\(\int \frac {(A+B x) (b x+c x^2)^{3/2}}{(d+e x)^7} \, dx\) [1181]

Optimal result

Integrand size = 26, antiderivative size = 402 \[ \int \frac {(A+B x) \left (b x+c x^2\right )^{3/2}}{(d+e x)^7} \, dx=-\frac {b^2 \left (24 A c^2 d^2-12 b c d (B d+2 A e)+b^2 e (5 B d+7 A e)\right ) (b d+(2 c d-b e) x) \sqrt {b x+c x^2}}{512 d^4 (c d-b e)^4 (d+e x)^2}+\frac {\left (24 A c^2 d^2-12 b c d (B d+2 A e)+b^2 e (5 B d+7 A e)\right ) (b d+(2 c d-b e) x) \left (b x+c x^2\right )^{3/2}}{192 d^3 (c d-b e)^3 (d+e x)^4}+\frac {(B d-A e) \left (b x+c x^2\right )^{5/2}}{6 d (c d-b e) (d+e x)^6}-\frac {(7 A e (2 c d-b e)-B d (2 c d+5 b e)) \left (b x+c x^2\right )^{5/2}}{60 d^2 (c d-b e)^2 (d+e x)^5}+\frac {b^4 \left (24 A c^2 d^2-12 b c d (B d+2 A e)+b^2 e (5 B d+7 A e)\right ) \text {arctanh}\left (\frac {b d+(2 c d-b e) x}{2 \sqrt {d} \sqrt {c d-b e} \sqrt {b x+c x^2}}\right )}{1024 d^{9/2} (c d-b e)^{9/2}} \]

[Out]

1/192*(24*A*c^2*d^2-12*b*c*d*(2*A*e+B*d)+b^2*e*(7*A*e+5*B*d))*(b*d+(-b*e+2*c*d)*x)*(c*x^2+b*x)^(3/2)/d^3/(-b*e
+c*d)^3/(e*x+d)^4+1/6*(-A*e+B*d)*(c*x^2+b*x)^(5/2)/d/(-b*e+c*d)/(e*x+d)^6-1/60*(7*A*e*(-b*e+2*c*d)-B*d*(5*b*e+
2*c*d))*(c*x^2+b*x)^(5/2)/d^2/(-b*e+c*d)^2/(e*x+d)^5+1/1024*b^4*(24*A*c^2*d^2-12*b*c*d*(2*A*e+B*d)+b^2*e*(7*A*
e+5*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^(9/2)/(-b*e+c*d)^(9/2
)-1/512*b^2*(24*A*c^2*d^2-12*b*c*d*(2*A*e+B*d)+b^2*e*(7*A*e+5*B*d))*(b*d+(-b*e+2*c*d)*x)*(c*x^2+b*x)^(1/2)/d^4
/(-b*e+c*d)^4/(e*x+d)^2

Rubi [A] (verified)

Time = 0.38 (sec) , antiderivative size = 402, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.192, Rules used = {848, 820, 734, 738, 212} \[ \int \frac {(A+B x) \left (b x+c x^2\right )^{3/2}}{(d+e x)^7} \, dx=\frac {b^4 \left (b^2 e (7 A e+5 B d)-12 b c d (2 A e+B d)+24 A c^2 d^2\right ) \text {arctanh}\left (\frac {x (2 c d-b e)+b d}{2 \sqrt {d} \sqrt {b x+c x^2} \sqrt {c d-b e}}\right )}{1024 d^{9/2} (c d-b e)^{9/2}}-\frac {b^2 \sqrt {b x+c x^2} (x (2 c d-b e)+b d) \left (b^2 e (7 A e+5 B d)-12 b c d (2 A e+B d)+24 A c^2 d^2\right )}{512 d^4 (d+e x)^2 (c d-b e)^4}+\frac {\left (b x+c x^2\right )^{3/2} (x (2 c d-b e)+b d) \left (b^2 e (7 A e+5 B d)-12 b c d (2 A e+B d)+24 A c^2 d^2\right )}{192 d^3 (d+e x)^4 (c d-b e)^3}-\frac {\left (b x+c x^2\right )^{5/2} (7 A e (2 c d-b e)-B d (5 b e+2 c d))}{60 d^2 (d+e x)^5 (c d-b e)^2}+\frac {\left (b x+c x^2\right )^{5/2} (B d-A e)}{6 d (d+e x)^6 (c d-b e)} \]

[In]

Int[((A + B*x)*(b*x + c*x^2)^(3/2))/(d + e*x)^7,x]

[Out]

-1/512*(b^2*(24*A*c^2*d^2 - 12*b*c*d*(B*d + 2*A*e) + b^2*e*(5*B*d + 7*A*e))*(b*d + (2*c*d - b*e)*x)*Sqrt[b*x +
c*x^2])/(d^4*(c*d - b*e)^4*(d + e*x)^2) + ((24*A*c^2*d^2 - 12*b*c*d*(B*d + 2*A*e) + b^2*e*(5*B*d + 7*A*e))*(b
*d + (2*c*d - b*e)*x)*(b*x + c*x^2)^(3/2))/(192*d^3*(c*d - b*e)^3*(d + e*x)^4) + ((B*d - A*e)*(b*x + c*x^2)^(5
/2))/(6*d*(c*d - b*e)*(d + e*x)^6) - ((7*A*e*(2*c*d - b*e) - B*d*(2*c*d + 5*b*e))*(b*x + c*x^2)^(5/2))/(60*d^2
*(c*d - b*e)^2*(d + e*x)^5) + (b^4*(24*A*c^2*d^2 - 12*b*c*d*(B*d + 2*A*e) + b^2*e*(5*B*d + 7*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])])/(1024*d^(9/2)*(c*d - b*e)^(9/2))

