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

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 = 0.450979, antiderivative size = 263, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077 \[ -\frac{2 \sqrt{d+e x} \left (2 A c e (2 c d-b e)-B \left (b^2 e^2-8 b c d e+10 c^2 d^2\right )\right )}{e^6}-\frac{2 \left (A e \left (b^2 e^2-6 b c d e+6 c^2 d^2\right )-B d \left (3 b^2 e^2-12 b c d e+10 c^2 d^2\right )\right )}{e^6 \sqrt{d+e x}}+\frac{2 d^2 (B d-A e) (c d-b e)^2}{5 e^6 (d+e x)^{5/2}}-\frac{2 c (d+e x)^{3/2} (-A c e-2 b B e+5 B c d)}{3 e^6}-\frac{2 d (c d-b e) (B d (5 c d-3 b e)-2 A e (2 c d-b e))}{3 e^6 (d+e x)^{3/2}}+\frac{2 B c^2 (d+e x)^{5/2}}{5 e^6} \]

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)

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Rubi in Sympy [A]  time = 99.5086, size = 289, normalized size = 1.1 \[ \frac{2 B c^{2} \left (d + e x\right )^{\frac{5}{2}}}{5 e^{6}} + \frac{2 c \left (d + e x\right )^{\frac{3}{2}} \left (A c e + 2 B b e - 5 B c d\right )}{3 e^{6}} - \frac{2 d^{2} \left (A e - B d\right ) \left (b e - c d\right )^{2}}{5 e^{6} \left (d + e x\right )^{\frac{5}{2}}} + \frac{2 d \left (b e - c d\right ) \left (2 A b e^{2} - 4 A c d e - 3 B b d e + 5 B c d^{2}\right )}{3 e^{6} \left (d + e x\right )^{\frac{3}{2}}} + \frac{2 \sqrt{d + e x} \left (2 A b c e^{2} - 4 A c^{2} d e + B b^{2} e^{2} - 8 B b c d e + 10 B c^{2} d^{2}\right )}{e^{6}} - \frac{2 \left (A b^{2} e^{3} - 6 A b c d e^{2} + 6 A c^{2} d^{2} e - 3 B b^{2} d e^{2} + 12 B b c d^{2} e - 10 B c^{2} d^{3}\right )}{e^{6} \sqrt{d + e x}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.570747, size = 224, normalized size = 0.85 \[ \frac{2 \sqrt{d+e x} \left (-\frac{15 \left (A e \left (b^2 e^2-6 b c d e+6 c^2 d^2\right )+B d \left (-3 b^2 e^2+12 b c d e-10 c^2 d^2\right )\right )}{d+e x}+\frac{3 d^2 (B d-A e) (c d-b e)^2}{(d+e x)^3}+c e x (5 A c e+10 b B e-19 B c d)-\frac{5 d (c d-b e) (2 A e (b e-2 c d)+B d (5 c d-3 b e))}{(d+e x)^2}+30 A b c e^2-55 A c^2 d e+15 b^2 B e^2-110 b B c d e+128 B c^2 d^2+3 B c^2 e^2 x^2\right )}{15 e^6} \]

Antiderivative was successfully verified.

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

[Out]

(2*Sqrt[d + e*x]*(128*B*c^2*d^2 - 110*b*B*c*d*e - 55*A*c^2*d*e + 15*b^2*B*e^2 +
30*A*b*c*e^2 + c*e*(-19*B*c*d + 10*b*B*e + 5*A*c*e)*x + 3*B*c^2*e^2*x^2 + (3*d^2
*(B*d - A*e)*(c*d - b*e)^2)/(d + e*x)^3 - (5*d*(c*d - b*e)*(B*d*(5*c*d - 3*b*e)
+ 2*A*e*(-2*c*d + b*e)))/(d + e*x)^2 - (15*(B*d*(-10*c^2*d^2 + 12*b*c*d*e - 3*b^
2*e^2) + A*e*(6*c^2*d^2 - 6*b*c*d*e + b^2*e^2)))/(d + e*x)))/(15*e^6)

