3.12.82 \(\int \frac {(A+B x) (b x+c x^2)^{3/2}}{(d+e x)^8} \, dx\) [1182]

Optimal. Leaf size=565 \[ -\frac {b^2 \left (48 A c^3 d^3-24 b c^2 d^2 (B d+3 A e)-b^3 e^2 (5 B d+9 A e)+2 b^2 c d e (10 B d+21 A e)\right ) (b d+(2 c d-b e) x) \sqrt {b x+c x^2}}{1024 d^5 (c d-b e)^5 (d+e x)^2}+\frac {\left (48 A c^3 d^3-24 b c^2 d^2 (B d+3 A e)-b^3 e^2 (5 B d+9 A e)+2 b^2 c d e (10 B d+21 A e)\right ) (b d+(2 c d-b e) x) \left (b x+c x^2\right )^{3/2}}{384 d^4 (c d-b e)^4 (d+e x)^4}+\frac {(B d-A e) \left (b x+c x^2\right )^{5/2}}{7 d (c d-b e) (d+e x)^7}-\frac {(9 A e (2 c d-b e)-B d (4 c d+5 b e)) \left (b x+c x^2\right )^{5/2}}{84 d^2 (c d-b e)^2 (d+e x)^6}+\frac {\left (B d \left (8 c^2 d^2+90 b c d e-35 b^2 e^2\right )-3 A e \left (68 c^2 d^2-68 b c d e+21 b^2 e^2\right )\right ) \left (b x+c x^2\right )^{5/2}}{840 d^3 (c d-b e)^3 (d+e x)^5}+\frac {b^4 \left (48 A c^3 d^3-24 b c^2 d^2 (B d+3 A e)-b^3 e^2 (5 B d+9 A e)+2 b^2 c d e (10 B d+21 A e)\right ) \tanh ^{-1}\left (\frac {b d+(2 c d-b e) x}{2 \sqrt {d} \sqrt {c d-b e} \sqrt {b x+c x^2}}\right )}{2048 d^{11/2} (c d-b e)^{11/2}} \]

[Out]

1/384*(48*A*c^3*d^3-24*b*c^2*d^2*(3*A*e+B*d)-b^3*e^2*(9*A*e+5*B*d)+2*b^2*c*d*e*(21*A*e+10*B*d))*(b*d+(-b*e+2*c
*d)*x)*(c*x^2+b*x)^(3/2)/d^4/(-b*e+c*d)^4/(e*x+d)^4+1/7*(-A*e+B*d)*(c*x^2+b*x)^(5/2)/d/(-b*e+c*d)/(e*x+d)^7-1/
84*(9*A*e*(-b*e+2*c*d)-B*d*(5*b*e+4*c*d))*(c*x^2+b*x)^(5/2)/d^2/(-b*e+c*d)^2/(e*x+d)^6+1/840*(B*d*(-35*b^2*e^2
+90*b*c*d*e+8*c^2*d^2)-3*A*e*(21*b^2*e^2-68*b*c*d*e+68*c^2*d^2))*(c*x^2+b*x)^(5/2)/d^3/(-b*e+c*d)^3/(e*x+d)^5+
1/2048*b^4*(48*A*c^3*d^3-24*b*c^2*d^2*(3*A*e+B*d)-b^3*e^2*(9*A*e+5*B*d)+2*b^2*c*d*e*(21*A*e+10*B*d))*arctanh(1
/2*(b*d+(-b*e+2*c*d)*x)/d^(1/2)/(-b*e+c*d)^(1/2)/(c*x^2+b*x)^(1/2))/d^(11/2)/(-b*e+c*d)^(11/2)-1/1024*b^2*(48*
A*c^3*d^3-24*b*c^2*d^2*(3*A*e+B*d)-b^3*e^2*(9*A*e+5*B*d)+2*b^2*c*d*e*(21*A*e+10*B*d))*(b*d+(-b*e+2*c*d)*x)*(c*
x^2+b*x)^(1/2)/d^5/(-b*e+c*d)^5/(e*x+d)^2

________________________________________________________________________________________

Rubi [A]
time = 0.65, antiderivative size = 565, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.192, Rules used = {848, 820, 734, 738, 212} \begin {gather*} \frac {\left (b x+c x^2\right )^{5/2} \left (B d \left (-35 b^2 e^2+90 b c d e+8 c^2 d^2\right )-3 A e \left (21 b^2 e^2-68 b c d e+68 c^2 d^2\right )\right )}{840 d^3 (d+e x)^5 (c d-b e)^3}-\frac {b^2 \sqrt {b x+c x^2} (x (2 c d-b e)+b d) \left (b^3 \left (-e^2\right ) (9 A e+5 B d)+2 b^2 c d e (21 A e+10 B d)-24 b c^2 d^2 (3 A e+B d)+48 A c^3 d^3\right )}{1024 d^5 (d+e x)^2 (c d-b e)^5}+\frac {\left (b x+c x^2\right )^{3/2} (x (2 c d-b e)+b d) \left (b^3 \left (-e^2\right ) (9 A e+5 B d)+2 b^2 c d e (21 A e+10 B d)-24 b c^2 d^2 (3 A e+B d)+48 A c^3 d^3\right )}{384 d^4 (d+e x)^4 (c d-b e)^4}+\frac {b^4 \left (b^3 \left (-e^2\right ) (9 A e+5 B d)+2 b^2 c d e (21 A e+10 B d)-24 b c^2 d^2 (3 A e+B d)+48 A c^3 d^3\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 )}{2048 d^{11/2} (c d-b e)^{11/2}}-\frac {\left (b x+c x^2\right )^{5/2} (9 A e (2 c d-b e)-B d (5 b e+4 c d))}{84 d^2 (d+e x)^6 (c d-b e)^2}+\frac {\left (b x+c x^2\right )^{5/2} (B d-A e)}{7 d (d+e x)^7 (c d-b e)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((A + B*x)*(b*x + c*x^2)^(3/2))/(d + e*x)^8,x]

