*c^3)*d^
7*e^3 - 15*(127*B*b^3*c + 96*A*b^2*c^2)*d^6*e^4 + 15*(7*B*b^4 + 15*A*b^3*c)*d^5*e^5 - 96*(B*c^4*d^4*e^6 - 2*B*
b*c^3*d^3*e^7 + B*b^2*c^2*d^2*e^8)*x^5 + 48*(12*B*c^4*d^5*e^5 - (33*B*b*c^3 + 4*A*c^4)*d^4*e^6 + 2*(15*B*b^2*c
^2 + 4*A*b*c^3)*d^3*e^7 - (9*B*b^3*c + 4*A*b^2*c^2)*d^2*e^8)*x^4 + (6000*B*c^4*d^6*e^4 - 15*A*b^4*d*e^9 - 8*(1
981*B*b*c^3 + 250*A*c^4)*d^5*e^5 + 2*(6995*B*b^2*c^2 + 2308*A*b*c^3)*d^4*e^6 - (4421*B*b^3*c + 3262*A*b^2*c^2)
*d^3*e^7 + (279*B*b^4 + 661*A*b^3*c)*d^2*e^8)*x^3 + (12480*B*c^4*d^7*e^3 + 73*A*b^4*d^2*e^8 - 40*(815*B*b*c^3
+ 104*A*c^4)*d^6*e^4 + 4*(7079*B*b^2*c^2 + 2370*A*b*c^3)*d^5*e^5 - (8707*B*b^3*c + 6452*A*b^2*c^2)*d^4*e^6 + (
511*B*b^4 + 1059*A*b^3*c)*d^3*e^7)*x^2 + 5*(2016*B*c^4*d^8*e^2 + 11*A*b^4*d^3*e^7 - 48*(109*B*b*c^3 + 14*A*c^4
)*d^7*e^3 + 2*(2251*B*b^2*c^2 + 760*A*b*c^3)*d^6*e^4 - (1363*B*b^3*c + 1022*A*b^2*c^2)*d^5*e^5 + (77*B*b^4 + 1
63*A*b^3*c)*d^4*e^6)*x)*sqrt(c*x^2 + b*x))/(c^2*d^8*e^7 - 2*b*c*d^7*e^8 + b^2*d^6*e^9 + (c^2*d^4*e^11 - 2*b*c*
d^3*e^12 + b^2*d^2*e^13)*x^4 + 4*(c^2*d^5*e^10 - 2*b*c*d^4*e^11 + b^2*d^3*e^12)*x^3 + 6*(c^2*d^6*e^9 - 2*b*c*d
^5*e^10 + b^2*d^4*e^11)*x^2 + 4*(c^2*d^7*e^8 - 2*b*c*d^6*e^9 + b^2*d^5*e^10)*x), -1/192*(15*(384*B*c^4*d^9 + A
*b^4*d^4*e^5 - 128*(7*B*b*c^3 + A*c^4)*d^8*e + 32*(21*B*b^2*c^2 + 8*A*b*c^3)*d^7*e^2 - 24*(7*B*b^3*c + 6*A*b^2
*c^2)*d^6*e^3 + (7*B*b^4 + 16*A*b^3*c)*d^5*e^4 + (384*B*c^4*d^5*e^4 + A*b^4*e^9 - 128*(7*B*b*c^3 + A*c^4)*d^4*
e^5 + 32*(21*B*b^2*c^2 + 8*A*b*c^3)*d^3*e^6 - 24*(7*B*b^3*c + 6*A*b^2*c^2)*d^2*e^7 + (7*B*b^4 + 16*A*b^3*c)*d*
e^8)*x^4 + 4*(384*B*c^4*d^6*e^3 + A*b^4*d*e^8 - 128*(7*B*b*c^3 + A*c^4)*d^5*e^4 + 32*(21*B*b^2*c^2 + 8*A*b*c^3
)*d^4*e^5 - 24*(7*B*b^3*c + 6*A*b^2*c^2)*d^3*e^6 + (7*B*b^4 + 16*A*b^3*c)*d^2*e^7)*x^3 + 6*(384*B*c^4*d^7*e^2
+ A*b^4*d^2*e^7 - 128*(7*B*b*c^3 + A*c^4)*d^6*e^3 + 32*(21*B*b^2*c^2 + 8*A*b*c^3)*d^5*e^4 - 24*(7*B*b^3*c + 6*
