3.1226 \(\int \frac {(A+B x) (b x+c x^2)^2}{(d+e x)^{3/2}} \, dx\)

Optimal. Leaf size=263 \[ -\frac {2 (d+e x)^{5/2} \left (2 A c e (2 c d-b e)-B \left (b^2 e^2-8 b c d e+10 c^2 d^2\right )\right )}{5 e^6}+\frac {2 (d+e x)^{3/2} \left (A e \left (b^2 e^2-6 b c d e+6 c^2 d^2\right )-B d \left (3 b^2 e^2-12 b c d e+10 c^2 d^2\right )\right )}{3 e^6}+\frac {2 d^2 (B d-A e) (c d-b e)^2}{e^6 \sqrt {d+e x}}-\frac {2 c (d+e x)^{7/2} (-A c e-2 b B e+5 B c d)}{7 e^6}+\frac {2 d \sqrt {d+e x} (c d-b e) (B d (5 c d-3 b e)-2 A e (2 c d-b e))}{e^6}+\frac {2 B c^2 (d+e x)^{9/2}}{9 e^6} \]

[Out]

2/3*(A*e*(b^2*e^2-6*b*c*d*e+6*c^2*d^2)-B*d*(3*b^2*e^2-12*b*c*d*e+10*c^2*d^2))*(e*x+d)^(3/2)/e^6-2/5*(2*A*c*e*(
-b*e+2*c*d)-B*(b^2*e^2-8*b*c*d*e+10*c^2*d^2))*(e*x+d)^(5/2)/e^6-2/7*c*(-A*c*e-2*B*b*e+5*B*c*d)*(e*x+d)^(7/2)/e
^6+2/9*B*c^2*(e*x+d)^(9/2)/e^6+2*d^2*(-A*e+B*d)*(-b*e+c*d)^2/e^6/(e*x+d)^(1/2)+2*d*(-b*e+c*d)*(B*d*(-3*b*e+5*c
*d)-2*A*e*(-b*e+2*c*d))*(e*x+d)^(1/2)/e^6

________________________________________________________________________________________

Rubi [A]  time = 0.16, antiderivative size = 263, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.038, Rules used = {771} \[ -\frac {2 (d+e x)^{5/2} \left (2 A c e (2 c d-b e)-B \left (b^2 e^2-8 b c d e+10 c^2 d^2\right )\right )}{5 e^6}+\frac {2 (d+e x)^{3/2} \left (A e \left (b^2 e^2-6 b c d e+6 c^2 d^2\right )-B d \left (3 b^2 e^2-12 b c d e+10 c^2 d^2\right )\right )}{3 e^6}+\frac {2 d^2 (B d-A e) (c d-b e)^2}{e^6 \sqrt {d+e x}}-\frac {2 c (d+e x)^{7/2} (-A c e-2 b B e+5 B c d)}{7 e^6}+\frac {2 d \sqrt {d+e x} (c d-b e) (B d (5 c d-3 b e)-2 A e (2 c d-b e))}{e^6}+\frac {2 B c^2 (d+e x)^{9/2}}{9 e^6} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*d^2*(B*d - A*e)*(c*d - b*e)^2)/(e^6*Sqrt[d + e*x]) + (2*d*(c*d - b*e)*(B*d*(5*c*d - 3*b*e) - 2*A*e*(2*c*d -
b*e))*Sqrt[d + e*x])/e^6 + (2*(A*e*(6*c^2*d^2 - 6*b*c*d*e + b^2*e^2) - B*d*(10*c^2*d^2 - 12*b*c*d*e + 3*b^2*e
^2))*(d + e*x)^(3/2))/(3*e^6) - (2*(2*A*c*e*(2*c*d - b*e) - B*(10*c^2*d^2 - 8*b*c*d*e + b^2*e^2))*(d + e*x)^(5
/2))/(5*e^6) - (2*c*(5*B*c*d - 2*b*B*e - A*c*e)*(d + e*x)^(7/2))/(7*e^6) + (2*B*c^2*(d + e*x)^(9/2))/(9*e^6)

Rule 771

Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> In
t[ExpandIntegrand[(d + e*x)^m*(f + g*x)*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && N
eQ[b^2 - 4*a*c, 0] && IntegerQ[p] && (GtQ[p, 0] || (EqQ[a, 0] && IntegerQ[m]))