[Out]

-1/1024*(b^2*(48*A*c^3*d^3 - 24*b*c^2*d^2*(B*d + 3*A*e) - b^3*e^2*(5*B*d + 9*A*e) + 2*b^2*c*d*e*(10*B*d + 21*A
*e))*(b*d + (2*c*d - b*e)*x)*Sqrt[b*x + c*x^2])/(d^5*(c*d - b*e)^5*(d + e*x)^2) + ((48*A*c^3*d^3 - 24*b*c^2*d^
2*(B*d + 3*A*e) - b^3*e^2*(5*B*d + 9*A*e) + 2*b^2*c*d*e*(10*B*d + 21*A*e))*(b*d + (2*c*d - b*e)*x)*(b*x + c*x^
2)^(3/2))/(384*d^4*(c*d - b*e)^4*(d + e*x)^4) + ((B*d - A*e)*(b*x + c*x^2)^(5/2))/(7*d*(c*d - b*e)*(d + e*x)^7
) - ((9*A*e*(2*c*d - b*e) - B*d*(4*c*d + 5*b*e))*(b*x + c*x^2)^(5/2))/(84*d^2*(c*d - b*e)^2*(d + e*x)^6) + ((B
*d*(8*c^2*d^2 + 90*b*c*d*e - 35*b^2*e^2) - 3*A*e*(68*c^2*d^2 - 68*b*c*d*e + 21*b^2*e^2))*(b*x + c*x^2)^(5/2))/
(840*d^3*(c*d - b*e)^3*(d + e*x)^5) + (b^4*(48*A*c^3*d^3 - 24*b*c^2*d^2*(B*d + 3*A*e) - b^3*e^2*(5*B*d + 9*A*e
) + 2*b^2*c*d*e*(10*B*d + 21*A*e))*ArcTanh[(b*d + (2*c*d - b*e)*x)/(2*Sqrt[d]*Sqrt[c*d - b*e]*Sqrt[b*x + c*x^2
])])/(2048*d^(11/2)*(c*d - b*e)^(11/2))

Rule 212

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 734

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(-(d + e*x)^(m + 1))
*(d*b - 2*a*e + (2*c*d - b*e)*x)*((a + b*x + c*x^2)^p/(2*(m + 1)*(c*d^2 - b*d*e + a*e^2))), x] + Dist[p*((b^2
- 4*a*c)/(2*(m + 1)*(c*d^2 - b*d*e + a*e^2))), Int[(d + e*x)^(m + 2)*(a + b*x + c*x^2)^(p - 1), x], x] /; Free
Q[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && NeQ[2*c*d - b*e, 0] && EqQ[m
+ 2*p + 2, 0] && GtQ[p, 0]

Rule 738

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 820

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Sim
p[(-(e*f - d*g))*(d + e*x)^(m + 1)*((a + b*x + c*x^2)^(p + 1)/(2*(p + 1)*(c*d^2 - b*d*e + a*e^2))), x] - Dist[
(b*(e*f + d*g) - 2*(c*d*f + a*e*g))/(2*(c*d^2 - b*d*e + a*e^2)), Int[(d + e*x)^(m + 1)*(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] && EqQ[S
implify[m + 2*p + 3], 0]

Rule 848

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Sim
p[(e*f - d*g)*(d + e*x)^(m + 1)*((a + b*x + c*x^2)^(p + 1)/((m + 1)*(c*d^2 - b*d*e + a*e^2))), x] + Dist[1/((m
+ 1)*(c*d^2 - b*d*e + a*e^2)), Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p*Simp[(c*d*f - f*b*e + a*e*g)*(m + 1)
+ b*(d*g - e*f)*(p + 1) - c*(e*f - d*g)*(m + 2*p + 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, p}, x] &&
NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && LtQ[m, -1] && (IntegerQ[m] || IntegerQ[p] || IntegersQ
[2*m, 2*p])