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Maple [A]  time = 0.011, size = 341, normalized size = 1.3 \[ -{\frac{-6\,B{c}^{2}{x}^{5}{e}^{5}-10\,A{c}^{2}{e}^{5}{x}^{4}-20\,Bbc{e}^{5}{x}^{4}+20\,B{c}^{2}d{e}^{4}{x}^{4}-60\,Abc{e}^{5}{x}^{3}+80\,A{c}^{2}d{e}^{4}{x}^{3}-30\,B{b}^{2}{e}^{5}{x}^{3}+160\,Bbcd{e}^{4}{x}^{3}-160\,B{c}^{2}{d}^{2}{e}^{3}{x}^{3}+30\,A{b}^{2}{e}^{5}{x}^{2}-360\,Abcd{e}^{4}{x}^{2}+480\,A{c}^{2}{d}^{2}{e}^{3}{x}^{2}-180\,B{b}^{2}d{e}^{4}{x}^{2}+960\,Bbc{d}^{2}{e}^{3}{x}^{2}-960\,B{c}^{2}{d}^{3}{e}^{2}{x}^{2}+40\,A{b}^{2}d{e}^{4}x-480\,Abc{d}^{2}{e}^{3}x+640\,A{c}^{2}{d}^{3}{e}^{2}x-240\,B{b}^{2}{d}^{2}{e}^{3}x+1280\,Bbc{d}^{3}{e}^{2}x-1280\,B{c}^{2}{d}^{4}ex+16\,A{b}^{2}{d}^{2}{e}^{3}-192\,Abc{d}^{3}{e}^{2}+256\,A{c}^{2}{d}^{4}e-96\,B{b}^{2}{d}^{3}{e}^{2}+512\,Bbc{d}^{4}e-512\,B{c}^{2}{d}^{5}}{15\,{e}^{6}} \left ( ex+d \right ) ^{-{\frac{5}{2}}}} \]

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+320*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+128*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 = 0.706224, size = 402, normalized size = 1.53 \[ \frac{2 \,{\left (\frac{3 \,{\left (e x + d\right )}^{\frac{5}{2}} B c^{2} - 5 \,{\left (5 \, B c^{2} d -{\left (2 \, B b c + A c^{2}\right )} e\right )}{\left (e x + d\right )}^{\frac{3}{2}} + 15 \,{\left (10 \, B c^{2} d^{2} - 4 \,{\left (2 \, B b c + A c^{2}\right )} d e +{\left (B b^{2} + 2 \, A b c\right )} e^{2}\right )} \sqrt{e x + d}}{e^{5}} + \frac{3 \, B c^{2} d^{5} - 3 \, A b^{2} d^{2} e^{3} - 3 \,{\left (2 \, B b c + A c^{2}\right )} d^{4} e + 3 \,{\left (B b^{2} + 2 \, A b c\right )} d^{3} e^{2} + 15 \,{\left (10 \, B c^{2} d^{3} - A b^{2} e^{3} - 6 \,{\left (2 \, B b c + A c^{2}\right )} d^{2} e + 3 \,{\left (B b^{2} + 2 \, A b c\right )} d e^{2}\right )}{\left (e x + d\right )}^{2} - 5 \,{\left (5 \, B c^{2} d^{4} - 2 \, A b^{2} d e^{3} - 4 \,{\left (2 \, B b c + A c^{2}\right )} d^{3} e + 3 \,{\left (B b^{2} + 2 \, A b c\right )} d^{2} e^{2}\right )}{\left (e x + d\right )}}{{\left (e x + d\right )}^{\frac{5}{2}} e^{5}}\right )}}{15 \, e} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x)^2*(B*x + A)/(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 = 0.270964, size = 420, normalized size = 1.6 \[ \frac{2 \,{\left (3 \, B c^{2} e^{5} x^{5} + 256 \, B c^{2} d^{5} - 8 \, A b^{2} d^{2} e^{3} - 128 \,{\left (2 \, B b c + A c^{2}\right )} d^{4} e + 48 \,{\left (B b^{2} + 2 \, A b c\right )} d^{3} e^{2} - 5 \,{\left (2 \, B c^{2} d e^{4} -{\left (2 \, B b c + A c^{2}\right )} e^{5}\right )} x^{4} + 5 \,{\left (16 \, B c^{2} d^{2} e^{3} - 8 \,{\left (2 \, B b c + A c^{2}\right )} d e^{4} + 3 \,{\left (B b^{2} + 2 \, A b c\right )} e^{5}\right )} x^{3} + 15 \,{\left (32 \, B c^{2} d^{3} e^{2} - A b^{2} e^{5} - 16 \,{\left (2 \, B b c + A c^{2}\right )} d^{2} e^{3} + 6 \,{\left (B b^{2} + 2 \, A b c\right )} d e^{4}\right )} x^{2} + 20 \,{\left (32 \, B c^{2} d^{4} e - A b^{2} d e^{4} - 16 \,{\left (2 \, B b c + A c^{2}\right )} d^{3} e^{2} + 6 \,{\left (B b^{2} + 2 \, A b c\right )} d^{2} e^{3}\right )} x\right )}}{15 \,{\left (e^{8} x^{2} + 2 \, d e^{7} x + d^{2} e^{6}\right )} \sqrt{e x + d}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x)^2*(B*x + A)/(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)/((e^8*x^2 + 2*d*e^7*x + d^2*e^6)*sq
rt(e*x + d))