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*} \text {integral}& = \frac {(B d-A e) \left (b x+c x^2\right )^{5/2}}{6 d (c d-b e) (d+e x)^6}-\frac {\int \frac {\left (\frac {1}{2} (-12 A c d+b (5 B d+7 A e))-c (B d-A e) x\right ) \left (b x+c x^2\right )^{3/2}}{(d+e x)^6} \, dx}{6 d (c d-b e)} \\ & = \frac {(B d-A e) \left (b x+c x^2\right )^{5/2}}{6 d (c d-b e) (d+e x)^6}-\frac {(7 A e (2 c d-b e)-B d (2 c d+5 b e)) \left (b x+c x^2\right )^{5/2}}{60 d^2 (c d-b e)^2 (d+e x)^5}+\frac {\left (24 A c^2 d^2-12 b c d (B d+2 A e)+b^2 e (5 B d+7 A e)\right ) \int \frac {\left (b x+c x^2\right )^{3/2}}{(d+e x)^5} \, dx}{24 d^2 (c d-b e)^2} \\ & = \frac {\left (24 A c^2 d^2-12 b c d (B d+2 A e)+b^2 e (5 B d+7 A e)\right ) (b d+(2 c d-b e) x) \left (b x+c x^2\right )^{3/2}}{192 d^3 (c d-b e)^3 (d+e x)^4}+\frac {(B d-A e) \left (b x+c x^2\right )^{5/2}}{6 d (c d-b e) (d+e x)^6}-\frac {(7 A e (2 c d-b e)-B d (2 c d+5 b e)) \left (b x+c x^2\right )^{5/2}}{60 d^2 (c d-b e)^2 (d+e x)^5}-\frac {\left (b^2 \left (24 A c^2 d^2-12 b c d (B d+2 A e)+b^2 e (5 B d+7 A e)\right )\right ) \int \frac {\sqrt {b x+c x^2}}{(d+e x)^3} \, dx}{128 d^3 (c d-b e)^3} \\ & = -\frac {b^2 \left (24 A c^2 d^2-12 b c d (B d+2 A e)+b^2 e (5 B d+7 A e)\right ) (b d+(2 c d-b e) x) \sqrt {b x+c x^2}}{512 d^4 (c d-b e)^4 (d+e x)^2}+\frac {\left (24 A c^2 d^2-12 b c d (B d+2 A e)+b^2 e (5 B d+7 A e)\right ) (b d+(2 c d-b e) x) \left (b x+c x^2\right )^{3/2}}{192 d^3 (c d-b e)^3 (d+e x)^4}+\frac {(B d-A e) \left (b x+c x^2\right )^{5/2}}{6 d (c d-b e) (d+e x)^6}-\frac {(7 A e (2 c d-b e)-B d (2 c d+5 b e)) \left (b x+c x^2\right )^{5/2}}{60 d^2 (c d-b e)^2 (d+e x)^5}+\frac {\left (b^4 \left (24 A c^2 d^2-12 b c d (B d+2 A e)+b^2 e (5 B d+7 A e)\right )\right ) \int \frac {1}{(d+e x) \sqrt {b x+c x^2}} \, dx}{1024 d^4 (c d-b e)^4} \\ & = -\frac {b^2 \left (24 A c^2 d^2-12 b c d (B d+2 A e)+b^2 e (5 B d+7 A e)\right ) (b d+(2 c d-b e) x) \sqrt {b x+c x^2}}{512 d^4 (c d-b e)^4 (d+e x)^2}+\frac {\left (24 A c^2 d^2-12 b c d (B d+2 A e)+b^2 e (5 B d+7 A e)\right ) (b d+(2 c d-b e) x) \left (b x+c x^2\right )^{3/2}}{192 d^3 (c d-b e)^3 (d+e x)^4}+\frac {(B d-A e) \left (b x+c x^2\right )^{5/2}}{6 d (c d-b e) (d+e x)^6}-\frac {(7 A e (2 c d-b e)-B d (2 c d+5 b e)) \left (b x+c x^2\right )^{5/2}}{60 d^2 (c d-b e)^2 (d+e x)^5}-\frac {\left (b^4 \left (24 A c^2 d^2-12 b c d (B d+2 A e)+b^2 e (5 B d+7 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 )}{512 d^4 (c d-b e)^4} \\ & = -\frac {b^2 \left (24 A c^2 d^2-12 b c d (B d+2 A e)+b^2 e (5 B d+7 A e)\right ) (b d+(2 c d-b e) x) \sqrt {b x+c x^2}}{512 d^4 (c d-b e)^4 (d+e x)^2}+\frac {\left (24 A c^2 d^2-12 b c d (B d+2 A e)+b^2 e (5 B d+7 A e)\right ) (b d+(2 c d-b e) x) \left (b x+c x^2\right )^{3/2}}{192 d^3 (c d-b e)^3 (d+e x)^4}+\frac {(B d-A e) \left (b x+c x^2\right )^{5/2}}{6 d (c d-b e) (d+e x)^6}-\frac {(7 A e (2 c d-b e)-B d (2 c d+5 b e)) \left (b x+c x^2\right )^{5/2}}{60 d^2 (c d-b e)^2 (d+e x)^5}+\frac {b^4 \left (24 A c^2 d^2-12 b c d (B d+2 A e)+b^2 e (5 B d+7 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 )}{1024 d^{9/2} (c d-b e)^{9/2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 10.94 (sec) , antiderivative size = 352, normalized size of antiderivative = 0.88 \[ \int \frac {(A+B x) \left (b x+c x^2\right )^{3/2}}{(d+e x)^7} \, dx=\frac {(x (b+c x))^{3/2} \left (\frac {(-B d+A e) x^{5/2} (b+c x)}{(d+e x)^6}-\frac {(7 A e (-2 c d+b e)+B d (2 c d+5 b e)) x^{5/2} (b+c x)}{10 d (c d-b e) (d+e x)^5}+\frac {\left (24 A c^2 d^2-12 b c d (B d+2 A e)+b^2 e (5 B d+7 A e)\right ) \left (-\frac {2 x^{3/2} (b+c x)^{5/2}}{(d+e x)^4}+\frac {b \sqrt {x} (b+c x)^{5/2}}{(c d-b e) (d+e x)^3}-\frac {b^2 \sqrt {x} \sqrt {b+c x} (5 b d+2 c d x+3 b e x)}{8 d^2 (c d-b e) (d+e x)^2}-\frac {3 b^4 \text {arctanh}\left (\frac {\sqrt {c d-b e} \sqrt {x}}{\sqrt {d} \sqrt {b+c x}}\right )}{8 d^{5/2} (c d-b e)^{3/2}}\right )}{32 d (c d-b e)^2 (b+c x)^{3/2}}\right )}{6 d (-c d+b e) x^{3/2}} \]

[In]

Integrate[((A + B*x)*(b*x + c*x^2)^(3/2))/(d + e*x)^7,x]

[Out]

((x*(b + c*x))^(3/2)*(((-(B*d) + A*e)*x^(5/2)*(b + c*x))/(d + e*x)^6 - ((7*A*e*(-2*c*d + b*e) + B*d*(2*c*d + 5
*b*e))*x^(5/2)*(b + c*x))/(10*d*(c*d - b*e)*(d + e*x)^5) + ((24*A*c^2*d^2 - 12*b*c*d*(B*d + 2*A*e) + b^2*e*(5*
B*d + 7*A*e))*((-2*x^(3/2)*(b + c*x)^(5/2))/(d + e*x)^4 + (b*Sqrt[x]*(b + c*x)^(5/2))/((c*d - b*e)*(d + e*x)^3
) - (b^2*Sqrt[x]*Sqrt[b + c*x]*(5*b*d + 2*c*d*x + 3*b*e*x))/(8*d^2*(c*d - b*e)*(d + e*x)^2) - (3*b^4*ArcTanh[(
Sqrt[c*d - b*e]*Sqrt[x])/(Sqrt[d]*Sqrt[b + c*x])])/(8*d^(5/2)*(c*d - b*e)^(3/2))))/(32*d*(c*d - b*e)^2*(b + c*
x)^(3/2))))/(6*d*(-(c*d) + b*e)*x^(3/2))

Maple [A] (verified)