Rubi steps

\begin {align*} \int \frac {(A+B x) \left (b x+c x^2\right )^{3/2}}{(d+e x)^8} \, dx &=\frac {(B d-A e) \left (b x+c x^2\right )^{5/2}}{7 d (c d-b e) (d+e x)^7}-\frac {\int \frac {\left (\frac {1}{2} (-14 A c d+b (5 B d+9 A e))-2 c (B d-A e) x\right ) \left (b x+c x^2\right )^{3/2}}{(d+e x)^7} \, dx}{7 d (c d-b e)}\\ &=\frac {(B d-A e) \left (b x+c x^2\right )^{5/2}}{7 d (c d-b e) (d+e x)^7}-\frac {(9 A e (2 c d-b e)-B d (4 c d+5 b e)) \left (b x+c x^2\right )^{5/2}}{84 d^2 (c d-b e)^2 (d+e x)^6}+\frac {\int \frac {\left (\frac {1}{4} \left (168 A c^2 d^2+7 b^2 e (5 B d+9 A e)-2 b c d (40 B d+93 A e)\right )-\frac {1}{2} c (9 A e (2 c d-b e)-B d (4 c d+5 b e)) x\right ) \left (b x+c x^2\right )^{3/2}}{(d+e x)^6} \, dx}{42 d^2 (c d-b e)^2}\\ &=\frac {(B d-A e) \left (b x+c x^2\right )^{5/2}}{7 d (c d-b e) (d+e x)^7}-\frac {(9 A e (2 c d-b e)-B d (4 c d+5 b e)) \left (b x+c x^2\right )^{5/2}}{84 d^2 (c d-b e)^2 (d+e x)^6}+\frac {\left (B d \left (8 c^2 d^2+90 b c d e-35 b^2 e^2\right )-3 A e \left (68 c^2 d^2-68 b c d e+21 b^2 e^2\right )\right ) \left (b x+c x^2\right )^{5/2}}{840 d^3 (c d-b e)^3 (d+e x)^5}+\frac {\left (48 A c^3 d^3-24 b c^2 d^2 (B d+3 A e)-b^3 e^2 (5 B d+9 A e)+2 b^2 c d e (10 B d+21 A e)\right ) \int \frac {\left (b x+c x^2\right )^{3/2}}{(d+e x)^5} \, dx}{48 d^3 (c d-b e)^3}\\ &=\frac {\left (48 A c^3 d^3-24 b c^2 d^2 (B d+3 A e)-b^3 e^2 (5 B d+9 A e)+2 b^2 c d e (10 B d+21 A e)\right ) (b d+(2 c d-b e) x) \left (b x+c x^2\right )^{3/2}}{384 d^4 (c d-b e)^4 (d+e x)^4}+\frac {(B d-A e) \left (b x+c x^2\right )^{5/2}}{7 d (c d-b e) (d+e x)^7}-\frac {(9 A e (2 c d-b e)-B d (4 c d+5 b e)) \left (b x+c x^2\right )^{5/2}}{84 d^2 (c d-b e)^2 (d+e x)^6}+\frac {\left (B d \left (8 c^2 d^2+90 b c d e-35 b^2 e^2\right )-3 A e \left (68 c^2 d^2-68 b c d e+21 b^2 e^2\right )\right ) \left (b x+c x^2\right )^{5/2}}{840 d^3 (c d-b e)^3 (d+e x)^5}-\frac {\left (b^2 \left (48 A c^3 d^3-24 b c^2 d^2 (B d+3 A e)-b^3 e^2 (5 B d+9 A e)+2 b^2 c d e (10 B d+21 A e)\right )\right ) \int \frac {\sqrt {b x+c x^2}}{(d+e x)^3} \, dx}{256 d^4 (c d-b e)^4}\\ &=-\frac {b^2 \left (48 A c^3 d^3-24 b c^2 d^2 (B d+3 A e)-b^3 e^2 (5 B d+9 A e)+2 b^2 c d e (10 B d+21 A e)\right ) (b d+(2 c d-b e) x) \sqrt {b x+c x^2}}{1024 d^5 (c d-b e)^5 (d+e x)^2}+\frac {\left (48 A c^3 d^3-24 b c^2 d^2 (B d+3 A e)-b^3 e^2 (5 B d+9 A e)+2 b^2 c d e (10 B d+21 A e)\right ) (b d+(2 c d-b e) x) \left (b x+c x^2\right )^{3/2}}{384 d^4 (c d-b e)^4 (d+e x)^4}+\frac {(B d-A e) \left (b x+c x^2\right )^{5/2}}{7 d (c d-b e) (d+e x)^7}-\frac {(9 A e (2 c d-b e)-B d (4 c d+5 b e)) \left (b x+c x^2\right )^{5/2}}{84 d^2 (c d-b e)^2 (d+e x)^6}+\frac {\left (B d \left (8 c^2 d^2+90 b c d e-35 b^2 e^2\right )-3 A e \left (68 c^2 d^2-68 b c d e+21 b^2 e^2\right )\right ) \left (b x+c x^2\right )^{5/2}}{840 d^3 (c d-b e)^3 (d+e x)^5}+\frac {\left (b^4 \left (48 A c^3 d^3-24 b c^2 d^2 (B d+3 A e)-b^3 e^2 (5 B d+9 A e)+2 b^2 c d e (10 B d+21 A e)\right )\right ) \int \frac {1}{(d+e x) \sqrt {b x+c x^2}} \, dx}{2048 d^5 (c d-b e)^5}\\ &=-\frac {b^2 \left (48 A c^3 d^3-24 b c^2 d^2 (B d+3 A e)-b^3 e^2 (5 B d+9 A e)+2 b^2 c d e (10 B d+21 A e)\right ) (b d+(2 c d-b e) x) \sqrt {b x+c x^2}}{1024 d^5 (c d-b e)^5 (d+e x)^2}+\frac {\left (48 A c^3 d^3-24 b c^2 d^2 (B d+3 A e)-b^3 e^2 (5 B d+9 A e)+2 b^2 c d e (10 B d+21 A e)\right ) (b d+(2 c d-b e) x) \left (b x+c x^2\right )^{3/2}}{384 d^4 (c d-b e)^4 (d+e x)^4}+\frac {(B d-A e) \left (b x+c x^2\right )^{5/2}}{7 d (c d-b e) (d+e x)^7}-\frac {(9 A e (2 c d-b e)-B d (4 c d+5 b e)) \left (b x+c x^2\right )^{5/2}}{84 d^2 (c d-b e)^2 (d+e x)^6}+\frac {\left (B d \left (8 c^2 d^2+90 b c d e-35 b^2 e^2\right )-3 A e \left (68 c^2 d^2-68 b c d e+21 b^2 e^2\right )\right ) \left (b x+c x^2\right )^{5/2}}{840 d^3 (c d-b e)^3 (d+e x)^5}-\frac {\left (b^4 \left (48 A c^3 d^3-24 b c^2 d^2 (B d+3 A e)-b^3 e^2 (5 B d+9 A e)+2 b^2 c d e (10 B d+21 A e)\right )\right ) \text {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 )}{1024 d^5 (c d-b e)^5}\\ &=-\frac {b^2 \left (48 A c^3 d^3-24 b c^2 d^2 (B d+3 A e)-b^3 e^2 (5 B d+9 A e)+2 b^2 c d e (10 B d+21 A e)\right ) (b d+(2 c d-b e) x) \sqrt {b x+c x^2}}{1024 d^5 (c d-b e)^5 (d+e x)^2}+\frac {\left (48 A c^3 d^3-24 b c^2 d^2 (B d+3 A e)-b^3 e^2 (5 B d+9 A e)+2 b^2 c d e (10 B d+21 A e)\right ) (b d+(2 c d-b e) x) \left (b x+c x^2\right )^{3/2}}{384 d^4 (c d-b e)^4 (d+e x)^4}+\frac {(B d-A e) \left (b x+c x^2\right )^{5/2}}{7 d (c d-b e) (d+e x)^7}-\frac {(9 A e (2 c d-b e)-B d (4 c d+5 b e)) \left (b x+c x^2\right )^{5/2}}{84 d^2 (c d-b e)^2 (d+e x)^6}+\frac {\left (B d \left (8 c^2 d^2+90 b c d e-35 b^2 e^2\right )-3 A e \left (68 c^2 d^2-68 b c d e+21 b^2 e^2\right )\right ) \left (b x+c x^2\right )^{5/2}}{840 d^3 (c d-b e)^3 (d+e x)^5}+\frac {b^4 \left (48 A c^3 d^3-24 b c^2 d^2 (B d+3 A e)-b^3 e^2 (5 B d+9 A e)+2 b^2 c d e (10 B d+21 A e)\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 )}{2048 d^{11/2} (c d-b e)^{11/2}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]
time = 12.22, size = 505, normalized size = 0.89 \begin {gather*} \frac {(x (b+c x))^{3/2} \left (-\frac {(B d-A e) x^{5/2} (b+c x)}{(d+e x)^7}-\frac {(9 A e (-2 c d+b e)+B d (4 c d+5 b e)) x^{5/2} (b+c x)}{12 d (c d-b e) (d+e x)^6}-\frac {\left (B d \left (8 c^2 d^2+90 b c d e-35 b^2 e^2\right )-3 A e \left (68 c^2 d^2-68 b c d e+21 b^2 e^2\right )\right ) x^{5/2} (b+c x)}{120 d^2 (c d-b e)^2 (d+e x)^5}-\frac {7 \left (48 A c^3 d^3-24 b c^2 d^2 (B d+3 A e)-b^3 e^2 (5 B d+9 A e)+2 b^2 c d e (10 B d+21 A e)\right ) \left (16 d^{5/2} (c d-b e)^{3/2} x^{3/2} (b+c x)^{5/2}-b (d+e x) \left (8 d^{5/2} \sqrt {c d-b e} \sqrt {x} (b+c x)^{5/2}-b (d+e x) \left (2 d^{3/2} \sqrt {c d-b e} \sqrt {x} (b+c x)^{3/2}+3 b (d+e x) \left (\sqrt {d} \sqrt {c d-b e} \sqrt {x} \sqrt {b+c x}+b (d+e x) \tanh ^{-1}\left (\frac {\sqrt {c d-b e} \sqrt {x}}{\sqrt {d} \sqrt {b+c x}}\right )\right )\right )\right )\right )}{3072 d^{9/2} (c d-b e)^{9/2} (b+c x)^{3/2} (d+e x)^4}\right )}{7 d (-c d+b e) x^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((A + B*x)*(b*x + c*x^2)^(3/2))/(d + e*x)^8,x]