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Sympy [A]  time = 13.5832, size = 1833, normalized size = 6.97 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x+A)*(c*x**2+b*x)**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*sq
rt(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*sqr
t(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*s
qrt(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)) + 96*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)) + 180*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)) + 30*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/(1
5*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), True))

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GIAC/XCAS [A]  time = 0.300646, size = 576, normalized size = 2.19 \[ \frac{2}{15} \,{\left (3 \,{\left (x e + d\right )}^{\frac{5}{2}} B c^{2} e^{24} - 25 \,{\left (x e + d\right )}^{\frac{3}{2}} B c^{2} d e^{24} + 150 \, \sqrt{x e + d} B c^{2} d^{2} e^{24} + 10 \,{\left (x e + d\right )}^{\frac{3}{2}} B b c e^{25} + 5 \,{\left (x e + d\right )}^{\frac{3}{2}} A c^{2} e^{25} - 120 \, \sqrt{x e + d} B b c d e^{25} - 60 \, \sqrt{x e + d} A c^{2} d e^{25} + 15 \, \sqrt{x e + d} B b^{2} e^{26} + 30 \, \sqrt{x e + d} A b c e^{26}\right )} e^{\left (-30\right )} + \frac{2 \,{\left (150 \,{\left (x e + d\right )}^{2} B c^{2} d^{3} - 25 \,{\left (x e + d\right )} B c^{2} d^{4} + 3 \, B c^{2} d^{5} - 180 \,{\left (x e + d\right )}^{2} B b c d^{2} e - 90 \,{\left (x e + d\right )}^{2} A c^{2} d^{2} e + 40 \,{\left (x e + d\right )} B b c d^{3} e + 20 \,{\left (x e + d\right )} A c^{2} d^{3} e - 6 \, B b c d^{4} e - 3 \, A c^{2} d^{4} e + 45 \,{\left (x e + d\right )}^{2} B b^{2} d e^{2} + 90 \,{\left (x e + d\right )}^{2} A b c d e^{2} - 15 \,{\left (x e + d\right )} B b^{2} d^{2} e^{2} - 30 \,{\left (x e + d\right )} A b c d^{2} e^{2} + 3 \, B b^{2} d^{3} e^{2} + 6 \, A b c d^{3} e^{2} - 15 \,{\left (x e + d\right )}^{2} A b^{2} e^{3} + 10 \,{\left (x e + d\right )} A b^{2} d e^{3} - 3 \, A b^{2} d^{2} e^{3}\right )} e^{\left (-6\right )}}{15 \,{\left (x e + d\right )}^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x)^2*(B*x + A)/(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*sqr
t(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)