3.1858 \(\int \frac{(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^{7/2}} \, dx\)

Optimal. Leaf size=448 \[ -\frac{20 b^2 \sqrt{a^2+2 a b x+b^2 x^2} \sqrt{d+e x} (b d-a e)^2 (-a B e-A b e+2 b B d)}{e^7 (a+b x)}-\frac{10 b \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^3 (-a B e-2 A b e+3 b B d)}{e^7 (a+b x) \sqrt{d+e x}}+\frac{2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^4 (-a B e-5 A b e+6 b B d)}{3 e^7 (a+b x) (d+e x)^{3/2}}-\frac{2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^5 (B d-A e)}{5 e^7 (a+b x) (d+e x)^{5/2}}-\frac{2 b^4 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{5/2} (-5 a B e-A b e+6 b B d)}{5 e^7 (a+b x)}+\frac{10 b^3 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{3/2} (b d-a e) (-2 a B e-A b e+3 b B d)}{3 e^7 (a+b x)}+\frac{2 b^5 B \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{7/2}}{7 e^7 (a+b x)} \]

[Out]

(-2*(b*d - a*e)^5*(B*d - A*e)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(5*e^7*(a + b*x)*(d
+ e*x)^(5/2)) + (2*(b*d - a*e)^4*(6*b*B*d - 5*A*b*e - a*B*e)*Sqrt[a^2 + 2*a*b*x
+ b^2*x^2])/(3*e^7*(a + b*x)*(d + e*x)^(3/2)) - (10*b*(b*d - a*e)^3*(3*b*B*d -
2*A*b*e - a*B*e)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(e^7*(a + b*x)*Sqrt[d + e*x]) -
(20*b^2*(b*d - a*e)^2*(2*b*B*d - A*b*e - a*B*e)*Sqrt[d + e*x]*Sqrt[a^2 + 2*a*b*x
+ b^2*x^2])/(e^7*(a + b*x)) + (10*b^3*(b*d - a*e)*(3*b*B*d - A*b*e - 2*a*B*e)*(
d + e*x)^(3/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(3*e^7*(a + b*x)) - (2*b^4*(6*b*B*
d - A*b*e - 5*a*B*e)*(d + e*x)^(5/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(5*e^7*(a +
b*x)) + (2*b^5*B*(d + e*x)^(7/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(7*e^7*(a + b*x)
)

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Rubi [A]  time = 0.712615, antiderivative size = 448, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.057 \[ -\frac{20 b^2 \sqrt{a^2+2 a b x+b^2 x^2} \sqrt{d+e x} (b d-a e)^2 (-a B e-A b e+2 b B d)}{e^7 (a+b x)}-\frac{10 b \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^3 (-a B e-2 A b e+3 b B d)}{e^7 (a+b x) \sqrt{d+e x}}+\frac{2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^4 (-a B e-5 A b e+6 b B d)}{3 e^7 (a+b x) (d+e x)^{3/2}}-\frac{2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^5 (B d-A e)}{5 e^7 (a+b x) (d+e x)^{5/2}}-\frac{2 b^4 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{5/2} (-5 a B e-A b e+6 b B d)}{5 e^7 (a+b x)}+\frac{10 b^3 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{3/2} (b d-a e) (-2 a B e-A b e+3 b B d)}{3 e^7 (a+b x)}+\frac{2 b^5 B \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{7/2}}{7 e^7 (a+b x)} \]

Antiderivative was successfully verified.