Time = 1.10 (sec) , antiderivative size = 598, normalized size of antiderivative = 1.49

method	result	size
pseudoelliptic	\(-\frac {7 \left (\left (\frac {12 \left (2 A \,c^{2}-B b c \right ) d^{2}}{7}-\frac {24 \left (A c -\frac {5 B b}{24}\right ) e b d}{7}+A \,b^{2} e^{2}\right ) \left (e x +d \right )^{6} b^{4} \arctan \left (\frac {\sqrt {x \left (c x +b \right )}\, d}{x \sqrt {d \left (b e -c d \right )}}\right )+\sqrt {d \left (b e -c d \right )}\, \left (\frac {24 c \left (-\frac {b^{4} B}{2}+c \left (\frac {B x}{3}+A \right ) b^{3}-\frac {2 c^{2} x \left (\frac {2 B x}{5}+A \right ) b^{2}}{3}-8 c^{3} x^{2} \left (\frac {11 B x}{15}+A \right ) b -\frac {16 c^{4} \left (\frac {4 B x}{5}+A \right ) x^{3}}{3}\right ) d^{7}}{7}-\frac {24 \left (-\frac {5 B \,b^{5}}{24}+c \left (\frac {107 B x}{36}+A \right ) b^{4}-\frac {19 c^{2} x \left (\frac {89 B x}{285}+A \right ) b^{3}}{3}-\frac {448 c^{3} \left (\frac {171 B x}{224}+A \right ) x^{2} b^{2}}{15}-\frac {72 c^{4} x^{3} \left (\frac {80 B x}{81}+A \right ) b}{5}+\frac {32 c^{5} x^{4} \left (\frac {B x}{3}+A \right )}{15}\right ) e \,d^{6}}{7}+\left (\left (\frac {85 B x}{21}+A \right ) b^{5}-\frac {422 c x \left (\frac {1328 B x}{1055}+A \right ) b^{4}}{21}-\frac {2456 c^{2} \left (\frac {285 B x}{307}+A \right ) x^{2} b^{3}}{21}-\frac {816 c^{3} x^{3} \left (\frac {332 B x}{153}+A \right ) b^{2}}{35}+\frac {1984 c^{4} \left (\frac {13 B x}{31}+A \right ) x^{4} b}{105}-\frac {128 A \,c^{5} x^{5}}{105}\right ) e^{2} d^{5}+\frac {17 \left (\left (\frac {198 B x}{119}+A \right ) b^{4}+\frac {7144 c \left (\frac {69 B x}{94}+A \right ) x \,b^{3}}{595}-\frac {2232 c^{2} x^{2} \left (-\frac {37 B x}{93}+A \right ) b^{2}}{595}-\frac {736 c^{3} \left (\frac {51 B x}{46}+A \right ) x^{3} b}{595}+\frac {64 A \,c^{4} x^{4}}{119}\right ) x \,e^{3} b \,d^{4}}{3}-\frac {562 x^{2} \left (\left (\frac {165 B x}{281}+A \right ) b^{3}-\frac {642 c \left (\frac {185 B x}{963}+A \right ) x \,b^{2}}{281}+\frac {168 c^{2} x^{2} \left (-\frac {5 B x}{63}+A \right ) b}{281}+\frac {16 A \,c^{3} x^{3}}{281}\right ) e^{4} b^{2} d^{3}}{35}-\frac {66 x^{3} \left (\left (\frac {425 B x}{1386}+A \right ) b^{2}-\frac {824 c \left (\frac {65 B x}{824}+A \right ) x b}{693}+\frac {8 A \,c^{2} x^{2}}{63}\right ) e^{5} b^{3} d^{2}}{5}-\frac {17 x^{4} \left (\left (\frac {15 B x}{119}+A \right ) b -\frac {58 A c x}{119}\right ) e^{6} b^{4} d}{3}-A \,b^{5} e^{7} x^{5}\right ) \sqrt {x \left (c x +b \right )}\right )}{512 \sqrt {d \left (b e -c d \right )}\, \left (e x +d \right )^{6} \left (b e -c d \right )^{4} d^{4}}\)	\(598\)
default	\(\text {Expression too large to display}\)	\(13664\)

[In]

int((B*x+A)*(c*x^2+b*x)^(3/2)/(e*x+d)^7,x,method=_RETURNVERBOSE)

[Out]

-7/512/(d*(b*e-c*d))^(1/2)*((12/7*(2*A*c^2-B*b*c)*d^2-24/7*(A*c-5/24*B*b)*e*b*d+A*b^2*e^2)*(e*x+d)^6*b^4*arcta
n((x*(c*x+b))^(1/2)/x*d/(d*(b*e-c*d))^(1/2))+(d*(b*e-c*d))^(1/2)*(24/7*c*(-1/2*b^4*B+c*(1/3*B*x+A)*b^3-2/3*c^2
*x*(2/5*B*x+A)*b^2-8*c^3*x^2*(11/15*B*x+A)*b-16/3*c^4*(4/5*B*x+A)*x^3)*d^7-24/7*(-5/24*B*b^5+c*(107/36*B*x+A)*
b^4-19/3*c^2*x*(89/285*B*x+A)*b^3-448/15*c^3*(171/224*B*x+A)*x^2*b^2-72/5*c^4*x^3*(80/81*B*x+A)*b+32/15*c^5*x^
4*(1/3*B*x+A))*e*d^6+((85/21*B*x+A)*b^5-422/21*c*x*(1328/1055*B*x+A)*b^4-2456/21*c^2*(285/307*B*x+A)*x^2*b^3-8
16/35*c^3*x^3*(332/153*B*x+A)*b^2+1984/105*c^4*(13/31*B*x+A)*x^4*b-128/105*A*c^5*x^5)*e^2*d^5+17/3*((198/119*B
*x+A)*b^4+7144/595*c*(69/94*B*x+A)*x*b^3-2232/595*c^2*x^2*(-37/93*B*x+A)*b^2-736/595*c^3*(51/46*B*x+A)*x^3*b+6
4/119*A*c^4*x^4)*x*e^3*b*d^4-562/35*x^2*((165/281*B*x+A)*b^3-642/281*c*(185/963*B*x+A)*x*b^2+168/281*c^2*x^2*(
-5/63*B*x+A)*b+16/281*A*c^3*x^3)*e^4*b^2*d^3-66/5*x^3*((425/1386*B*x+A)*b^2-824/693*c*(65/824*B*x+A)*x*b+8/63*
A*c^2*x^2)*e^5*b^3*d^2-17/3*x^4*((15/119*B*x+A)*b-58/119*A*c*x)*e^6*b^4*d-A*b^5*e^7*x^5)*(x*(c*x+b))^(1/2))/(e
*x+d)^6/(b*e-c*d)^4/d^4

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1861 vs. \(2 (371) = 742\).

Time = 1.08 (sec) , antiderivative size = 3734, normalized size of antiderivative = 9.29 \[ \int \frac {(A+B x) \left (b x+c x^2\right )^{3/2}}{(d+e x)^7} \, dx=\text {Too large to display} \]

[In]

integrate((B*x+A)*(c*x^2+b*x)^(3/2)/(e*x+d)^7,x, algorithm="fricas")

[Out]