Rubi steps

\begin {align*} \int \frac {(A+B x) \left (b x+c x^2\right )^2}{(d+e x)^{3/2}} \, dx &=\int \left (-\frac {d^2 (B d-A e) (c d-b e)^2}{e^5 (d+e x)^{3/2}}+\frac {d (c d-b e) (B d (5 c d-3 b e)-2 A e (2 c d-b e))}{e^5 \sqrt {d+e x}}+\frac {\left (A e \left (6 c^2 d^2-6 b c d e+b^2 e^2\right )-B d \left (10 c^2 d^2-12 b c d e+3 b^2 e^2\right )\right ) \sqrt {d+e x}}{e^5}+\frac {\left (-2 A c e (2 c d-b e)+B \left (10 c^2 d^2-8 b c d e+b^2 e^2\right )\right ) (d+e x)^{3/2}}{e^5}+\frac {c (-5 B c d+2 b B e+A c e) (d+e x)^{5/2}}{e^5}+\frac {B c^2 (d+e x)^{7/2}}{e^5}\right ) \, dx\\ &=\frac {2 d^2 (B d-A e) (c d-b e)^2}{e^6 \sqrt {d+e x}}+\frac {2 d (c d-b e) (B d (5 c d-3 b e)-2 A e (2 c d-b e)) \sqrt {d+e x}}{e^6}+\frac {2 \left (A e \left (6 c^2 d^2-6 b c d e+b^2 e^2\right )-B d \left (10 c^2 d^2-12 b c d e+3 b^2 e^2\right )\right ) (d+e x)^{3/2}}{3 e^6}-\frac {2 \left (2 A c e (2 c d-b e)-B \left (10 c^2 d^2-8 b c d e+b^2 e^2\right )\right ) (d+e x)^{5/2}}{5 e^6}-\frac {2 c (5 B c d-2 b B e-A c e) (d+e x)^{7/2}}{7 e^6}+\frac {2 B c^2 (d+e x)^{9/2}}{9 e^6}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.17, size = 273, normalized size = 1.04 \[ \frac {2 B \left (63 b^2 e^2 \left (16 d^3+8 d^2 e x-2 d e^2 x^2+e^3 x^3\right )+18 b c e \left (-128 d^4-64 d^3 e x+16 d^2 e^2 x^2-8 d e^3 x^3+5 e^4 x^4\right )+5 c^2 \left (256 d^5+128 d^4 e x-32 d^3 e^2 x^2+16 d^2 e^3 x^3-10 d e^4 x^4+7 e^5 x^5\right )\right )-6 A e \left (35 b^2 e^2 \left (8 d^2+4 d e x-e^2 x^2\right )-42 b c e \left (16 d^3+8 d^2 e x-2 d e^2 x^2+e^3 x^3\right )+3 c^2 \left (128 d^4+64 d^3 e x-16 d^2 e^2 x^2+8 d e^3 x^3-5 e^4 x^4\right )\right )}{315 e^6 \sqrt {d+e x}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-6*A*e*(35*b^2*e^2*(8*d^2 + 4*d*e*x - e^2*x^2) - 42*b*c*e*(16*d^3 + 8*d^2*e*x - 2*d*e^2*x^2 + e^3*x^3) + 3*c^
2*(128*d^4 + 64*d^3*e*x - 16*d^2*e^2*x^2 + 8*d*e^3*x^3 - 5*e^4*x^4)) + 2*B*(63*b^2*e^2*(16*d^3 + 8*d^2*e*x - 2
*d*e^2*x^2 + e^3*x^3) + 18*b*c*e*(-128*d^4 - 64*d^3*e*x + 16*d^2*e^2*x^2 - 8*d*e^3*x^3 + 5*e^4*x^4) + 5*c^2*(2
56*d^5 + 128*d^4*e*x - 32*d^3*e^2*x^2 + 16*d^2*e^3*x^3 - 10*d*e^4*x^4 + 7*e^5*x^5)))/(315*e^6*Sqrt[d + e*x])