[In]  Int[((A + B*x)*(a^2 + 2*a*b*x + b^2*x^2)^(5/2))/(d + e*x)^(7/2),x]

[Out]

(-2*(b*d - a*e)^5*(B*d - A*e)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(5*e^7*(a + b*x)*(d
+ e*x)^(5/2)) + (2*(b*d - a*e)^4*(6*b*B*d - 5*A*b*e - a*B*e)*Sqrt[a^2 + 2*a*b*x
+ b^2*x^2])/(3*e^7*(a + b*x)*(d + e*x)^(3/2)) - (10*b*(b*d - a*e)^3*(3*b*B*d -
2*A*b*e - a*B*e)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(e^7*(a + b*x)*Sqrt[d + e*x]) -
(20*b^2*(b*d - a*e)^2*(2*b*B*d - A*b*e - a*B*e)*Sqrt[d + e*x]*Sqrt[a^2 + 2*a*b*x
+ b^2*x^2])/(e^7*(a + b*x)) + (10*b^3*(b*d - a*e)*(3*b*B*d - A*b*e - 2*a*B*e)*(
d + e*x)^(3/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(3*e^7*(a + b*x)) - (2*b^4*(6*b*B*
d - A*b*e - 5*a*B*e)*(d + e*x)^(5/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(5*e^7*(a +
b*x)) + (2*b^5*B*(d + e*x)^(7/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(7*e^7*(a + b*x)
)

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Rubi in Sympy [A]  time = 85.3507, size = 459, normalized size = 1.02 \[ \frac{32 b^{2} \sqrt{d + e x} \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{3}{2}} \left (7 A b e + 5 B a e - 12 B b d\right )}{21 e^{4} \left (a e - b d\right )} + \frac{64 b^{2} \left (3 a + 3 b x\right ) \sqrt{d + e x} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}} \left (7 A b e + 5 B a e - 12 B b d\right )}{105 e^{5}} + \frac{256 b^{2} \sqrt{d + e x} \left (a e - b d\right ) \sqrt{a^{2} + 2 a b x + b^{2} x^{2}} \left (7 A b e + 5 B a e - 12 B b d\right )}{105 e^{6}} + \frac{512 b^{2} \sqrt{d + e x} \left (a e - b d\right )^{2} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}} \left (7 A b e + 5 B a e - 12 B b d\right )}{105 e^{7} \left (a + b x\right )} - \frac{4 b \left (a + b x\right ) \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{3}{2}} \left (7 A b e + 5 B a e - 12 B b d\right )}{3 e^{3} \sqrt{d + e x} \left (a e - b d\right )} - \frac{\left (2 a + 2 b x\right ) \left (A e - B d\right ) \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{5}{2}}}{5 e \left (d + e x\right )^{\frac{5}{2}} \left (a e - b d\right )} - \frac{2 \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{5}{2}} \left (7 A b e + 5 B a e - 12 B b d\right )}{15 e^{2} \left (d + e x\right )^{\frac{3}{2}} \left (a e - b d\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((B*x+A)*(b**2*x**2+2*a*b*x+a**2)**(5/2)/(e*x+d)**(7/2),x)

[Out]

32*b**2*sqrt(d + e*x)*(a**2 + 2*a*b*x + b**2*x**2)**(3/2)*(7*A*b*e + 5*B*a*e - 1
2*B*b*d)/(21*e**4*(a*e - b*d)) + 64*b**2*(3*a + 3*b*x)*sqrt(d + e*x)*sqrt(a**2 +
2*a*b*x + b**2*x**2)*(7*A*b*e + 5*B*a*e - 12*B*b*d)/(105*e**5) + 256*b**2*sqrt(
d + e*x)*(a*e - b*d)*sqrt(a**2 + 2*a*b*x + b**2*x**2)*(7*A*b*e + 5*B*a*e - 12*B*
b*d)/(105*e**6) + 512*b**2*sqrt(d + e*x)*(a*e - b*d)**2*sqrt(a**2 + 2*a*b*x + b*
*2*x**2)*(7*A*b*e + 5*B*a*e - 12*B*b*d)/(105*e**7*(a + b*x)) - 4*b*(a + b*x)*(a*
*2 + 2*a*b*x + b**2*x**2)**(3/2)*(7*A*b*e + 5*B*a*e - 12*B*b*d)/(3*e**3*sqrt(d +
e*x)*(a*e - b*d)) - (2*a + 2*b*x)*(A*e - B*d)*(a**2 + 2*a*b*x + b**2*x**2)**(5/
2)/(5*e*(d + e*x)**(5/2)*(a*e - b*d)) - 2*(a**2 + 2*a*b*x + b**2*x**2)**(5/2)*(7
*A*b*e + 5*B*a*e - 12*B*b*d)/(15*e**2*(d + e*x)**(3/2)*(a*e - b*d))