[1/15360*(15*(7*A*b^6*d^6*e^2 - 12*(B*b^5*c - 2*A*b^4*c^2)*d^8 + (5*B*b^6 - 24*A*b^5*c)*d^7*e + (7*A*b^6*e^8 -
12*(B*b^5*c - 2*A*b^4*c^2)*d^2*e^6 + (5*B*b^6 - 24*A*b^5*c)*d*e^7)*x^6 + 6*(7*A*b^6*d*e^7 - 12*(B*b^5*c - 2*A
*b^4*c^2)*d^3*e^5 + (5*B*b^6 - 24*A*b^5*c)*d^2*e^6)*x^5 + 15*(7*A*b^6*d^2*e^6 - 12*(B*b^5*c - 2*A*b^4*c^2)*d^4
*e^4 + (5*B*b^6 - 24*A*b^5*c)*d^3*e^5)*x^4 + 20*(7*A*b^6*d^3*e^5 - 12*(B*b^5*c - 2*A*b^4*c^2)*d^5*e^3 + (5*B*b
^6 - 24*A*b^5*c)*d^4*e^4)*x^3 + 15*(7*A*b^6*d^4*e^4 - 12*(B*b^5*c - 2*A*b^4*c^2)*d^6*e^2 + (5*B*b^6 - 24*A*b^5
*c)*d^5*e^3)*x^2 + 6*(7*A*b^6*d^5*e^3 - 12*(B*b^5*c - 2*A*b^4*c^2)*d^7*e + (5*B*b^6 - 24*A*b^5*c)*d^6*e^2)*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*(105*
A*b^6*d^6*e^3 + 180*(B*b^4*c^2 - 2*A*b^3*c^3)*d^9 - 15*(17*B*b^5*c - 48*A*b^4*c^2)*d^8*e + 15*(5*B*b^6 - 31*A*
b^5*c)*d^7*e^2 + (256*B*c^6*d^8*e - 105*A*b^6*d*e^8 - 64*(17*B*b*c^5 - 2*A*c^6)*d^7*e^2 + 16*(103*B*b^2*c^4 -
28*A*b*c^5)*d^6*e^3 - 32*(28*B*b^3*c^3 - 13*A*b^2*c^4)*d^5*e^4 - 10*(5*B*b^4*c^2 - 8*A*b^3*c^3)*d^4*e^5 + (205
*B*b^5*c - 466*A*b^4*c^2)*d^3*e^6 - 5*(15*B*b^6 - 79*A*b^5*c)*d^2*e^7)*x^5 + (1536*B*c^6*d^9 - 595*A*b^6*d^2*e
^7 - 256*(26*B*b*c^5 - 3*A*c^6)*d^8*e + 64*(163*B*b^2*c^4 - 43*A*b*c^5)*d^7*e^2 - 40*(155*B*b^3*c^3 - 68*A*b^2
*c^4)*d^6*e^3 + 4*(37*B*b^4*c^2 + 68*A*b^3*c^3)*d^5*e^4 + (1165*B*b^5*c - 2656*A*b^4*c^2)*d^4*e^5 - (425*B*b^6
- 2243*A*b^5*c)*d^3*e^6)*x^4 - 6*(231*A*b^6*d^3*e^6 - 32*(11*B*b*c^5 + 10*A*c^6)*d^9 + 8*(215*B*b^2*c^4 + 148
*A*b*c^5)*d^8*e - 4*(817*B*b^3*c^3 + 318*A*b^2*c^4)*d^7*e^2 + 2*(1387*B*b^4*c^2 + 18*A*b^3*c^3)*d^6*e^3 - (103
9*B*b^5*c - 1014*A*b^4*c^2)*d^5*e^4 + 3*(55*B*b^6 - 291*A*b^5*c)*d^4*e^5)*x^3 - 2*(843*A*b^6*d^4*e^5 - 48*(B*b
^2*c^4 + 30*A*b*c^5)*d^9 + 4*(101*B*b^3*c^3 + 1704*A*b^2*c^4)*d^8*e - 4*(421*B*b^4*c^2 + 2879*A*b^3*c^3)*d^7*e
^2 + (1823*B*b^5*c + 9712*A*b^4*c^2)*d^6*e^3 - 5*(99*B*b^6 + 883*A*b^5*c)*d^5*e^4)*x^2 + 5*(119*A*b^6*d^5*e^4
- 24*(B*b^3*c^3 - 2*A*b^2*c^4)*d^9 + 14*(17*B*b^4*c^2 - 36*A*b^3*c^3)*d^8*e - (299*B*b^5*c - 878*A*b^4*c^2)*d^
7*e^2 + (85*B*b^6 - 541*A*b^5*c)*d^6*e^3)*x)*sqrt(c*x^2 + b*x))/(c^5*d^16 - 5*b*c^4*d^15*e + 10*b^2*c^3*d^14*e
^2 - 10*b^3*c^2*d^13*e^3 + 5*b^4*c*d^12*e^4 - b^5*d^11*e^5 + (c^5*d^10*e^6 - 5*b*c^4*d^9*e^7 + 10*b^2*c^3*d^8*
e^8 - 10*b^3*c^2*d^7*e^9 + 5*b^4*c*d^6*e^10 - b^5*d^5*e^11)*x^6 + 6*(c^5*d^11*e^5 - 5*b*c^4*d^10*e^6 + 10*b^2*
c^3*d^9*e^7 - 10*b^3*c^2*d^8*e^8 + 5*b^4*c*d^7*e^9 - b^5*d^6*e^10)*x^5 + 15*(c^5*d^12*e^4 - 5*b*c^4*d^11*e^5 +
10*b^2*c^3*d^10*e^6 - 10*b^3*c^2*d^9*e^7 + 5*b^4*c*d^8*e^8 - b^5*d^7*e^9)*x^4 + 20*(c^5*d^13*e^3 - 5*b*c^4*d^
12*e^4 + 10*b^2*c^3*d^11*e^5 - 10*b^3*c^2*d^10*e^6 + 5*b^4*c*d^9*e^7 - b^5*d^8*e^8)*x^3 + 15*(c^5*d^14*e^2 - 5
*b*c^4*d^13*e^3 + 10*b^2*c^3*d^12*e^4 - 10*b^3*c^2*d^11*e^5 + 5*b^4*c*d^10*e^6 - b^5*d^9*e^7)*x^2 + 6*(c^5*d^1
5*e - 5*b*c^4*d^14*e^2 + 10*b^2*c^3*d^13*e^3 - 10*b^3*c^2*d^12*e^4 + 5*b^4*c*d^11*e^5 - b^5*d^10*e^6)*x), 1/76
80*(15*(7*A*b^6*d^6*e^2 - 12*(B*b^5*c - 2*A*b^4*c^2)*d^8 + (5*B*b^6 - 24*A*b^5*c)*d^7*e + (7*A*b^6*e^8 - 12*(B
*b^5*c - 2*A*b^4*c^2)*d^2*e^6 + (5*B*b^6 - 24*A*b^5*c)*d*e^7)*x^6 + 6*(7*A*b^6*d*e^7 - 12*(B*b^5*c - 2*A*b^4*c
^2)*d^3*e^5 + (5*B*b^6 - 24*A*b^5*c)*d^2*e^6)*x^5 + 15*(7*A*b^6*d^2*e^6 - 12*(B*b^5*c - 2*A*b^4*c^2)*d^4*e^4 +
(5*B*b^6 - 24*A*b^5*c)*d^3*e^5)*x^4 + 20*(7*A*b^6*d^3*e^5 - 12*(B*b^5*c - 2*A*b^4*c^2)*d^5*e^3 + (5*B*b^6 - 2
4*A*b^5*c)*d^4*e^4)*x^3 + 15*(7*A*b^6*d^4*e^4 - 12*(B*b^5*c - 2*A*b^4*c^2)*d^6*e^2 + (5*B*b^6 - 24*A*b^5*c)*d^
5*e^3)*x^2 + 6*(7*A*b^6*d^5*e^3 - 12*(B*b^5*c - 2*A*b^4*c^2)*d^7*e + (5*B*b^6 - 24*A*b^5*c)*d^6*e^2)*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)) + (105*A*b^6*d^6*e^3 + 180*(B*b
^4*c^2 - 2*A*b^3*c^3)*d^9 - 15*(17*B*b^5*c - 48*A*b^4*c^2)*d^8*e + 15*(5*B*b^6 - 31*A*b^5*c)*d^7*e^2 + (256*B*
c^6*d^8*e - 105*A*b^6*d*e^8 - 64*(17*B*b*c^5 - 2*A*c^6)*d^7*e^2 + 16*(103*B*b^2*c^4 - 28*A*b*c^5)*d^6*e^3 - 32
*(28*B*b^3*c^3 - 13*A*b^2*c^4)*d^5*e^4 - 10*(5*B*b^4*c^2 - 8*A*b^3*c^3)*d^4*e^5 + (205*B*b^5*c - 466*A*b^4*c^2
)*d^3*e^6 - 5*(15*B*b^6 - 79*A*b^5*c)*d^2*e^7)*x^5 + (1536*B*c^6*d^9 - 595*A*b^6*d^2*e^7 - 256*(26*B*b*c^5 - 3
*A*c^6)*d^8*e + 64*(163*B*b^2*c^4 - 43*A*b*c^5)*d^7*e^2 - 40*(155*B*b^3*c^3 - 68*A*b^2*c^4)*d^6*e^3 + 4*(37*B*
b^4*c^2 + 68*A*b^3*c^3)*d^5*e^4 + (1165*B*b^5*c - 2656*A*b^4*c^2)*d^4*e^5 - (425*B*b^6 - 2243*A*b^5*c)*d^3*e^6
)*x^4 - 6*(231*A*b^6*d^3*e^6 - 32*(11*B*b*c^5 + 10*A*c^6)*d^9 + 8*(215*B*b^2*c^4 + 148*A*b*c^5)*d^8*e - 4*(817
*B*b^3*c^3 + 318*A*b^2*c^4)*d^7*e^2 + 2*(1387*B*b^4*c^2 + 18*A*b^3*c^3)*d^6*e^3 - (1039*B*b^5*c - 1014*A*b^4*c
^2)*d^5*e^4 + 3*(55*B*b^6 - 291*A*b^5*c)*d^4*e^5)*x^3 - 2*(843*A*b^6*d^4*e^5 - 48*(B*b^2*c^4 + 30*A*b*c^5)*d^9
+ 4*(101*B*b^3*c^3 + 1704*A*b^2*c^4)*d^8*e - 4*(421*B*b^4*c^2 + 2879*A*b^3*c^3)*d^7*e^2 + (1823*B*b^5*c + 971
2*A*b^4*c^2)*d^6*e^3 - 5*(99*B*b^6 + 883*A*b^5*c)*d^5*e^4)*x^2 + 5*(119*A*b^6*d^5*e^4 - 24*(B*b^3*c^3 - 2*A*b^
2*c^4)*d^9 + 14*(17*B*b^4*c^2 - 36*A*b^3*c^3)*d^8*e - (299*B*b^5*c - 878*A*b^4*c^2)*d^7*e^2 + (85*B*b^6 - 541*
A*b^5*c)*d^6*e^3)*x)*sqrt(c*x^2 + b*x))/(c^5*d^16 - 5*b*c^4*d^15*e + 10*b^2*c^3*d^14*e^2 - 10*b^3*c^2*d^13*e^3
+ 5*b^4*c*d^12*e^4 - b^5*d^11*e^5 + (c^5*d^10*e^6 - 5*b*c^4*d^9*e^7 + 10*b^2*c^3*d^8*e^8 - 10*b^3*c^2*d^7*e^9
+ 5*b^4*c*d^6*e^10 - b^5*d^5*e^11)*x^6 + 6*(c^5*d^11*e^5 - 5*b*c^4*d^10*e^6 + 10*b^2*c^3*d^9*e^7 - 10*b^3*c^2
*d^8*e^8 + 5*b^4*c*d^7*e^9 - b^5*d^6*e^10)*x^5 + 15*(c^5*d^12*e^4 - 5*b*c^4*d^11*e^5 + 10*b^2*c^3*d^10*e^6 - 1
0*b^3*c^2*d^9*e^7 + 5*b^4*c*d^8*e^8 - b^5*d^7*e^9)*x^4 + 20*(c^5*d^13*e^3 - 5*b*c^4*d^12*e^4 + 10*b^2*c^3*d^11
*e^5 - 10*b^3*c^2*d^10*e^6 + 5*b^4*c*d^9*e^7 - b^5*d^8*e^8)*x^3 + 15*(c^5*d^14*e^2 - 5*b*c^4*d^13*e^3 + 10*b^2
*c^3*d^12*e^4 - 10*b^3*c^2*d^11*e^5 + 5*b^4*c*d^10*e^6 - b^5*d^9*e^7)*x^2 + 6*(c^5*d^15*e - 5*b*c^4*d^14*e^2 +
10*b^2*c^3*d^13*e^3 - 10*b^3*c^2*d^12*e^4 + 5*b^4*c*d^11*e^5 - b^5*d^10*e^6)*x)]