[Out]

((x*(b + c*x))^(3/2)*(-(((B*d - A*e)*x^(5/2)*(b + c*x))/(d + e*x)^7) - ((9*A*e*(-2*c*d + b*e) + B*d*(4*c*d + 5
*b*e))*x^(5/2)*(b + c*x))/(12*d*(c*d - b*e)*(d + e*x)^6) - ((B*d*(8*c^2*d^2 + 90*b*c*d*e - 35*b^2*e^2) - 3*A*e
*(68*c^2*d^2 - 68*b*c*d*e + 21*b^2*e^2))*x^(5/2)*(b + c*x))/(120*d^2*(c*d - b*e)^2*(d + e*x)^5) - (7*(48*A*c^3
*d^3 - 24*b*c^2*d^2*(B*d + 3*A*e) - b^3*e^2*(5*B*d + 9*A*e) + 2*b^2*c*d*e*(10*B*d + 21*A*e))*(16*d^(5/2)*(c*d
- b*e)^(3/2)*x^(3/2)*(b + c*x)^(5/2) - b*(d + e*x)*(8*d^(5/2)*Sqrt[c*d - b*e]*Sqrt[x]*(b + c*x)^(5/2) - b*(d +
e*x)*(2*d^(3/2)*Sqrt[c*d - b*e]*Sqrt[x]*(b + c*x)^(3/2) + 3*b*(d + e*x)*(Sqrt[d]*Sqrt[c*d - b*e]*Sqrt[x]*Sqrt
[b + c*x] + b*(d + e*x)*ArcTanh[(Sqrt[c*d - b*e]*Sqrt[x])/(Sqrt[d]*Sqrt[b + c*x])])))))/(3072*d^(9/2)*(c*d - b
*e)^(9/2)*(b + c*x)^(3/2)*(d + e*x)^4)))/(7*d*(-(c*d) + b*e)*x^(3/2))

________________________________________________________________________________________

Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(22884\) vs. \(2(531)=1062\).
time = 0.66, size = 22885, normalized size = 40.50

method	result	size
default	\(\text {Expression too large to display}\)	\(22885\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x+A)*(c*x^2+b*x)^(3/2)/(e*x+d)^8,x,method=_RETURNVERBOSE)

[Out]

result too large to display

________________________________________________________________________________________

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)^(3/2)/(e*x+d)^8,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(c*d-%e*b>0)', see `assume?` fo
r more detai

________________________________________________________________________________________

Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 2706 vs. \(2 (559) = 1118\).
time = 4.19, size = 5424, normalized size = 9.60 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x^2+b*x)^(3/2)/(e*x+d)^8,x, algorithm="fricas")