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Mathematica [A]  time = 0.785061, size = 489, normalized size = 1.09 \[ -\frac{2 \sqrt{(a+b x)^2} \left (7 a^5 e^5 (3 A e+2 B d+5 B e x)+35 a^4 b e^4 \left (A e (2 d+5 e x)+B \left (8 d^2+20 d e x+15 e^2 x^2\right )\right )-70 a^3 b^2 e^3 \left (3 B \left (16 d^3+40 d^2 e x+30 d e^2 x^2+5 e^3 x^3\right )-A e \left (8 d^2+20 d e x+15 e^2 x^2\right )\right )+70 a^2 b^3 e^2 \left (B \left (128 d^4+320 d^3 e x+240 d^2 e^2 x^2+40 d e^3 x^3-5 e^4 x^4\right )-3 A e \left (16 d^3+40 d^2 e x+30 d e^2 x^2+5 e^3 x^3\right )\right )-35 a b^4 e \left (A e \left (-128 d^4-320 d^3 e x-240 d^2 e^2 x^2-40 d e^3 x^3+5 e^4 x^4\right )+B \left (256 d^5+640 d^4 e x+480 d^3 e^2 x^2+80 d^2 e^3 x^3-10 d e^4 x^4+3 e^5 x^5\right )\right )+b^5 \left (3 B \left (1024 d^6+2560 d^5 e x+1920 d^4 e^2 x^2+320 d^3 e^3 x^3-40 d^2 e^4 x^4+12 d e^5 x^5-5 e^6 x^6\right )-7 A e \left (256 d^5+640 d^4 e x+480 d^3 e^2 x^2+80 d^2 e^3 x^3-10 d e^4 x^4+3 e^5 x^5\right )\right )\right )}{105 e^7 (a+b x) (d+e x)^{5/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[((A + B*x)*(a^2 + 2*a*b*x + b^2*x^2)^(5/2))/(d + e*x)^(7/2),x]

[Out]

(-2*Sqrt[(a + b*x)^2]*(7*a^5*e^5*(2*B*d + 3*A*e + 5*B*e*x) + 35*a^4*b*e^4*(A*e*(
2*d + 5*e*x) + B*(8*d^2 + 20*d*e*x + 15*e^2*x^2)) - 70*a^3*b^2*e^3*(-(A*e*(8*d^2
+ 20*d*e*x + 15*e^2*x^2)) + 3*B*(16*d^3 + 40*d^2*e*x + 30*d*e^2*x^2 + 5*e^3*x^3
)) + 70*a^2*b^3*e^2*(-3*A*e*(16*d^3 + 40*d^2*e*x + 30*d*e^2*x^2 + 5*e^3*x^3) + B
*(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)) - 35*a*b^
4*e*(A*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) +
B*(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)) + b^5*(-7*A*e*(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) + 3*B*(1024*d^6 + 2560*d^5*e*x + 1920*d^4*e^2*x^2
+ 320*d^3*e^3*x^3 - 40*d^2*e^4*x^4 + 12*d*e^5*x^5 - 5*e^6*x^6))))/(105*e^7*(a +
b*x)*(d + e*x)^(5/2))