Sympy [F]

\[ \int \frac {(A+B x) \left (b x+c x^2\right )^{3/2}}{(d+e x)^7} \, dx=\int \frac {\left (x \left (b + c x\right )\right )^{\frac {3}{2}} \left (A + B x\right )}{\left (d + e x\right )^{7}}\, dx \]

[In]

integrate((B*x+A)*(c*x**2+b*x)**(3/2)/(e*x+d)**7,x)

[Out]

Integral((x*(b + c*x))**(3/2)*(A + B*x)/(d + e*x)**7, x)

Maxima [F(-2)]

Exception generated. \[ \int \frac {(A+B x) \left (b x+c x^2\right )^{3/2}}{(d+e x)^7} \, dx=\text {Exception raised: ValueError} \]

[In]

integrate((B*x+A)*(c*x^2+b*x)^(3/2)/(e*x+d)^7,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 detail

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 5881 vs. \(2 (371) = 742\).

Time = 0.41 (sec) , antiderivative size = 5881, normalized size of antiderivative = 14.63 \[ \int \frac {(A+B x) \left (b x+c x^2\right )^{3/2}}{(d+e x)^7} \, dx=\text {Too large to display} \]

[In]

integrate((B*x+A)*(c*x^2+b*x)^(3/2)/(e*x+d)^7,x, algorithm="giac")

[Out]