________________________________________________________________________________________

fricas [A]  time = 0.61, size = 299, normalized size = 1.14 \[ \frac {2 \, {\left (35 \, B c^{2} e^{5} x^{5} + 1280 \, B c^{2} d^{5} - 840 \, A b^{2} d^{2} e^{3} - 1152 \, {\left (2 \, B b c + A c^{2}\right )} d^{4} e + 1008 \, {\left (B b^{2} + 2 \, A b c\right )} d^{3} e^{2} - 5 \, {\left (10 \, B c^{2} d e^{4} - 9 \, {\left (2 \, B b c + A c^{2}\right )} e^{5}\right )} x^{4} + {\left (80 \, B c^{2} d^{2} e^{3} - 72 \, {\left (2 \, B b c + A c^{2}\right )} d e^{4} + 63 \, {\left (B b^{2} + 2 \, A b c\right )} e^{5}\right )} x^{3} - {\left (160 \, B c^{2} d^{3} e^{2} - 105 \, A b^{2} e^{5} - 144 \, {\left (2 \, B b c + A c^{2}\right )} d^{2} e^{3} + 126 \, {\left (B b^{2} + 2 \, A b c\right )} d e^{4}\right )} x^{2} + 4 \, {\left (160 \, B c^{2} d^{4} e - 105 \, A b^{2} d e^{4} - 144 \, {\left (2 \, B b c + A c^{2}\right )} d^{3} e^{2} + 126 \, {\left (B b^{2} + 2 \, A b c\right )} d^{2} e^{3}\right )} x\right )} \sqrt {e x + d}}{315 \, {\left (e^{7} x + d e^{6}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/315*(35*B*c^2*e^5*x^5 + 1280*B*c^2*d^5 - 840*A*b^2*d^2*e^3 - 1152*(2*B*b*c + A*c^2)*d^4*e + 1008*(B*b^2 + 2*
A*b*c)*d^3*e^2 - 5*(10*B*c^2*d*e^4 - 9*(2*B*b*c + A*c^2)*e^5)*x^4 + (80*B*c^2*d^2*e^3 - 72*(2*B*b*c + A*c^2)*d
*e^4 + 63*(B*b^2 + 2*A*b*c)*e^5)*x^3 - (160*B*c^2*d^3*e^2 - 105*A*b^2*e^5 - 144*(2*B*b*c + A*c^2)*d^2*e^3 + 12
6*(B*b^2 + 2*A*b*c)*d*e^4)*x^2 + 4*(160*B*c^2*d^4*e - 105*A*b^2*d*e^4 - 144*(2*B*b*c + A*c^2)*d^3*e^2 + 126*(B
*b^2 + 2*A*b*c)*d^2*e^3)*x)*sqrt(e*x + d)/(e^7*x + d*e^6)

________________________________________________________________________________________

giac [A]  time = 0.56, size = 441, normalized size = 1.68 \[ \frac {2}{315} \, {\left (35 \, {\left (x e + d\right )}^{\frac {9}{2}} B c^{2} e^{48} - 225 \, {\left (x e + d\right )}^{\frac {7}{2}} B c^{2} d e^{48} + 630 \, {\left (x e + d\right )}^{\frac {5}{2}} B c^{2} d^{2} e^{48} - 1050 \, {\left (x e + d\right )}^{\frac {3}{2}} B c^{2} d^{3} e^{48} + 1575 \, \sqrt {x e + d} B c^{2} d^{4} e^{48} + 90 \, {\left (x e + d\right )}^{\frac {7}{2}} B b c e^{49} + 45 \, {\left (x e + d\right )}^{\frac {7}{2}} A c^{2} e^{49} - 504 \, {\left (x e + d\right )}^{\frac {5}{2}} B b c d e^{49} - 252 \, {\left (x e + d\right )}^{\frac {5}{2}} A c^{2} d e^{49} + 1260 \, {\left (x e + d\right )}^{\frac {3}{2}} B b c d^{2} e^{49} + 630 \, {\left (x e + d\right )}^{\frac {3}{2}} A c^{2} d^{2} e^{49} - 2520 \, \sqrt {x e + d} B b c d^{3} e^{49} - 1260 \, \sqrt {x e + d} A c^{2} d^{3} e^{49} + 63 \, {\left (x e + d\right )}^{\frac {5}{2}} B b^{2} e^{50} + 126 \, {\left (x e + d\right )}^{\frac {5}{2}} A b c e^{50} - 315 \, {\left (x e + d\right )}^{\frac {3}{2}} B b^{2} d e^{50} - 630 \, {\left (x e + d\right )}^{\frac {3}{2}} A b c d e^{50} + 945 \, \sqrt {x e + d} B b^{2} d^{2} e^{50} + 1890 \, \sqrt {x e + d} A b c d^{2} e^{50} + 105 \, {\left (x e + d\right )}^{\frac {3}{2}} A b^{2} e^{51} - 630 \, \sqrt {x e + d} A b^{2} d e^{51}\right )} e^{\left (-54\right )} + \frac {2 \, {\left (B c^{2} d^{5} - 2 \, B b c d^{4} e - A c^{2} d^{4} e + B b^{2} d^{3} e^{2} + 2 \, A b c d^{3} e^{2} - A b^{2} d^{2} e^{3}\right )} e^{\left (-6\right )}}{\sqrt {x e + d}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/315*(35*(x*e + d)^(9/2)*B*c^2*e^48 - 225*(x*e + d)^(7/2)*B*c^2*d*e^48 + 630*(x*e + d)^(5/2)*B*c^2*d^2*e^48 -
1050*(x*e + d)^(3/2)*B*c^2*d^3*e^48 + 1575*sqrt(x*e + d)*B*c^2*d^4*e^48 + 90*(x*e + d)^(7/2)*B*b*c*e^49 + 45*
(x*e + d)^(7/2)*A*c^2*e^49 - 504*(x*e + d)^(5/2)*B*b*c*d*e^49 - 252*(x*e + d)^(5/2)*A*c^2*d*e^49 + 1260*(x*e +
d)^(3/2)*B*b*c*d^2*e^49 + 630*(x*e + d)^(3/2)*A*c^2*d^2*e^49 - 2520*sqrt(x*e + d)*B*b*c*d^3*e^49 - 1260*sqrt(
x*e + d)*A*c^2*d^3*e^49 + 63*(x*e + d)^(5/2)*B*b^2*e^50 + 126*(x*e + d)^(5/2)*A*b*c*e^50 - 315*(x*e + d)^(3/2)
*B*b^2*d*e^50 - 630*(x*e + d)^(3/2)*A*b*c*d*e^50 + 945*sqrt(x*e + d)*B*b^2*d^2*e^50 + 1890*sqrt(x*e + d)*A*b*c
*d^2*e^50 + 105*(x*e + d)^(3/2)*A*b^2*e^51 - 630*sqrt(x*e + d)*A*b^2*d*e^51)*e^(-54) + 2*(B*c^2*d^5 - 2*B*b*c*
d^4*e - A*c^2*d^4*e + B*b^2*d^3*e^2 + 2*A*b*c*d^3*e^2 - A*b^2*d^2*e^3)*e^(-6)/sqrt(x*e + d)