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Maple [A]  time = 0.015, size = 689, normalized size = 1.5 \[ -{\frac{-30\,B{x}^{6}{b}^{5}{e}^{6}-42\,A{x}^{5}{b}^{5}{e}^{6}-210\,B{x}^{5}a{b}^{4}{e}^{6}+72\,B{x}^{5}{b}^{5}d{e}^{5}-350\,A{x}^{4}a{b}^{4}{e}^{6}+140\,A{x}^{4}{b}^{5}d{e}^{5}-700\,B{x}^{4}{a}^{2}{b}^{3}{e}^{6}+700\,B{x}^{4}a{b}^{4}d{e}^{5}-240\,B{x}^{4}{b}^{5}{d}^{2}{e}^{4}-2100\,A{x}^{3}{a}^{2}{b}^{3}{e}^{6}+2800\,A{x}^{3}a{b}^{4}d{e}^{5}-1120\,A{x}^{3}{b}^{5}{d}^{2}{e}^{4}-2100\,B{x}^{3}{a}^{3}{b}^{2}{e}^{6}+5600\,B{x}^{3}{a}^{2}{b}^{3}d{e}^{5}-5600\,B{x}^{3}a{b}^{4}{d}^{2}{e}^{4}+1920\,B{x}^{3}{b}^{5}{d}^{3}{e}^{3}+2100\,A{x}^{2}{a}^{3}{b}^{2}{e}^{6}-12600\,A{x}^{2}{a}^{2}{b}^{3}d{e}^{5}+16800\,A{x}^{2}a{b}^{4}{d}^{2}{e}^{4}-6720\,A{x}^{2}{b}^{5}{d}^{3}{e}^{3}+1050\,B{x}^{2}{a}^{4}b{e}^{6}-12600\,B{x}^{2}{a}^{3}{b}^{2}d{e}^{5}+33600\,B{x}^{2}{a}^{2}{b}^{3}{d}^{2}{e}^{4}-33600\,B{x}^{2}a{b}^{4}{d}^{3}{e}^{3}+11520\,B{x}^{2}{b}^{5}{d}^{4}{e}^{2}+350\,Ax{a}^{4}b{e}^{6}+2800\,Ax{a}^{3}{b}^{2}d{e}^{5}-16800\,Ax{a}^{2}{b}^{3}{d}^{2}{e}^{4}+22400\,Axa{b}^{4}{d}^{3}{e}^{3}-8960\,Ax{b}^{5}{d}^{4}{e}^{2}+70\,Bx{a}^{5}{e}^{6}+1400\,Bx{a}^{4}bd{e}^{5}-16800\,Bx{a}^{3}{b}^{2}{d}^{2}{e}^{4}+44800\,Bx{a}^{2}{b}^{3}{d}^{3}{e}^{3}-44800\,Bxa{b}^{4}{d}^{4}{e}^{2}+15360\,Bx{b}^{5}{d}^{5}e+42\,A{a}^{5}{e}^{6}+140\,Ad{e}^{5}{a}^{4}b+1120\,A{a}^{3}{b}^{2}{d}^{2}{e}^{4}-6720\,A{a}^{2}{b}^{3}{d}^{3}{e}^{3}+8960\,Aa{b}^{4}{d}^{4}{e}^{2}-3584\,A{b}^{5}{d}^{5}e+28\,Bd{e}^{5}{a}^{5}+560\,B{a}^{4}b{d}^{2}{e}^{4}-6720\,B{a}^{3}{b}^{2}{d}^{3}{e}^{3}+17920\,B{a}^{2}{b}^{3}{d}^{4}{e}^{2}-17920\,Ba{b}^{4}{d}^{5}e+6144\,B{b}^{5}{d}^{6}}{105\, \left ( bx+a \right ) ^{5}{e}^{7}} \left ( \left ( bx+a \right ) ^{2} \right ) ^{{\frac{5}{2}}} \left ( ex+d \right ) ^{-{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x+A)*(b^2*x^2+2*a*b*x+a^2)^(5/2)/(e*x+d)^(7/2),x)

[Out]