-1/512*(12*B*b^5*c*d^2 - 24*A*b^4*c^2*d^2 - 5*B*b^6*d*e + 24*A*b^5*c*d*e - 7*A*b^6*e^2)*arctan(-((sqrt(c)*x -
sqrt(c*x^2 + b*x))*e + sqrt(c)*d)/sqrt(-c*d^2 + b*d*e))/((c^4*d^8 - 4*b*c^3*d^7*e + 6*b^2*c^2*d^6*e^2 - 4*b^3*
c*d^5*e^3 + b^4*d^4*e^4)*sqrt(-c*d^2 + b*d*e)) + 1/7680*(180*(sqrt(c)*x - sqrt(c*x^2 + b*x))^11*B*b^5*c*d^2*e^
10 - 360*(sqrt(c)*x - sqrt(c*x^2 + b*x))^11*A*b^4*c^2*d^2*e^10 - 75*(sqrt(c)*x - sqrt(c*x^2 + b*x))^11*B*b^6*d
*e^11 + 360*(sqrt(c)*x - sqrt(c*x^2 + b*x))^11*A*b^5*c*d*e^11 - 105*(sqrt(c)*x - sqrt(c*x^2 + b*x))^11*A*b^6*e
^12 + 15360*(sqrt(c)*x - sqrt(c*x^2 + b*x))^10*B*c^(13/2)*d^8*e^4 - 61440*(sqrt(c)*x - sqrt(c*x^2 + b*x))^10*B
*b*c^(11/2)*d^7*e^5 + 92160*(sqrt(c)*x - sqrt(c*x^2 + b*x))^10*B*b^2*c^(9/2)*d^6*e^6 - 61440*(sqrt(c)*x - sqrt
(c*x^2 + b*x))^10*B*b^3*c^(7/2)*d^5*e^7 + 15360*(sqrt(c)*x - sqrt(c*x^2 + b*x))^10*B*b^4*c^(5/2)*d^4*e^8 + 198
0*(sqrt(c)*x - sqrt(c*x^2 + b*x))^10*B*b^5*c^(3/2)*d^3*e^9 - 3960*(sqrt(c)*x - sqrt(c*x^2 + b*x))^10*A*b^4*c^(
5/2)*d^3*e^9 - 825*(sqrt(c)*x - sqrt(c*x^2 + b*x))^10*B*b^6*sqrt(c)*d^2*e^10 + 3960*(sqrt(c)*x - sqrt(c*x^2 +
b*x))^10*A*b^5*c^(3/2)*d^2*e^10 - 1155*(sqrt(c)*x - sqrt(c*x^2 + b*x))^10*A*b^6*sqrt(c)*d*e^11 + 40960*(sqrt(c
)*x - sqrt(c*x^2 + b*x))^9*B*c^7*d^9*e^3 - 117760*(sqrt(c)*x - sqrt(c*x^2 + b*x))^9*B*b*c^6*d^8*e^4 + 20480*(s
qrt(c)*x - sqrt(c*x^2 + b*x))^9*A*c^7*d^8*e^4 + 61440*(sqrt(c)*x - sqrt(c*x^2 + b*x))^9*B*b^2*c^5*d^7*e^5 - 81
920*(sqrt(c)*x - sqrt(c*x^2 + b*x))^9*A*b*c^6*d^7*e^5 + 112640*(sqrt(c)*x - sqrt(c*x^2 + b*x))^9*B*b^3*c^4*d^6
*e^6 + 122880*(sqrt(c)*x - sqrt(c*x^2 + b*x))^9*A*b^2*c^5*d^6*e^6 - 143360*(sqrt(c)*x - sqrt(c*x^2 + b*x))^9*B
*b^4*c^3*d^5*e^7 - 81920*(sqrt(c)*x - sqrt(c*x^2 + b*x))^9*A*b^3*c^4*d^5*e^7 + 54960*(sqrt(c)*x - sqrt(c*x^2 +
b*x))^9*B*b^5*c^2*d^4*e^8 + 2720*(sqrt(c)*x - sqrt(c*x^2 + b*x))^9*A*b^4*c^3*d^4*e^8 - 2680*(sqrt(c)*x - sqrt
(c*x^2 + b*x))^9*B*b^6*c*d^3*e^9 + 15720*(sqrt(c)*x - sqrt(c*x^2 + b*x))^9*A*b^5*c^2*d^3*e^9 - 425*(sqrt(c)*x
- sqrt(c*x^2 + b*x))^9*B*b^7*d^2*e^10 - 3140*(sqrt(c)*x - sqrt(c*x^2 + b*x))^9*A*b^6*c*d^2*e^10 - 595*(sqrt(c)
*x - sqrt(c*x^2 + b*x))^9*A*b^7*d*e^11 + 61440*(sqrt(c)*x - sqrt(c*x^2 + b*x))^8*B*c^(15/2)*d^10*e^2 - 138240*
(sqrt(c)*x - sqrt(c*x^2 + b*x))^8*B*b*c^(13/2)*d^9*e^3 + 30720*(sqrt(c)*x - sqrt(c*x^2 + b*x))^8*A*c^(15/2)*d^
9*e^3 - 46080*(sqrt(c)*x - sqrt(c*x^2 + b*x))^8*A*b*c^(13/2)*d^8*e^4 + 153600*(sqrt(c)*x - sqrt(c*x^2 + b*x))^
8*B*b^3*c^(9/2)*d^7*e^5 - 122880*(sqrt(c)*x - sqrt(c*x^2 + b*x))^8*A*b^2*c^(11/2)*d^7*e^5 + 337920*(sqrt(c)*x
- sqrt(c*x^2 + b*x))^8*A*b^3*c^(9/2)*d^6*e^6 - 117720*(sqrt(c)*x - sqrt(c*x^2 + b*x))^8*B*b^5*c^(5/2)*d^5*e^7
- 317520*(sqrt(c)*x - sqrt(c*x^2 + b*x))^8*A*b^4*c^(7/2)*d^5*e^7 + 62070*(sqrt(c)*x - sqrt(c*x^2 + b*x))^8*B*b
^6*c^(3/2)*d^4*e^8 + 99480*(sqrt(c)*x - sqrt(c*x^2 + b*x))^8*A*b^5*c^(5/2)*d^4*e^8 - 3825*(sqrt(c)*x - sqrt(c*
x^2 + b*x))^8*B*b^7*sqrt(c)*d^3*e^9 + 6390*(sqrt(c)*x - sqrt(c*x^2 + b*x))^8*A*b^6*c^(3/2)*d^3*e^9 - 5355*(sqr
t(c)*x - sqrt(c*x^2 + b*x))^8*A*b^7*sqrt(c)*d^2*e^10 + 49152*(sqrt(c)*x - sqrt(c*x^2 + b*x))^7*B*c^8*d^11*e -
36864*(sqrt(c)*x - sqrt(c*x^2 + b*x))^7*B*b*c^7*d^10*e^2 + 24576*(sqrt(c)*x - sqrt(c*x^2 + b*x))^7*A*c^8*d^10*
e^2 - 211968*(sqrt(c)*x - sqrt(c*x^2 + b*x))^7*B*b^2*c^6*d^9*e^3 + 276480*(sqrt(c)*x - sqrt(c*x^2 + b*x))^7*B*
b^3*c^5*d^8*e^4 - 119808*(sqrt(c)*x - sqrt(c*x^2 + b*x))^7*A*b^2*c^6*d^8*e^4 + 30720*(sqrt(c)*x - sqrt(c*x^2 +
b*x))^7*B*b^4*c^4*d^7*e^5 - 12288*(sqrt(c)*x - sqrt(c*x^2 + b*x))^7*A*b^3*c^5*d^7*e^5 - 85536*(sqrt(c)*x - sq