________________________________________________________________________________________

maple [A]  time = 0.05, size = 341, normalized size = 1.30 \[ -\frac {2 \left (-35 B \,c^{2} x^{5} e^{5}-45 A \,c^{2} e^{5} x^{4}-90 B b c \,e^{5} x^{4}+50 B \,c^{2} d \,e^{4} x^{4}-126 A b c \,e^{5} x^{3}+72 A \,c^{2} d \,e^{4} x^{3}-63 B \,b^{2} e^{5} x^{3}+144 B b c d \,e^{4} x^{3}-80 B \,c^{2} d^{2} e^{3} x^{3}-105 A \,b^{2} e^{5} x^{2}+252 A b c d \,e^{4} x^{2}-144 A \,c^{2} d^{2} e^{3} x^{2}+126 B \,b^{2} d \,e^{4} x^{2}-288 B b c \,d^{2} e^{3} x^{2}+160 B \,c^{2} d^{3} e^{2} x^{2}+420 A \,b^{2} d \,e^{4} x -1008 A b c \,d^{2} e^{3} x +576 A \,c^{2} d^{3} e^{2} x -504 B \,b^{2} d^{2} e^{3} x +1152 B b c \,d^{3} e^{2} x -640 B \,c^{2} d^{4} e x +840 A \,b^{2} d^{2} e^{3}-2016 A b c \,d^{3} e^{2}+1152 A \,c^{2} d^{4} e -1008 B \,b^{2} d^{3} e^{2}+2304 B b c \,d^{4} e -1280 B \,c^{2} d^{5}\right )}{315 \sqrt {e x +d}\, e^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/315*(-35*B*c^2*e^5*x^5-45*A*c^2*e^5*x^4-90*B*b*c*e^5*x^4+50*B*c^2*d*e^4*x^4-126*A*b*c*e^5*x^3+72*A*c^2*d*e^
4*x^3-63*B*b^2*e^5*x^3+144*B*b*c*d*e^4*x^3-80*B*c^2*d^2*e^3*x^3-105*A*b^2*e^5*x^2+252*A*b*c*d*e^4*x^2-144*A*c^
2*d^2*e^3*x^2+126*B*b^2*d*e^4*x^2-288*B*b*c*d^2*e^3*x^2+160*B*c^2*d^3*e^2*x^2+420*A*b^2*d*e^4*x-1008*A*b*c*d^2
*e^3*x+576*A*c^2*d^3*e^2*x-504*B*b^2*d^2*e^3*x+1152*B*b*c*d^3*e^2*x-640*B*c^2*d^4*e*x+840*A*b^2*d^2*e^3-2016*A
*b*c*d^3*e^2+1152*A*c^2*d^4*e-1008*B*b^2*d^3*e^2+2304*B*b*c*d^4*e-1280*B*c^2*d^5)/(e*x+d)^(1/2)/e^6

