3.1228 \(\int \frac{(A+B x) (b x+c x^2)^2}{(d+e x)^{7/2}} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.156702, antiderivative size = 263, normalized size of antiderivative = 1., 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} \[ -\frac{2 \sqrt{d+e x} \left (2 A c e (2 c d-b e)-B \left (b^2 e^2-8 b c d e+10 c^2 d^2\right )\right )}{e^6}-\frac{2 \left (A e \left (b^2 e^2-6 b c d e+6 c^2 d^2\right )-B d \left (3 b^2 e^2-12 b c d e+10 c^2 d^2\right )\right )}{e^6 \sqrt{d+e x}}+\frac{2 d^2 (B d-A e) (c d-b e)^2}{5 e^6 (d+e x)^{5/2}}-\frac{2 c (d+e x)^{3/2} (-A c e-2 b B e+5 B c d)}{3 e^6}-\frac{2 d (c d-b e) (B d (5 c d-3 b e)-2 A e (2 c d-b e))}{3 e^6 (d+e x)^{3/2}}+\frac{2 B c^2 (d+e x)^{5/2}}{5 e^6} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Mathematica [A]  time = 0.16642, size = 272, normalized size = 1.03 \[ \frac{2 \left (A e \left (-b^2 e^2 \left (8 d^2+20 d e x+15 e^2 x^2\right )+6 b c e \left (40 d^2 e x+16 d^3+30 d e^2 x^2+5 e^3 x^3\right )+c^2 \left (-\left (240 d^2 e^2 x^2+320 d^3 e x+128 d^4+40 d e^3 x^3-5 e^4 x^4\right )\right )\right )+B \left (3 b^2 e^2 \left (40 d^2 e x+16 d^3+30 d e^2 x^2+5 e^3 x^3\right )-2 b c e \left (240 d^2 e^2 x^2+320 d^3 e x+128 d^4+40 d e^3 x^3-5 e^4 x^4\right )+c^2 \left (480 d^3 e^2 x^2+80 d^2 e^3 x^3+640 d^4 e x+256 d^5-10 d e^4 x^4+3 e^5 x^5\right )\right )\right )}{15 e^6 (d+e x)^{5/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(A*e*(-(b^2*e^2*(8*d^2 + 20*d*e*x + 15*e^2*x^2)) + 6*b*c*e*(16*d^3 + 40*d^2*e*x + 30*d*e^2*x^2 + 5*e^3*x^3)
- c^2*(128*d^4 + 320*d^3*e*x + 240*d^2*e^2*x^2 + 40*d*e^3*x^3 - 5*e^4*x^4)) + B*(3*b^2*e^2*(16*d^3 + 40*d^2*e
*x + 30*d*e^2*x^2 + 5*e^3*x^3) - 2*b*c*e*(128*d^4 + 320*d^3*e*x + 240*d^2*e^2*x^2 + 40*d*e^3*x^3 - 5*e^4*x^4)
+ c^2*(256*d^5 + 640*d^4*e*x + 480*d^3*e^2*x^2 + 80*d^2*e^3*x^3 - 10*d*e^4*x^4 + 3*e^5*x^5))))/(15*e^6*(d + e*
x)^(5/2))

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Maple [A]  time = 0.007, size = 341, normalized size = 1.3 \begin{align*} -{\frac{-6\,B{c}^{2}{x}^{5}{e}^{5}-10\,A{c}^{2}{e}^{5}{x}^{4}-20\,Bbc{e}^{5}{x}^{4}+20\,B{c}^{2}d{e}^{4}{x}^{4}-60\,Abc{e}^{5}{x}^{3}+80\,A{c}^{2}d{e}^{4}{x}^{3}-30\,B{b}^{2}{e}^{5}{x}^{3}+160\,Bbcd{e}^{4}{x}^{3}-160\,B{c}^{2}{d}^{2}{e}^{3}{x}^{3}+30\,A{b}^{2}{e}^{5}{x}^{2}-360\,Abcd{e}^{4}{x}^{2}+480\,A{c}^{2}{d}^{2}{e}^{3}{x}^{2}-180\,B{b}^{2}d{e}