A*b^2*c^2)*d^4*e^5 + (7*B*b^4 + 16*A*b^3*c)*d^3*e^6)*x^2 + 4*(384*B*c^4*d^8*e + A*b^4*d^3*e^6 - 128*(7*B*b*c^3
+ A*c^4)*d^7*e^2 + 32*(21*B*b^2*c^2 + 8*A*b*c^3)*d^6*e^3 - 24*(7*B*b^3*c + 6*A*b^2*c^2)*d^5*e^4 + (7*B*b^4 +
16*A*b^3*c)*d^4*e^5)*x)*sqrt(-c*d^2 + b*d*e)*arctan(-sqrt(-c*d^2 + b*d*e)*sqrt(c*x^2 + b*x)/((c*d - b*e)*x)) +
240*(24*B*c^4*d^10 - 4*(17*B*b*c^3 + 2*A*c^4)*d^9*e + (67*B*b^2*c^2 + 20*A*b*c^3)*d^8*e^2 - 2*(13*B*b^3*c + 8
*A*b^2*c^2)*d^7*e^3 + (3*B*b^4 + 4*A*b^3*c)*d^6*e^4 + (24*B*c^4*d^6*e^4 - 4*(17*B*b*c^3 + 2*A*c^4)*d^5*e^5 + (
67*B*b^2*c^2 + 20*A*b*c^3)*d^4*e^6 - 2*(13*B*b^3*c + 8*A*b^2*c^2)*d^3*e^7 + (3*B*b^4 + 4*A*b^3*c)*d^2*e^8)*x^4
+ 4*(24*B*c^4*d^7*e^3 - 4*(17*B*b*c^3 + 2*A*c^4)*d^6*e^4 + (67*B*b^2*c^2 + 20*A*b*c^3)*d^5*e^5 - 2*(13*B*b^3*
c + 8*A*b^2*c^2)*d^4*e^6 + (3*B*b^4 + 4*A*b^3*c)*d^3*e^7)*x^3 + 6*(24*B*c^4*d^8*e^2 - 4*(17*B*b*c^3 + 2*A*c^4)
*d^7*e^3 + (67*B*b^2*c^2 + 20*A*b*c^3)*d^6*e^4 - 2*(13*B*b^3*c + 8*A*b^2*c^2)*d^5*e^5 + (3*B*b^4 + 4*A*b^3*c)*
d^4*e^6)*x^2 + 4*(24*B*c^4*d^9*e - 4*(17*B*b*c^3 + 2*A*c^4)*d^8*e^2 + (67*B*b^2*c^2 + 20*A*b*c^3)*d^7*e^3 - 2*
(13*B*b^3*c + 8*A*b^2*c^2)*d^6*e^4 + (3*B*b^4 + 4*A*b^3*c)*d^5*e^5)*x)*sqrt(-c)*arctan(sqrt(c*x^2 + b*x)*sqrt(
-c)/(c*x)) + (2880*B*c^4*d^9*e + 15*A*b^4*d^4*e^6 - 240*(31*B*b*c^3 + 4*A*c^4)*d^8*e^2 + 120*(53*B*b^2*c^2 + 1
8*A*b*c^3)*d^7*e^3 - 15*(127*B*b^3*c + 96*A*b^2*c^2)*d^6*e^4 + 15*(7*B*b^4 + 15*A*b^3*c)*d^5*e^5 - 96*(B*c^4*d
^4*e^6 - 2*B*b*c^3*d^3*e^7 + B*b^2*c^2*d^2*e^8)*x^5 + 48*(12*B*c^4*d^5*e^5 - (33*B*b*c^3 + 4*A*c^4)*d^4*e^6 +
2*(15*B*b^2*c^2 + 4*A*b*c^3)*d^3*e^7 - (9*B*b^3*c + 4*A*b^2*c^2)*d^2*e^8)*x^4 + (6000*B*c^4*d^6*e^4 - 15*A*b^4
*d*e^9 - 8*(1981*B*b*c^3 + 250*A*c^4)*d^5*e^5 + 2*(6995*B*b^2*c^2 + 2308*A*b*c^3)*d^4*e^6 - (4421*B*b^3*c + 32
62*A*b^2*c^2)*d^3*e^7 + (279*B*b^4 + 661*A*b^3*c)*d^2*e^8)*x^3 + (12480*B*c^4*d^7*e^3 + 73*A*b^4*d^2*e^8 - 40*
(815*B*b*c^3 + 104*A*c^4)*d^6*e^4 + 4*(7079*B*b^2*c^2 + 2370*A*b*c^3)*d^5*e^5 - (8707*B*b^3*c + 6452*A*b^2*c^2