[Out]

[-1/215040*(105*(9*A*b^7*x^7*e^10 + 24*(B*b^5*c^2 - 2*A*b^4*c^3)*d^10 + (63*A*b^7*d*x^6 + (5*B*b^7 - 42*A*b^6*
c)*d*x^7)*e^9 + (189*A*b^7*d^2*x^5 - 4*(5*B*b^6*c - 18*A*b^5*c^2)*d^2*x^7 + 7*(5*B*b^7 - 42*A*b^6*c)*d^2*x^6)*
e^8 + (315*A*b^7*d^3*x^4 + 24*(B*b^5*c^2 - 2*A*b^4*c^3)*d^3*x^7 - 28*(5*B*b^6*c - 18*A*b^5*c^2)*d^3*x^6 + 21*(
5*B*b^7 - 42*A*b^6*c)*d^3*x^5)*e^7 + 7*(45*A*b^7*d^4*x^3 + 24*(B*b^5*c^2 - 2*A*b^4*c^3)*d^4*x^6 - 12*(5*B*b^6*
c - 18*A*b^5*c^2)*d^4*x^5 + 5*(5*B*b^7 - 42*A*b^6*c)*d^4*x^4)*e^6 + 7*(27*A*b^7*d^5*x^2 + 72*(B*b^5*c^2 - 2*A*
b^4*c^3)*d^5*x^5 - 20*(5*B*b^6*c - 18*A*b^5*c^2)*d^5*x^4 + 5*(5*B*b^7 - 42*A*b^6*c)*d^5*x^3)*e^5 + 7*(9*A*b^7*
d^6*x + 120*(B*b^5*c^2 - 2*A*b^4*c^3)*d^6*x^4 - 20*(5*B*b^6*c - 18*A*b^5*c^2)*d^6*x^3 + 3*(5*B*b^7 - 42*A*b^6*
c)*d^6*x^2)*e^4 + (9*A*b^7*d^7 + 840*(B*b^5*c^2 - 2*A*b^4*c^3)*d^7*x^3 - 84*(5*B*b^6*c - 18*A*b^5*c^2)*d^7*x^2
+ 7*(5*B*b^7 - 42*A*b^6*c)*d^7*x)*e^3 + (504*(B*b^5*c^2 - 2*A*b^4*c^3)*d^8*x^2 - 28*(5*B*b^6*c - 18*A*b^5*c^2
)*d^8*x + (5*B*b^7 - 42*A*b^6*c)*d^8)*e^2 + 4*(42*(B*b^5*c^2 - 2*A*b^4*c^3)*d^9*x - (5*B*b^6*c - 18*A*b^5*c^2)
*d^9)*e)*sqrt(c*d^2 - b*d*e)*log((2*c*d*x - b*x*e + b*d + 2*sqrt(c*d^2 - b*d*e)*sqrt(c*x^2 + b*x))/(x*e + d))
- 2*(21504*B*c^7*d^11*x^4 + 945*A*b^7*d*x^6*e^10 + 2688*(11*B*b*c^6 + 10*A*c^7)*d^11*x^3 + 1344*(B*b^2*c^5 + 3
0*A*b*c^6)*d^11*x^2 - 1680*(B*b^3*c^4 - 2*A*b^2*c^5)*d^11*x + 2520*(B*b^4*c^3 - 2*A*b^3*c^4)*d^11 + 525*(12*A*
b^7*d^2*x^5 + (B*b^7 - 9*A*b^6*c)*d^2*x^6)*e^9 + 7*(2547*A*b^7*d^3*x^4 - (325*B*b^6*c - 1272*A*b^5*c^2)*d^3*x^
6 + 2*(250*B*b^7 - 2253*A*b^6*c)*d^3*x^5)*e^8 + (27648*A*b^7*d^4*x^3 + 6*(525*B*b^5*c^2 - 1174*A*b^4*c^3)*d^4*
x^6 - 10*(1519*B*b^6*c - 5955*A*b^5*c^2)*d^4*x^5 + (9905*B*b^7 - 89403*A*b^6*c)*d^4*x^4)*e^7 + 3*(8393*A*b^7*d
^5*x^2 - 40*(5*B*b^4*c^3 - 8*A*b^3*c^4)*d^5*x^6 + 2*(3515*B*b^5*c^2 - 7878*A*b^4*c^3)*d^5*x^5 - 11*(1305*B*b^6
*c - 5126*A*b^5*c^2)*d^5*x^4 + 20*(256*B*b^7 - 2315*A*b^6*c)*d^5*x^3)*e^6 - (6300*A*b^7*d^6*x + 96*(59*B*b^3*c
^4 - 34*A*b^2*c^5)*d^6*x^6 + 24*(57*B*b^4*c^3 - 212*A*b^3*c^4)*d^6*x^5 - 48*(1207*B*b^5*c^2 - 2785*A*b^4*c^3)*
d^6*x^4 + 4*(25265*B*b^6*c - 65643*A*b^5*c^2)*d^6*x^3 + 7*(1415*B*b^7 + 21837*A*b^6*c)*d^6*x^2)*e^5 - (945*A*b
^7*d^7 - 384*(23*B*b^2*c^5 - 8*A*b*c^6)*d^7*x^6 + 96*(459*B*b^3*c^4 - 254*A*b^2*c^5)*d^7*x^5 - 24*(439*B*b^4*c
^3 + 316*A*b^3*c^4)*d^7*x^4 - 12*(24291*B*b^5*c^2 - 16846*A*b^4*c^3)*d^7*x^3 - 35*(1481*B*b^6*c + 11292*A*b^5*
c^2)*d^7*x^2 + 70*(50*B*b^7 - 519*A*b^6*c)*d^7*x)*e^4 - 3*(128*(13*B*b*c^6 - 2*A*c^7)*d^8*x^6 - 64*(335*B*b^2*