rt(c*x^2 + b*x))^7*B*b^5*c^3*d^6*e^6 + 336960*(sqrt(c)*x - sqrt(c*x^2 + b*x))^7*A*b^4*c^4*d^6*e^6 - 18408*(sqr
t(c)*x - sqrt(c*x^2 + b*x))^7*B*b^6*c^2*d^5*e^7 - 419328*(sqrt(c)*x - sqrt(c*x^2 + b*x))^7*A*b^5*c^3*d^5*e^7 +
32064*(sqrt(c)*x - sqrt(c*x^2 + b*x))^7*B*b^7*c*d^4*e^8 + 170520*(sqrt(c)*x - sqrt(c*x^2 + b*x))^7*A*b^6*c^2*
d^4*e^8 - 990*(sqrt(c)*x - sqrt(c*x^2 + b*x))^7*B*b^8*d^3*e^9 - 13896*(sqrt(c)*x - sqrt(c*x^2 + b*x))^7*A*b^7*
c*d^3*e^9 - 1386*(sqrt(c)*x - sqrt(c*x^2 + b*x))^7*A*b^8*d^2*e^10 + 16384*(sqrt(c)*x - sqrt(c*x^2 + b*x))^6*B*
c^(17/2)*d^12 + 69632*(sqrt(c)*x - sqrt(c*x^2 + b*x))^6*B*b*c^(15/2)*d^11*e + 8192*(sqrt(c)*x - sqrt(c*x^2 + b
*x))^6*A*c^(17/2)*d^11*e - 254976*(sqrt(c)*x - sqrt(c*x^2 + b*x))^6*B*b^2*c^(13/2)*d^10*e^2 + 40960*(sqrt(c)*x
- sqrt(c*x^2 + b*x))^6*A*b*c^(15/2)*d^10*e^2 + 92160*(sqrt(c)*x - sqrt(c*x^2 + b*x))^6*B*b^3*c^(11/2)*d^9*e^3
- 101376*(sqrt(c)*x - sqrt(c*x^2 + b*x))^6*A*b^2*c^(13/2)*d^9*e^3 + 230400*(sqrt(c)*x - sqrt(c*x^2 + b*x))^6*
B*b^4*c^(9/2)*d^8*e^4 - 55296*(sqrt(c)*x - sqrt(c*x^2 + b*x))^6*A*b^3*c^(11/2)*d^8*e^4 - 84192*(sqrt(c)*x - sq
rt(c*x^2 + b*x))^6*B*b^5*c^(7/2)*d^7*e^5 + 100800*(sqrt(c)*x - sqrt(c*x^2 + b*x))^6*A*b^4*c^(9/2)*d^7*e^5 - 58
936*(sqrt(c)*x - sqrt(c*x^2 + b*x))^6*B*b^6*c^(5/2)*d^6*e^6 + 95424*(sqrt(c)*x - sqrt(c*x^2 + b*x))^6*A*b^5*c^
(7/2)*d^6*e^6 + 29608*(sqrt(c)*x - sqrt(c*x^2 + b*x))^6*B*b^7*c^(3/2)*d^5*e^7 - 242840*(sqrt(c)*x - sqrt(c*x^2
+ b*x))^6*A*b^6*c^(5/2)*d^5*e^7 + 8430*(sqrt(c)*x - sqrt(c*x^2 + b*x))^6*B*b^8*sqrt(c)*d^4*e^8 + 115328*(sqrt
(c)*x - sqrt(c*x^2 + b*x))^6*A*b^7*c^(3/2)*d^4*e^8 - 9702*(sqrt(c)*x - sqrt(c*x^2 + b*x))^6*A*b^8*sqrt(c)*d^3*
e^9 + 49152*(sqrt(c)*x - sqrt(c*x^2 + b*x))^5*B*b*c^8*d^12 - 36864*(sqrt(c)*x - sqrt(c*x^2 + b*x))^5*B*b^2*c^7
*d^11*e + 24576*(sqrt(c)*x - sqrt(c*x^2 + b*x))^5*A*b*c^8*d^11*e - 211968*(sqrt(c)*x - sqrt(c*x^2 + b*x))^5*B*
b^3*c^6*d^10*e^2 + 276480*(sqrt(c)*x - sqrt(c*x^2 + b*x))^5*B*b^4*c^5*d^9*e^3 - 119808*(sqrt(c)*x - sqrt(c*x^2
+ b*x))^5*A*b^3*c^6*d^9*e^3 + 34560*(sqrt(c)*x - sqrt(c*x^2 + b*x))^5*B*b^5*c^4*d^8*e^4 + 41472*(sqrt(c)*x -
sqrt(c*x^2 + b*x))^5*A*b^4*c^5*d^8*e^4 - 82656*(sqrt(c)*x - sqrt(c*x^2 + b*x))^5*B*b^6*c^3*d^7*e^5 + 87360*(sq
rt(c)*x - sqrt(c*x^2 + b*x))^5*A*b^5*c^4*d^7*e^5 - 1128*(sqrt(c)*x - sqrt(c*x^2 + b*x))^5*B*b^7*c^2*d^6*e^6 -
71808*(sqrt(c)*x - sqrt(c*x^2 + b*x))^5*A*b^6*c^3*d^6*e^6 + 19944*(sqrt(c)*x - sqrt(c*x^2 + b*x))^5*B*b^8*c*d^
5*e^7 - 47400*(sqrt(c)*x - sqrt(c*x^2 + b*x))^5*A*b^7*c^2*d^5*e^7 + 990*(sqrt(c)*x - sqrt(c*x^2 + b*x))^5*B*b^
9*d^4*e^8 + 38784*(sqrt(c)*x - sqrt(c*x^2 + b*x))^5*A*b^8*c*d^4*e^8 - 1686*(sqrt(c)*x - sqrt(c*x^2 + b*x))^5*A
*b^9*d^3*e^9 + 61440*(sqrt(c)*x - sqrt(c*x^2 + b*x))^4*B*b^2*c^(15/2)*d^12 - 138240*(sqrt(c)*x - sqrt(c*x^2 +
b*x))^4*B*b^3*c^(13/2)*d^11*e + 30720*(sqrt(c)*x - sqrt(c*x^2 + b*x))^4*A*b^2*c^(15/2)*d^11*e - 46080*(sqrt(c)
*x - sqrt(c*x^2 + b*x))^4*A*b^3*c^(13/2)*d^10*e^2 + 164160*(sqrt(c)*x - sqrt(c*x^2 + b*x))^4*B*b^5*c^(9/2)*d^9
*e^3 - 51840*(sqrt(c)*x - sqrt(c*x^2 + b*x))^4*A*b^4*c^(11/2)*d^9*e^3 - 38160*(sqrt(c)*x - sqrt(c*x^2 + b*x))^
4*B*b^6*c^(7/2)*d^8*e^4 + 60480*(sqrt(c)*x - sqrt(c*x^2 + b*x))^4*A*b^5*c^(9/2)*d^8*e^4 - 34920*(sqrt(c)*x - s
qrt(c*x^2 + b*x))^4*B*b^7*c^(5/2)*d^7*e^5 + 15120*(sqrt(c)*x - sqrt(c*x^2 + b*x))^4*A*b^6*c^(7/2)*d^7*e^5 + 15
420*(sqrt(c)*x - sqrt(c*x^2 + b*x))^4*B*b^8*c^(3/2)*d^6*e^6 - 64440*(sqrt(c)*x - sqrt(c*x^2 + b*x))^4*A*b^7*c^
(5/2)*d^6*e^6 + 4950*(sqrt(c)*x - sqrt(c*x^2 + b*x))^4*B*b^9*sqrt(c)*d^5*e^7 + 14460*(sqrt(c)*x - sqrt(c*x^2 +