)*d^4*e^6 + (511*B*b^4 + 1059*A*b^3*c)*d^3*e^7)*x^2 + 5*(2016*B*c^4*d^8*e^2 + 11*A*b^4*d^3*e^7 - 48*(109*B*b*c
^3 + 14*A*c^4)*d^7*e^3 + 2*(2251*B*b^2*c^2 + 760*A*b*c^3)*d^6*e^4 - (1363*B*b^3*c + 1022*A*b^2*c^2)*d^5*e^5 +
(77*B*b^4 + 163*A*b^3*c)*d^4*e^6)*x)*sqrt(c*x^2 + b*x))/(c^2*d^8*e^7 - 2*b*c*d^7*e^8 + b^2*d^6*e^9 + (c^2*d^4*
e^11 - 2*b*c*d^3*e^12 + b^2*d^2*e^13)*x^4 + 4*(c^2*d^5*e^10 - 2*b*c*d^4*e^11 + b^2*d^3*e^12)*x^3 + 6*(c^2*d^6*
e^9 - 2*b*c*d^5*e^10 + b^2*d^4*e^11)*x^2 + 4*(c^2*d^7*e^8 - 2*b*c*d^6*e^9 + b^2*d^5*e^10)*x)]
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giac [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((B*x+A)*(c*x^2+b*x)^(5/2)/(e*x+d)^5,x, algorithm="giac")
[Out]
Timed out
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maple [B] time = 0.09, size = 23819, normalized size = 37.63 \begin {gather*} \text {output too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((B*x+A)*(c*x^2+b*x)^(5/2)/(e*x+d)^5,x)
[Out]
result too large to display
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maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((B*x+A)*(c*x^2+b*x)^(5/2)/(e*x+d)^5,x, algorithm="maxima")
[Out]
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(b*e-c*d>0)', see `assume?` for
more details)Is b*e-c*d zero or nonzero?
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\left (c\,x^2+b\,x\right )}^{5/2}\,\left (A+B\,x\right )}{{\left (d+e\,x\right )}^5} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(((b*x + c*x^2)^(5/2)*(A + B*x))/(d + e*x)^5,x)
[Out]
int(((b*x + c*x^2)^(5/2)*(A + B*x))/(d + e*x)^5, x)
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (x \left (b + c x\right )\right )^{\frac {5}{2}} \left (A + B x\right )}{\left (d + e x\right )^{5}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((B*x+A)*(c*x**2+b*x)**(5/2)/(e*x+d)**5,x)
[Out]
Integral((x*(b + c*x))**(5/2)*(A + B*x)/(d + e*x)**5, x)
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