c^5 - 114*A*b*c^6)*d^8*x^5 + 16*(3161*B*b^3*c^4 - 1658*A*b^2*c^5)*d^8*x^4 + 16*(9883*B*b^4*c^3 + 187*A*b^3*c^4
)*d^8*x^3 + 14*(2357*B*b^5*c^2 + 13498*A*b^4*c^3)*d^8*x^2 - 70*(85*B*b^6*c - 397*A*b^5*c^2)*d^8*x + 35*(5*B*b^
7 - 51*A*b^6*c)*d^8)*e^3 + (1024*B*c^7*d^9*x^6 - 128*(277*B*b*c^6 - 42*A*c^7)*d^9*x^5 + 192*(1059*B*b^2*c^5 -
350*A*b*c^6)*d^9*x^4 + 16*(25775*B*b^3*c^4 + 9282*A*b^2*c^5)*d^9*x^3 + 56*(1237*B*b^4*c^3 + 8718*A*b^3*c^4)*d^
9*x^2 - 210*(155*B*b^5*c^2 - 438*A*b^4*c^3)*d^9*x + 105*(25*B*b^6*c - 114*A*b^5*c^2)*d^9)*e^2 + 28*(256*B*c^7*
d^10*x^5 - 32*(121*B*b*c^6 - 18*A*c^7)*d^10*x^4 - 2064*(3*B*b^2*c^5 + 2*A*b*c^6)*d^10*x^3 - 44*(11*B*b^3*c^4 +
186*A*b^2*c^5)*d^10*x^2 + 10*(71*B*b^4*c^3 - 150*A*b^3*c^4)*d^10*x - 15*(11*B*b^5*c^2 - 30*A*b^4*c^3)*d^10)*e
)*sqrt(c*x^2 + b*x))/(c^6*d^19 + b^6*d^6*x^7*e^13 - (6*b^5*c*d^7*x^7 - 7*b^6*d^7*x^6)*e^12 + 3*(5*b^4*c^2*d^8*
x^7 - 14*b^5*c*d^8*x^6 + 7*b^6*d^8*x^5)*e^11 - (20*b^3*c^3*d^9*x^7 - 105*b^4*c^2*d^9*x^6 + 126*b^5*c*d^9*x^5 -
35*b^6*d^9*x^4)*e^10 + 5*(3*b^2*c^4*d^10*x^7 - 28*b^3*c^3*d^10*x^6 + 63*b^4*c^2*d^10*x^5 - 42*b^5*c*d^10*x^4
+ 7*b^6*d^10*x^3)*e^9 - 3*(2*b*c^5*d^11*x^7 - 35*b^2*c^4*d^11*x^6 + 140*b^3*c^3*d^11*x^5 - 175*b^4*c^2*d^11*x^
4 + 70*b^5*c*d^11*x^3 - 7*b^6*d^11*x^2)*e^8 + (c^6*d^12*x^7 - 42*b*c^5*d^12*x^6 + 315*b^2*c^4*d^12*x^5 - 700*b
^3*c^3*d^12*x^4 + 525*b^4*c^2*d^12*x^3 - 126*b^5*c*d^12*x^2 + 7*b^6*d^12*x)*e^7 + (7*c^6*d^13*x^6 - 126*b*c^5*
d^13*x^5 + 525*b^2*c^4*d^13*x^4 - 700*b^3*c^3*d^13*x^3 + 315*b^4*c^2*d^13*x^2 - 42*b^5*c*d^13*x + b^6*d^13)*e^
6 + 3*(7*c^6*d^14*x^5 - 70*b*c^5*d^14*x^4 + 175*b^2*c^4*d^14*x^3 - 140*b^3*c^3*d^14*x^2 + 35*b^4*c^2*d^14*x -
2*b^5*c*d^14)*e^5 + 5*(7*c^6*d^15*x^4 - 42*b*c^5*d^15*x^3 + 63*b^2*c^4*d^15*x^2 - 28*b^3*c^3*d^15*x + 3*b^4*c^
2*d^15)*e^4 + (35*c^6*d^16*x^3 - 126*b*c^5*d^16*x^2 + 105*b^2*c^4*d^16*x - 20*b^3*c^3*d^16)*e^3 + 3*(7*c^6*d^1
7*x^2 - 14*b*c^5*d^17*x + 5*b^2*c^4*d^17)*e^2 + (7*c^6*d^18*x - 6*b*c^5*d^18)*e), -1/107520*(105*(9*A*b^7*x^7*
e^10 + 24*(B*b^5*c^2 - 2*A*b^4*c^3)*d^10 + (63*A*b^7*d*x^6 + (5*B*b^7 - 42*A*b^6*c)*d*x^7)*e^9 + (189*A*b^7*d^
2*x^5 - 4*(5*B*b^6*c - 18*A*b^5*c^2)*d^2*x^7 + 7*(5*B*b^7 - 42*A*b^6*c)*d^2*x^6)*e^8 + (315*A*b^7*d^3*x^4 + 24
*(B*b^5*c^2 - 2*A*b^4*c^3)*d^3*x^7 - 28*(5*B*b^6*c - 18*A*b^5*c^2)*d^3*x^6 + 21*(5*B*b^7 - 42*A*b^6*c)*d^3*x^5
)*e^7 + 7*(45*A*b^7*d^4*x^3 + 24*(B*b^5*c^2 - 2*A*b^4*c^3)*d^4*x^6 - 12*(5*B*b^6*c - 18*A*b^5*c^2)*d^4*x^5 + 5
*(5*B*b^7 - 42*A*b^6*c)*d^4*x^4)*e^6 + 7*(27*A*...