________________________________________________________________________________________

maxima [A]  time = 0.49, size = 299, normalized size = 1.14 \[ \frac {2 \, {\left (\frac {35 \, {\left (e x + d\right )}^{\frac {9}{2}} B c^{2} - 45 \, {\left (5 \, B c^{2} d - {\left (2 \, B b c + A c^{2}\right )} e\right )} {\left (e x + d\right )}^{\frac {7}{2}} + 63 \, {\left (10 \, B c^{2} d^{2} - 4 \, {\left (2 \, B b c + A c^{2}\right )} d e + {\left (B b^{2} + 2 \, A b c\right )} e^{2}\right )} {\left (e x + d\right )}^{\frac {5}{2}} - 105 \, {\left (10 \, B c^{2} d^{3} - A b^{2} e^{3} - 6 \, {\left (2 \, B b c + A c^{2}\right )} d^{2} e + 3 \, {\left (B b^{2} + 2 \, A b c\right )} d e^{2}\right )} {\left (e x + d\right )}^{\frac {3}{2}} + 315 \, {\left (5 \, B c^{2} d^{4} - 2 \, A b^{2} d e^{3} - 4 \, {\left (2 \, B b c + A c^{2}\right )} d^{3} e + 3 \, {\left (B b^{2} + 2 \, A b c\right )} d^{2} e^{2}\right )} \sqrt {e x + d}}{e^{5}} + \frac {315 \, {\left (B c^{2} d^{5} - A b^{2} d^{2} e^{3} - {\left (2 \, B b c + A c^{2}\right )} d^{4} e + {\left (B b^{2} + 2 \, A b c\right )} d^{3} e^{2}\right )}}{\sqrt {e x + d} e^{5}}\right )}}{315 \, e} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/315*((35*(e*x + d)^(9/2)*B*c^2 - 45*(5*B*c^2*d - (2*B*b*c + A*c^2)*e)*(e*x + d)^(7/2) + 63*(10*B*c^2*d^2 - 4
*(2*B*b*c + A*c^2)*d*e + (B*b^2 + 2*A*b*c)*e^2)*(e*x + d)^(5/2) - 105*(10*B*c^2*d^3 - A*b^2*e^3 - 6*(2*B*b*c +
A*c^2)*d^2*e + 3*(B*b^2 + 2*A*b*c)*d*e^2)*(e*x + d)^(3/2) + 315*(5*B*c^2*d^4 - 2*A*b^2*d*e^3 - 4*(2*B*b*c + A
*c^2)*d^3*e + 3*(B*b^2 + 2*A*b*c)*d^2*e^2)*sqrt(e*x + d))/e^5 + 315*(B*c^2*d^5 - A*b^2*d^2*e^3 - (2*B*b*c + A*
c^2)*d^4*e + (B*b^2 + 2*A*b*c)*d^3*e^2)/(sqrt(e*x + d)*e^5))/e