b*x))^4*A*b^8*c^(3/2)*d^5*e^7 + 6930*(sqrt(c)*x - sqrt(c*x^2 + b*x))^4*A*b^9*sqrt(c)*d^4*e^8 + 40960*(sqrt(c)
*x - sqrt(c*x^2 + b*x))^3*B*b^3*c^7*d^12 - 117760*(sqrt(c)*x - sqrt(c*x^2 + b*x))^3*B*b^4*c^6*d^11*e + 20480*(
sqrt(c)*x - sqrt(c*x^2 + b*x))^3*A*b^3*c^7*d^11*e + 80640*(sqrt(c)*x - sqrt(c*x^2 + b*x))^3*B*b^5*c^5*d^10*e^2
- 43520*(sqrt(c)*x - sqrt(c*x^2 + b*x))^3*A*b^4*c^6*d^10*e^2 + 36160*(sqrt(c)*x - sqrt(c*x^2 + b*x))^3*B*b^6*
c^4*d^9*e^3 - 3840*(sqrt(c)*x - sqrt(c*x^2 + b*x))^3*A*b^5*c^5*d^9*e^3 - 25600*(sqrt(c)*x - sqrt(c*x^2 + b*x))
^3*B*b^7*c^3*d^8*e^4 + 33920*(sqrt(c)*x - sqrt(c*x^2 + b*x))^3*A*b^6*c^4*d^8*e^4 - 4680*(sqrt(c)*x - sqrt(c*x^
2 + b*x))^3*B*b^8*c^2*d^7*e^5 - 18080*(sqrt(c)*x - sqrt(c*x^2 + b*x))^3*A*b^7*c^3*d^7*e^5 + 7180*(sqrt(c)*x -
sqrt(c*x^2 + b*x))^3*B*b^9*c*d^6*e^6 - 16320*(sqrt(c)*x - sqrt(c*x^2 + b*x))^3*A*b^8*c^2*d^6*e^6 + 425*(sqrt(c
)*x - sqrt(c*x^2 + b*x))^3*B*b^10*d^5*e^7 + 9440*(sqrt(c)*x - sqrt(c*x^2 + b*x))^3*A*b^9*c*d^5*e^7 + 595*(sqrt
(c)*x - sqrt(c*x^2 + b*x))^3*A*b^10*d^4*e^8 + 15360*(sqrt(c)*x - sqrt(c*x^2 + b*x))^2*B*b^4*c^(13/2)*d^12 - 48
384*(sqrt(c)*x - sqrt(c*x^2 + b*x))^2*B*b^5*c^(11/2)*d^11*e + 7680*(sqrt(c)*x - sqrt(c*x^2 + b*x))^2*A*b^4*c^(
13/2)*d^11*e + 43968*(sqrt(c)*x - sqrt(c*x^2 + b*x))^2*B*b^6*c^(9/2)*d^10*e^2 - 18432*(sqrt(c)*x - sqrt(c*x^2
+ b*x))^2*A*b^5*c^(11/2)*d^10*e^2 + 96*(sqrt(c)*x - sqrt(c*x^2 + b*x))^2*B*b^7*c^(7/2)*d^9*e^3 + 3840*(sqrt(c)
*x - sqrt(c*x^2 + b*x))^2*A*b^6*c^(9/2)*d^9*e^3 - 8280*(sqrt(c)*x - sqrt(c*x^2 + b*x))^2*B*b^8*c^(5/2)*d^8*e^4
+ 11136*(sqrt(c)*x - sqrt(c*x^2 + b*x))^2*A*b^7*c^(7/2)*d^8*e^4 + 1740*(sqrt(c)*x - sqrt(c*x^2 + b*x))^2*B*b^
9*c^(3/2)*d^7*e^5 - 12384*(sqrt(c)*x - sqrt(c*x^2 + b*x))^2*A*b^8*c^(5/2)*d^7*e^5 + 1275*(sqrt(c)*x - sqrt(c*x
^2 + b*x))^2*B*b^10*sqrt(c)*d^6*e^6 + 600*(sqrt(c)*x - sqrt(c*x^2 + b*x))^2*A*b^9*c^(3/2)*d^6*e^6 + 1785*(sqrt
(c)*x - sqrt(c*x^2 + b*x))^2*A*b^10*sqrt(c)*d^5*e^7 + 3072*(sqrt(c)*x - sqrt(c*x^2 + b*x))*B*b^5*c^6*d^12 - 99
84*(sqrt(c)*x - sqrt(c*x^2 + b*x))*B*b^6*c^5*d^11*e + 1536*(sqrt(c)*x - sqrt(c*x^2 + b*x))*A*b^5*c^6*d^11*e +
9792*(sqrt(c)*x - sqrt(c*x^2 + b*x))*B*b^7*c^4*d^10*e^2 - 3840*(sqrt(c)*x - sqrt(c*x^2 + b*x))*A*b^6*c^5*d^10*
e^2 - 960*(sqrt(c)*x - sqrt(c*x^2 + b*x))*B*b^8*c^3*d^9*e^3 + 1152*(sqrt(c)*x - sqrt(c*x^2 + b*x))*A*b^7*c^4*d
^9*e^3 - 1560*(sqrt(c)*x - sqrt(c*x^2 + b*x))*B*b^9*c^2*d^8*e^4 + 2112*(sqrt(c)*x - sqrt(c*x^2 + b*x))*A*b^8*c
^3*d^8*e^4 + 720*(sqrt(c)*x - sqrt(c*x^2 + b*x))*B*b^10*c*d^7*e^5 - 3120*(sqrt(c)*x - sqrt(c*x^2 + b*x))*A*b^9
*c^2*d^7*e^5 + 75*(sqrt(c)*x - sqrt(c*x^2 + b*x))*B*b^11*d^6*e^6 + 900*(sqrt(c)*x - sqrt(c*x^2 + b*x))*A*b^10*
c*d^6*e^6 + 105*(sqrt(c)*x - sqrt(c*x^2 + b*x))*A*b^11*d^5*e^7 + 256*B*b^6*c^(11/2)*d^12 - 832*B*b^7*c^(9/2)*d
^11*e + 128*A*b^6*c^(11/2)*d^11*e + 816*B*b^8*c^(7/2)*d^10*e^2 - 320*A*b^7*c^(9/2)*d^10*e^2 - 80*B*b^9*c^(5/2)
*d^9*e^3 + 96*A*b^8*c^(7/2)*d^9*e^3 - 130*B*b^10*c^(3/2)*d^8*e^4 + 176*A*b^9*c^(5/2)*d^8*e^4 + 75*B*b^11*sqrt(
c)*d^7*e^5 - 290*A*b^10*c^(3/2)*d^7*e^5 + 105*A*b^11*sqrt(c)*d^6*e^6)/((c^4*d^8*e^5 - 4*b*c^3*d^7*e^6 + 6*b^2*
c^2*d^6*e^7 - 4*b^3*c*d^5*e^8 + b^4*d^4*e^9)*((sqrt(c)*x - sqrt(c*x^2 + b*x))^2*e + 2*(sqrt(c)*x - sqrt(c*x^2
+ b*x))*sqrt(c)*d + b*d)^6)

Mupad [F(-1)]

Timed out. \[ \int \frac {(A+B x) \left (b x+c x^2\right )^{3/2}}{(d+e x)^7} \, dx=\int \frac {{\left (c\,x^2+b\,x\right )}^{3/2}\,\left (A+B\,x\right )}{{\left (d+e\,x\right )}^7} \,d x \]

[In]

int(((b*x + c*x^2)^(3/2)*(A + B*x))/(d + e*x)^7,x)

[Out]

int(((b*x + c*x^2)^(3/2)*(A + B*x))/(d + e*x)^7, x)