________________________________________________________________________________________

Sympy [F(-1)] Timed out
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)**(3/2)/(e*x+d)**8,x)

[Out]

Timed out

________________________________________________________________________________________

Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 7914 vs. \(2 (559) = 1118\).
time = 2.61, size = 7914, normalized size = 14.01 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x^2+b*x)^(3/2)/(e*x+d)^8,x, algorithm="giac")

[Out]

-1/1024*(24*B*b^5*c^2*d^3 - 48*A*b^4*c^3*d^3 - 20*B*b^6*c*d^2*e + 72*A*b^5*c^2*d^2*e + 5*B*b^7*d*e^2 - 42*A*b^
6*c*d*e^2 + 9*A*b^7*e^3)*arctan(-((sqrt(c)*x - sqrt(c*x^2 + b*x))*e + sqrt(c)*d)/sqrt(-c*d^2 + b*d*e))/((c^5*d
^10 - 5*b*c^4*d^9*e + 10*b^2*c^3*d^8*e^2 - 10*b^3*c^2*d^7*e^3 + 5*b^4*c*d^6*e^4 - b^5*d^5*e^5)*sqrt(-c*d^2 + b
*d*e)) + 1/107520*(458752*(sqrt(c)*x - sqrt(c*x^2 + b*x))^8*B*c^(19/2)*d^13*e + 131072*(sqrt(c)*x - sqrt(c*x^2
+ b*x))^7*B*c^10*d^14 + 688128*(sqrt(c)*x - sqrt(c*x^2 + b*x))^9*B*c^9*d^12*e^2 + 868352*(sqrt(c)*x - sqrt(c*
x^2 + b*x))^7*B*b*c^9*d^13*e + 98304*(sqrt(c)*x - sqrt(c*x^2 + b*x))^7*A*c^10*d^13*e + 458752*(sqrt(c)*x - sqr
t(c*x^2 + b*x))^6*B*b*c^(19/2)*d^14 + 573440*(sqrt(c)*x - sqrt(c*x^2 + b*x))^10*B*c^(17/2)*d^11*e^3 - 57344*(s
qrt(c)*x - sqrt(c*x^2 + b*x))^8*B*b*c^(17/2)*d^12*e^2 + 344064*(sqrt(c)*x - sqrt(c*x^2 + b*x))^8*A*c^(19/2)*d^
12*e^2 - 57344*(sqrt(c)*x - sqrt(c*x^2 + b*x))^6*B*b^2*c^(17/2)*d^13*e + 344064*(sqrt(c)*x - sqrt(c*x^2 + b*x)
)^6*A*b*c^(19/2)*d^13*e + 688128*(sqrt(c)*x - sqrt(c*x^2 + b*x))^5*B*b^2*c^9*d^14 + 286720*(sqrt(c)*x - sqrt(c
*x^2 + b*x))^11*B*c^8*d^10*e^4 - 1519616*(sqrt(c)*x - sqrt(c*x^2 + b*x))^9*B*b*c^8*d^11*e^3 + 516096*(sqrt(c)*
x - sqrt(c*x^2 + b*x))^9*A*c^9*d^11*e^3 - 2990080*(sqrt(c)*x - sqrt(c*x^2 + b*x))^7*B*b^2*c^8*d^12*e^2 + 73728
0*(sqrt(c)*x - sqrt(c*x^2 + b*x))^7*A*b*c^9*d^12*e^2 - 1519616*(sqrt(c)*x - sqrt(c*x^2 + b*x))^5*B*b^3*c^8*d^1
3*e + 516096*(sqrt(c)*x - sqrt(c*x^2 + b*x))^5*A*b^2*c^9*d^13*e + 573440*(sqrt(c)*x - sqrt(c*x^2 + b*x))^4*B*b
^3*c^(17/2)*d^14 - 1792000*(sqrt(c)*x - sqrt(c*x^2 + b*x))^10*B*b*c^(15/2)*d^10*e^4 + 430080*(sqrt(c)*x - sqrt
(c*x^2 + b*x))^10*A*c^(17/2)*d^10*e^4 - 3627008*(sqrt(c)*x - sqrt(c*x^2 + b*x))^8*B*b^2*c^(15/2)*d^11*e^3 + 25
8048*(sqrt(c)*x - sqrt(c*x^2 + b*x))^8*A*b*c^(17/2)*d^11*e^3 - 3627008*(sqrt(c)*x - sqrt(c*x^2 + b*x))^6*B*b^3
*c^(15/2)*d^12*e^2 + 258048*(sqrt(c)*x - sqrt(c*x^2 + b*x))^6*A*b^2*c^(17/2)*d^12*e^2 - 1792000*(sqrt(c)*x - s
qrt(c*x^2 + b*x))^4*B*b^4*c^(15/2)*d^13*e + 430080*(sqrt(c)*x - sqrt(c*x^2 + b*x))^4*A*b^3*c^(17/2)*d^13*e + 2
86720*(sqrt(c)*x - sqrt(c*x^2 + b*x))^3*B*b^4*c^8*d^14 - 1433600*(sqrt(c)*x - sqrt(c*x^2 + b*x))^11*B*b*c^7*d^
9*e^5 - 960512*(sqrt(c)*x - sqrt(c*x^2 + b*x))^9*B*b^2*c^7*d^10*e^4 - 688128*(sqrt(c)*x - sqrt(c*x^2 + b*x))^9
*A*b*c^8*d^10*e^4 + 55296*(sqrt(c)*x - sqrt(c*x^2 + b*x))^7*B*b^3*c^7*d^11*e^3 - 1683456*(sqrt(c)*x - sqrt(c*x