^{4}{x}^{2}+960\,Bbc{d}^{2}{e}^{3}{x}^{2}-960\,B{c}^{2}{d}^{3}{e}^{2}{x}^{2}+40\,A{b}^{2}d{e}^{4}x-480\,Abc{d}^{2}{e}^{3}x+640\,A{c}^{2}{d}^{3}{e}^{2}x-240\,B{b}^{2}{d}^{2}{e}^{3}x+1280\,Bbc{d}^{3}{e}^{2}x-1280\,B{c}^{2}{d}^{4}ex+16\,A{b}^{2}{d}^{2}{e}^{3}-192\,Abc{d}^{3}{e}^{2}+256\,A{c}^{2}{d}^{4}e-96\,B{b}^{2}{d}^{3}{e}^{2}+512\,Bbc{d}^{4}e-512\,B{c}^{2}{d}^{5}}{15\,{e}^{6}} \left ( ex+d \right ) ^{-{\frac{5}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/15*(-3*B*c^2*e^5*x^5-5*A*c^2*e^5*x^4-10*B*b*c*e^5*x^4+10*B*c^2*d*e^4*x^4-30*A*b*c*e^5*x^3+40*A*c^2*d*e^4*x^
3-15*B*b^2*e^5*x^3+80*B*b*c*d*e^4*x^3-80*B*c^2*d^2*e^3*x^3+15*A*b^2*e^5*x^2-180*A*b*c*d*e^4*x^2+240*A*c^2*d^2*
e^3*x^2-90*B*b^2*d*e^4*x^2+480*B*b*c*d^2*e^3*x^2-480*B*c^2*d^3*e^2*x^2+20*A*b^2*d*e^4*x-240*A*b*c*d^2*e^3*x+32
0*A*c^2*d^3*e^2*x-120*B*b^2*d^2*e^3*x+640*B*b*c*d^3*e^2*x-640*B*c^2*d^4*e*x+8*A*b^2*d^2*e^3-96*A*b*c*d^3*e^2+1
28*A*c^2*d^4*e-48*B*b^2*d^3*e^2+256*B*b*c*d^4*e-256*B*c^2*d^5)/(e*x+d)^(5/2)/e^6

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Maxima [A]  time = 1.07537, size = 402, normalized size = 1.53 \begin{align*} \frac{2 \,{\left (\frac{3 \,{\left (e x + d\right )}^{\frac{5}{2}} B c^{2} - 5 \,{\left (5 \, B c^{2} d -{\left (2 \, B b c + A c^{2}\right )} e\right )}{\left (e x + d\right )}^{\frac{3}{2}} + 15 \,{\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 )} \sqrt{e x + d}}{e^{5}} + \frac{3 \, B c^{2} d^{5} - 3 \, A b^{2} d^{2} e^{3} - 3 \,{\left (2 \, B b c + A c^{2}\right )} d^{4} e + 3 \,{\left (B b^{2} + 2 \, A b c\right )} d^{3} e^{2} + 15 \,{\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 )}^{2} - 5 \,{\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 )}{\left (e x + d\right )}}{{\left (e x + d\right )}^{\frac{5}{2}} e^{5}}\right )}}{15 \, e} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/15*((3*(e*x + d)^(5/2)*B*c^2 - 5*(5*B*c^2*d - (2*B*b*c + A*c^2)*e)*(e*x + d)^(3/2) + 15*(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)*sqrt(e*x + d))/e^5 + (3*B*c^2*d^5 - 3*A*b^2*d^2*e^3 - 3*(2*B*b*c
+ A*c^2)*d^4*e + 3*(B*b^2 + 2*A*b*c)*d^3*e^2 + 15*(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)^2 - 5*(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)*(e*x + d))/((e*x + d)^(5/2)*e^5))/e