________________________________________________________________________________________

mupad [B]  time = 1.53, size = 296, normalized size = 1.13 \[ \frac {{\left (d+e\,x\right )}^{7/2}\,\left (2\,A\,c^2\,e-10\,B\,c^2\,d+4\,B\,b\,c\,e\right )}{7\,e^6}+\frac {{\left (d+e\,x\right )}^{3/2}\,\left (-6\,B\,b^2\,d\,e^2+2\,A\,b^2\,e^3+24\,B\,b\,c\,d^2\,e-12\,A\,b\,c\,d\,e^2-20\,B\,c^2\,d^3+12\,A\,c^2\,d^2\,e\right )}{3\,e^6}+\frac {{\left (d+e\,x\right )}^{5/2}\,\left (2\,B\,b^2\,e^2-16\,B\,b\,c\,d\,e+4\,A\,b\,c\,e^2+20\,B\,c^2\,d^2-8\,A\,c^2\,d\,e\right )}{5\,e^6}+\frac {2\,B\,b^2\,d^3\,e^2-2\,A\,b^2\,d^2\,e^3-4\,B\,b\,c\,d^4\,e+4\,A\,b\,c\,d^3\,e^2+2\,B\,c^2\,d^5-2\,A\,c^2\,d^4\,e}{e^6\,\sqrt {d+e\,x}}+\frac {2\,B\,c^2\,{\left (d+e\,x\right )}^{9/2}}{9\,e^6}-\frac {2\,d\,\left (b\,e-c\,d\right )\,\sqrt {d+e\,x}\,\left (2\,A\,b\,e^2+5\,B\,c\,d^2-4\,A\,c\,d\,e-3\,B\,b\,d\,e\right )}{e^6} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((d + e*x)^(7/2)*(2*A*c^2*e - 10*B*c^2*d + 4*B*b*c*e))/(7*e^6) + ((d + e*x)^(3/2)*(2*A*b^2*e^3 - 20*B*c^2*d^3
+ 12*A*c^2*d^2*e - 6*B*b^2*d*e^2 - 12*A*b*c*d*e^2 + 24*B*b*c*d^2*e))/(3*e^6) + ((d + e*x)^(5/2)*(2*B*b^2*e^2 +
20*B*c^2*d^2 + 4*A*b*c*e^2 - 8*A*c^2*d*e - 16*B*b*c*d*e))/(5*e^6) + (2*B*c^2*d^5 - 2*A*c^2*d^4*e - 2*A*b^2*d^
2*e^3 + 2*B*b^2*d^3*e^2 - 4*B*b*c*d^4*e + 4*A*b*c*d^3*e^2)/(e^6*(d + e*x)^(1/2)) + (2*B*c^2*(d + e*x)^(9/2))/(
9*e^6) - (2*d*(b*e - c*d)*(d + e*x)^(1/2)*(2*A*b*e^2 + 5*B*c*d^2 - 4*A*c*d*e - 3*B*b*d*e))/e^6

________________________________________________________________________________________

sympy [A]  time = 62.03, size = 321, normalized size = 1.22 \[ \frac {2 B c^{2} \left (d + e x\right )^{\frac {9}{2}}}{9 e^{6}} + \frac {2 d^{2} \left (- A e + B d\right ) \left (b e - c d\right )^{2}}{e^{6} \sqrt {d + e x}} + \frac {\left (d + e x\right )^{\frac {7}{2}} \left (2 A c^{2} e + 4 B b c e - 10 B c^{2} d\right )}{7 e^{6}} + \frac {\left (d + e x\right )^{\frac {5}{2}} \left (4 A b c e^{2} - 8 A c^{2} d e + 2 B b^{2} e^{2} - 16 B b c d e + 20 B c^{2} d^{2}\right )}{5 e^{6}} + \frac {\left (d + e x\right )^{\frac {3}{2}} \left (2 A b^{2} e^{3} - 12 A b c d e^{2} + 12 A c^{2} d^{2} e - 6 B b^{2} d e^{2} + 24 B b c d^{2} e - 20 B c^{2} d^{3}\right )}{3 e^{6}} + \frac {\sqrt {d + e x} \left (- 4 A b^{2} d e^{3} + 12 A b c d^{2} e^{2} - 8 A c^{2} d^{3} e + 6 B b^{2} d^{2} e^{2} - 16 B b c d^{3} e + 10 B c^{2} d^{4}\right )}{e^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*B*c**2*(d + e*x)**(9/2)/(9*e**6) + 2*d**2*(-A*e + B*d)*(b*e - c*d)**2/(e**6*sqrt(d + e*x)) + (d + e*x)**(7/2
)*(2*A*c**2*e + 4*B*b*c*e - 10*B*c**2*d)/(7*e**6) + (d + e*x)**(5/2)*(4*A*b*c*e**2 - 8*A*c**2*d*e + 2*B*b**2*e
**2 - 16*B*b*c*d*e + 20*B*c**2*d**2)/(5*e**6) + (d + e*x)**(3/2)*(2*A*b**2*e**3 - 12*A*b*c*d*e**2 + 12*A*c**2*
d**2*e - 6*B*b**2*d*e**2 + 24*B*b*c*d**2*e - 20*B*c**2*d**3)/(3*e**6) + sqrt(d + e*x)*(-4*A*b**2*d*e**3 + 12*A
*b*c*d**2*e**2 - 8*A*c**2*d**3*e + 6*B*b**2*d**2*e**2 - 16*B*b*c*d**3*e + 10*B*c**2*d**4)/e**6

________________________________________________________________________________________