^2 + b*x))^7*A*b^2*c^8*d^11*e^3 - 960512*(sqrt(c)*x - sqrt(c*x^2 + b*x))^5*B*b^4*c^7*d^12*e^2 - 688128*(sqrt(c
)*x - sqrt(c*x^2 + b*x))^5*A*b^3*c^8*d^12*e^2 - 1025024*(sqrt(c)*x - sqrt(c*x^2 + b*x))^3*B*b^5*c^7*d^13*e + 2
15040*(sqrt(c)*x - sqrt(c*x^2 + b*x))^3*A*b^4*c^8*d^13*e + 86016*(sqrt(c)*x - sqrt(c*x^2 + b*x))^2*B*b^5*c^(15
/2)*d^14 + 358400*(sqrt(c)*x - sqrt(c*x^2 + b*x))^10*B*b^2*c^(13/2)*d^9*e^5 - 2150400*(sqrt(c)*x - sqrt(c*x^2
+ b*x))^10*A*b*c^(15/2)*d^9*e^5 + 4515840*(sqrt(c)*x - sqrt(c*x^2 + b*x))^8*B*b^3*c^(13/2)*d^10*e^4 - 2795520*
(sqrt(c)*x - sqrt(c*x^2 + b*x))^8*A*b^2*c^(15/2)*d^10*e^4 + 4515840*(sqrt(c)*x - sqrt(c*x^2 + b*x))^6*B*b^4*c^
(13/2)*d^11*e^3 - 2795520*(sqrt(c)*x - sqrt(c*x^2 + b*x))^6*A*b^3*c^(15/2)*d^11*e^3 + 1078784*(sqrt(c)*x - sqr
t(c*x^2 + b*x))^4*B*b^5*c^(13/2)*d^12*e^2 - 967680*(sqrt(c)*x - sqrt(c*x^2 + b*x))^4*A*b^4*c^(15/2)*d^12*e^2 -
326144*(sqrt(c)*x - sqrt(c*x^2 + b*x))^2*B*b^6*c^(13/2)*d^13*e + 64512*(sqrt(c)*x - sqrt(c*x^2 + b*x))^2*A*b^
5*c^(15/2)*d^13*e + 14336*(sqrt(c)*x - sqrt(c*x^2 + b*x))*B*b^6*c^7*d^14 + 2867200*(sqrt(c)*x - sqrt(c*x^2 + b
*x))^11*B*b^2*c^6*d^8*e^6 + 3512320*(sqrt(c)*x - sqrt(c*x^2 + b*x))^9*B*b^3*c^6*d^9*e^5 - 4300800*(sqrt(c)*x -
sqrt(c*x^2 + b*x))^9*A*b^2*c^7*d^9*e^5 + 5806080*(sqrt(c)*x - sqrt(c*x^2 + b*x))^7*B*b^4*c^6*d^10*e^4 - 21196
80*(sqrt(c)*x - sqrt(c*x^2 + b*x))^7*A*b^3*c^7*d^10*e^4 + 4076800*(sqrt(c)*x - sqrt(c*x^2 + b*x))^5*B*b^5*c^6*
d^11*e^3 - 1774080*(sqrt(c)*x - sqrt(c*x^2 + b*x))^5*A*b^4*c^7*d^11*e^3 + 1028608*(sqrt(c)*x - sqrt(c*x^2 + b*
x))^3*B*b^6*c^6*d^12*e^2 - 580608*(sqrt(c)*x - sqrt(c*x^2 + b*x))^3*A*b^5*c^7*d^12*e^2 - 55552*(sqrt(c)*x - sq
rt(c*x^2 + b*x))*B*b^7*c^6*d^13*e + 10752*(sqrt(c)*x - sqrt(c*x^2 + b*x))*A*b^6*c^7*d^13*e + 1024*B*b^7*c^(13/
2)*d^14 + 5017600*(sqrt(c)*x - sqrt(c*x^2 + b*x))^10*B*b^3*c^(11/2)*d^8*e^6 + 4300800*(sqrt(c)*x - sqrt(c*x^2
+ b*x))^10*A*b^2*c^(13/2)*d^8*e^6 + 1720320*(sqrt(c)*x - sqrt(c*x^2 + b*x))^8*B*b^4*c^(11/2)*d^9*e^5 - 1935360
*(sqrt(c)*x - sqrt(c*x^2 + b*x))^8*A*b^3*c^(13/2)*d^9*e^5 + 1975680*(sqrt(c)*x - sqrt(c*x^2 + b*x))^6*B*b^5*c^
(11/2)*d^10*e^4 + 564480*(sqrt(c)*x - sqrt(c*x^2 + b*x))^6*A*b^4*c^(13/2)*d^10*e^4 + 1511552*(sqrt(c)*x - sqrt
(c*x^2 + b*x))^4*B*b^6*c^(11/2)*d^11*e^3 - 413952*(sqrt(c)*x - sqrt(c*x^2 + b*x))^4*A*b^5*c^(13/2)*d^11*e^3 +
380800*(sqrt(c)*x - sqrt(c*x^2 + b*x))^2*B*b^7*c^(11/2)*d^12*e^2 - 188160*(sqrt(c)*x - sqrt(c*x^2 + b*x))^2*A*
b^6*c^(13/2)*d^12*e^2 - 3968*B*b^8*c^(11/2)*d^13*e + 768*A*b^7*c^(13/2)*d^13*e - 2867200*(sqrt(c)*x - sqrt(c*x
^2 + b*x))^11*B*b^3*c^5*d^7*e^7 + 1863680*(sqrt...

________________________________________________________________________________________

Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\left (c\,x^2+b\,x\right )}^{3/2}\,\left (A+B\,x\right )}{{\left (d+e\,x\right )}^8} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((b*x + c*x^2)^(3/2)*(A + B*x))/(d + e*x)^8,x)

[Out]

int(((b*x + c*x^2)^(3/2)*(A + B*x))/(d + e*x)^8, x)

________________________________________________________________________________________