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Fricas [A]  time = 1.75711, size = 686, normalized size = 2.61 \begin{align*} \frac{2 \,{\left (3 \, B c^{2} e^{5} x^{5} + 256 \, B c^{2} d^{5} - 8 \, A b^{2} d^{2} e^{3} - 128 \,{\left (2 \, B b c + A c^{2}\right )} d^{4} e + 48 \,{\left (B b^{2} + 2 \, A b c\right )} d^{3} e^{2} - 5 \,{\left (2 \, B c^{2} d e^{4} -{\left (2 \, B b c + A c^{2}\right )} e^{5}\right )} x^{4} + 5 \,{\left (16 \, B c^{2} d^{2} e^{3} - 8 \,{\left (2 \, B b c + A c^{2}\right )} d e^{4} + 3 \,{\left (B b^{2} + 2 \, A b c\right )} e^{5}\right )} x^{3} + 15 \,{\left (32 \, B c^{2} d^{3} e^{2} - A b^{2} e^{5} - 16 \,{\left (2 \, B b c + A c^{2}\right )} d^{2} e^{3} + 6 \,{\left (B b^{2} + 2 \, A b c\right )} d e^{4}\right )} x^{2} + 20 \,{\left (32 \, B c^{2} d^{4} e - A b^{2} d e^{4} - 16 \,{\left (2 \, B b c + A c^{2}\right )} d^{3} e^{2} + 6 \,{\left (B b^{2} + 2 \, A b c\right )} d^{2} e^{3}\right )} x\right )} \sqrt{e x + d}}{15 \,{\left (e^{9} x^{3} + 3 \, d e^{8} x^{2} + 3 \, d^{2} e^{7} x + d^{3} e^{6}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/15*(3*B*c^2*e^5*x^5 + 256*B*c^2*d^5 - 8*A*b^2*d^2*e^3 - 128*(2*B*b*c + A*c^2)*d^4*e + 48*(B*b^2 + 2*A*b*c)*d
^3*e^2 - 5*(2*B*c^2*d*e^4 - (2*B*b*c + A*c^2)*e^5)*x^4 + 5*(16*B*c^2*d^2*e^3 - 8*(2*B*b*c + A*c^2)*d*e^4 + 3*(
B*b^2 + 2*A*b*c)*e^5)*x^3 + 15*(32*B*c^2*d^3*e^2 - A*b^2*e^5 - 16*(2*B*b*c + A*c^2)*d^2*e^3 + 6*(B*b^2 + 2*A*b
*c)*d*e^4)*x^2 + 20*(32*B*c^2*d^4*e - A*b^2*d*e^4 - 16*(2*B*b*c + A*c^2)*d^3*e^2 + 6*(B*b^2 + 2*A*b*c)*d^2*e^3
)*x)*sqrt(e*x + d)/(e^9*x^3 + 3*d*e^8*x^2 + 3*d^2*e^7*x + d^3*e^6)

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Sympy [A]  time = 5.69962, size = 1833, normalized size = 6.97 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((-16*A*b**2*d**2*e**3/(15*d**2*e**6*sqrt(d + e*x) + 30*d*e**7*x*sqrt(d + e*x) + 15*e**8*x**2*sqrt(d
+ e*x)) - 40*A*b**2*d*e**4*x/(15*d**2*e**6*sqrt(d + e*x) + 30*d*e**7*x*sqrt(d + e*x) + 15*e**8*x**2*sqrt(d + e
*x)) - 30*A*b**2*e**5*x**2/(15*d**2*e**6*sqrt(d + e*x) + 30*d*e**7*x*sqrt(d + e*x) + 15*e**8*x**2*sqrt(d + e*x
)) + 192*A*b*c*d**3*e**2/(15*d**2*e**6*sqrt(d + e*x) + 30*d*e**7*x*sqrt(d + e*x) + 15*e**8*x**2*sqrt(d + e*x))
+ 480*A*b*c*d**2*e**3*x/(15*d**2*e**6*sqrt(d + e*x) + 30*d*e**7*x*sqrt(d + e*x) + 15*e**8*x**2*sqrt(d + e*x))
+ 360*A*b*c*d*e**4*x**2/(15*d**2*e**6*sqrt(d + e*x) + 30*d*e**7*x*sqrt(d + e*x) + 15*e**8*x**2*sqrt(d + e*x))
+ 60*A*b*c*e**5*x**3/(15*d**2*e**6*sqrt(d + e*x) + 30*d*e**7*x*sqrt(d + e*x) + 15*e**8*x**2*sqrt(d + e*x)) -
256*A*c**2*d**4*e/(15*d**2*e**6*sqrt(d + e*x) + 30*d*e**7*x*sqrt(d + e*x) + 15*e**8*x**2*sqrt(d + e*x)) - 640*
A*c**2*d**3*e**2*x/(15*d**2*e**6*sqrt(d + e*x) + 30*d*e**7*x*sqrt(d + e*x) + 15*e**8*x**2*sqrt(d + e*x)) - 480
*A*c**2*d**2*e**3*x**2/(15*d**2*e**6*sqrt(d + e*x) + 30*d*e**7*x*sqrt(d + e*x) + 15*e**8*x**2*sqrt(d + e*x)) -
80*A*c**2*d*e**4*x**3/(15*d**2*e**6*sqrt(d + e*x) + 30*d*e**7*x*sqrt(d + e*x) + 15*e**8*x**2*sqrt(d + e*x)) +
10*A*c**2*e**5*x**4/(15*d**2*e**6*sqrt(d + e*x) + 30*d*e**7*x*sqrt(d + e*x) + 15*e**8*x**2*sqrt(d + e*x)) + 9
6*B*b**2*d**3*e**2/(15*d**2*e**6*sqrt(d + e*x) + 30*d*e**7*x*sqrt(d + e*x) + 15*e**8*x**2*sqrt(d + e*x)) + 240
*B*b**2*d**2*e**3*x/(15*d**2*e**6*sqrt(d + e*x) + 30*d*e**7*x*sqrt(d + e*x) + 15*e**8*x**2*sqrt(d + e*x)) + 18
0*B*b**2*d*e**4*x**2/(15*d**2*e**6*sqrt(d + e*x) + 30*d*e**7*x*sqrt(d + e*x) + 15*e**8*x**2*sqrt(d + e*x)) + 3
0*B*b**2*e**5*x**3/(15*d**2*e**6*sqrt(d + e*x) + 30*d*e**7*x*sqrt(d + e*x) + 15*e**8*x**2*sqrt(d + e*x)) - 512
*B*b*c*d**4*e/(15*d**2*e**6*sqrt(d + e*x) + 30*d*e**7*x*sqrt(d + e*x) + 15*e**8*x**2*sqrt(d + e*x)) - 1280*B*b
*c*d**3*e**2*x/(15*d**2*e**6*sqrt(d + e*x) + 30*d*e**7*x*sqrt(d + e*x) + 15*e**8*x**2*sqrt(d + e*x)) - 960*B*b
*c*d**2*e**3*x**2/(15*d**2*e**6*sqrt(d + e*x) + 30*d*e**7*x*sqrt(d + e*x) + 15*e**8*x**2*sqrt(d + e*x)) - 160*
B*b*c*d*e**4*x**3/(15*d**2*e**6*sqrt(d + e*x) + 30*d*e**7*x*sqrt(d + e*x) + 15*e**8*x**2*sqrt(d + e*x)) + 20*B
*b*c*e**5*x**4/(15*d**2*e**6*sqrt(d + e*x) + 30*d*e**7*x*sqrt(d + e*x) + 15*e**8*x**2*sqrt(d + e*x)) + 512*B*c
**2*d**5/(15*d**2*e**6*sqrt(d + e*x) + 30*d*e**7*x*sqrt(d + e*x) + 15*e**8*x**2*sqrt(d + e*x)) + 1280*B*c**2*d
**4*e*x/(15*d**2*e**6*sqrt(d + e*x) + 30*d*e**7*x*sqrt(d + e*x) + 15*e**8*x**2*sqrt(d + e*x)) + 960*B*c**2*d**
3*e**2*x**2/(15*d**2*e**6*sqrt(d + e*x) + 30*d*e**7*x*sqrt(d + e*x) + 15*e**8*x**2*sqrt(d + e*x)) + 160*B*c**2
*d**2*e**3*x**3/(15*d**2*e**6*sqrt(d + e*x) + 30*d*e**7*x*sqrt(d + e*x) + 15*e**8*x**2*sqrt(d + e*x)) - 20*B*c
**2*d*e**4*x**4/(15*d**2*e**6*sqrt(d + e*x) + 30*d*e**7*x*sqrt(d + e*x) + 15*e**8*x**2*sqrt(d + e*x)) + 6*B*c*
*2*e**5*x**5/(15*d**2*e**6*sqrt(d + e*x) + 30*d*e**7*x*sqrt(d + e*x) + 15*e**8*x**2*sqrt(d + e*x)), Ne(e, 0)),
((A*b**2*x**3/3 + A*b*c*x**4/2 + A*c**2*x**5/5 + B*b**2*x**4/4 + 2*B*b*c*x**5/5 + B*c**2*x**6/6)/d**(7/2), Tr
ue))

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Giac [A]  time = 1.36752, size = 576, normalized size = 2.19 \begin{align*} \frac{2}{15} \,{\left (3 \,{\left (x e + d\right )}^{\frac{5}{2}} B c^{2} e^{24} - 25 \,{\left (x e + d\right )}^{\frac{3}{2}} B c^{2} d e^{24} + 150 \, \sqrt{x e + d} B c^{2} d^{2} e^{24} + 10 \,{\left (x e + d\right )}^{\frac{3}{2}} B b c e^{25} + 5 \,{\left (x e + d\right )}^{\frac{3}{2}} A c^{2} e^{25} - 120 \, \sqrt{x e + d} B b c d e^{25} - 60 \, \sqrt{x e + d} A c^{2} d e^{25} + 15 \, \sqrt{x e + d} B b^{2} e^{26} + 30 \, \sqrt{x e + d} A b c e^{26}\right )} e^{\left (-30\right )} + \frac{2 \,{\left (150 \,{\left (x e + d\right )}^{2} B c^{2} d^{3} - 25 \,{\left (x e + d\right )} B c^{2} d^{4} + 3 \, B c^{2} d^{5} - 180 \,{\left (x e + d\right )}^{2} B b c d^{2} e - 90 \,{\left (x e + d\right )}^{2} A c^{2} d^{2} e + 40 \,{\left (x e + d\right )} B b c d^{3} e + 20 \,{\left (x e + d\right )} A c^{2} d^{3} e - 6 \, B b c d^{4} e - 3 \, A c^{2} d^{4} e + 45 \,{\left (x e + d\right )}^{2} B b^{2} d e^{2} + 90 \,{\left (x e + d\right )}^{2} A b c d e^{2} - 15 \,{\left (x e + d\right )} B b^{2} d^{2} e^{2} - 30 \,{\left (x e + d\right )} A b c d^{2} e^{2} + 3 \, B b^{2} d^{3} e^{2} + 6 \, A b c d^{3} e^{2} - 15 \,{\left (x e + d\right )}^{2} A b^{2} e^{3} + 10 \,{\left (x e + d\right )} A b^{2} d e^{3} - 3 \, A b^{2} d^{2} e^{3}\right )} e^{\left (-6\right )}}{15 \,{\left (x e + d\right )}^{\frac{5}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/15*(3*(x*e + d)^(5/2)*B*c^2*e^24 - 25*(x*e + d)^(3/2)*B*c^2*d*e^24 + 150*sqrt(x*e + d)*B*c^2*d^2*e^24 + 10*(
x*e + d)^(3/2)*B*b*c*e^25 + 5*(x*e + d)^(3/2)*A*c^2*e^25 - 120*sqrt(x*e + d)*B*b*c*d*e^25 - 60*sqrt(x*e + d)*A
*c^2*d*e^25 + 15*sqrt(x*e + d)*B*b^2*e^26 + 30*sqrt(x*e + d)*A*b*c*e^26)*e^(-30) + 2/15*(150*(x*e + d)^2*B*c^2
*d^3 - 25*(x*e + d)*B*c^2*d^4 + 3*B*c^2*d^5 - 180*(x*e + d)^2*B*b*c*d^2*e - 90*(x*e + d)^2*A*c^2*d^2*e + 40*(x
*e + d)*B*b*c*d^3*e + 20*(x*e + d)*A*c^2*d^3*e - 6*B*b*c*d^4*e - 3*A*c^2*d^4*e + 45*(x*e + d)^2*B*b^2*d*e^2 +
90*(x*e + d)^2*A*b*c*d*e^2 - 15*(x*e + d)*B*b^2*d^2*e^2 - 30*(x*e + d)*A*b*c*d^2*e^2 + 3*B*b^2*d^3*e^2 + 6*A*b
*c*d^3*e^2 - 15*(x*e + d)^2*A*b^2*e^3 + 10*(x*e + d)*A*b^2*d*e^3 - 3*A*b^2*d^2*e^3)*e^(-6)/(x*e + d)^(5/2)