-2/105/(e*x+d)^(5/2)*(-15*B*b^5*e^6*x^6-21*A*b^5*e^6*x^5-105*B*a*b^4*e^6*x^5+36*
B*b^5*d*e^5*x^5-175*A*a*b^4*e^6*x^4+70*A*b^5*d*e^5*x^4-350*B*a^2*b^3*e^6*x^4+350
*B*a*b^4*d*e^5*x^4-120*B*b^5*d^2*e^4*x^4-1050*A*a^2*b^3*e^6*x^3+1400*A*a*b^4*d*e
^5*x^3-560*A*b^5*d^2*e^4*x^3-1050*B*a^3*b^2*e^6*x^3+2800*B*a^2*b^3*d*e^5*x^3-280
0*B*a*b^4*d^2*e^4*x^3+960*B*b^5*d^3*e^3*x^3+1050*A*a^3*b^2*e^6*x^2-6300*A*a^2*b^
3*d*e^5*x^2+8400*A*a*b^4*d^2*e^4*x^2-3360*A*b^5*d^3*e^3*x^2+525*B*a^4*b*e^6*x^2-
6300*B*a^3*b^2*d*e^5*x^2+16800*B*a^2*b^3*d^2*e^4*x^2-16800*B*a*b^4*d^3*e^3*x^2+5
760*B*b^5*d^4*e^2*x^2+175*A*a^4*b*e^6*x+1400*A*a^3*b^2*d*e^5*x-8400*A*a^2*b^3*d^
2*e^4*x+11200*A*a*b^4*d^3*e^3*x-4480*A*b^5*d^4*e^2*x+35*B*a^5*e^6*x+700*B*a^4*b*
d*e^5*x-8400*B*a^3*b^2*d^2*e^4*x+22400*B*a^2*b^3*d^3*e^3*x-22400*B*a*b^4*d^4*e^2
*x+7680*B*b^5*d^5*e*x+21*A*a^5*e^6+70*A*a^4*b*d*e^5+560*A*a^3*b^2*d^2*e^4-3360*A
*a^2*b^3*d^3*e^3+4480*A*a*b^4*d^4*e^2-1792*A*b^5*d^5*e+14*B*a^5*d*e^5+280*B*a^4*
b*d^2*e^4-3360*B*a^3*b^2*d^3*e^3+8960*B*a^2*b^3*d^4*e^2-8960*B*a*b^4*d^5*e+3072*
B*b^5*d^6)*((b*x+a)^2)^(5/2)/e^7/(b*x+a)^5

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Maxima [A]  time = 0.753266, size = 873, normalized size = 1.95 \[ \frac{2 \,{\left (3 \, b^{5} e^{5} x^{5} + 256 \, b^{5} d^{5} - 640 \, a b^{4} d^{4} e + 480 \, a^{2} b^{3} d^{3} e^{2} - 80 \, a^{3} b^{2} d^{2} e^{3} - 10 \, a^{4} b d e^{4} - 3 \, a^{5} e^{5} - 5 \,{\left (2 \, b^{5} d e^{4} - 5 \, a b^{4} e^{5}\right )} x^{4} + 10 \,{\left (8 \, b^{5} d^{2} e^{3} - 20 \, a b^{4} d e^{4} + 15 \, a^{2} b^{3} e^{5}\right )} x^{3} + 30 \,{\left (16 \, b^{5} d^{3} e^{2} - 40 \, a b^{4} d^{2} e^{3} + 30 \, a^{2} b^{3} d e^{4} - 5 \, a^{3} b^{2} e^{5}\right )} x^{2} + 5 \,{\left (128 \, b^{5} d^{4} e - 320 \, a b^{4} d^{3} e^{2} + 240 \, a^{2} b^{3} d^{2} e^{3} - 40 \, a^{3} b^{2} d e^{4} - 5 \, a^{4} b e^{5}\right )} x\right )} A}{15 \,{\left (e^{8} x^{2} + 2 \, d e^{7} x + d^{2} e^{6}\right )} \sqrt{e x + d}} + \frac{2 \,{\left (15 \, b^{5} e^{6} x^{6} - 3072 \, b^{5} d^{6} + 8960 \, a b^{4} d^{5} e - 8960 \, a^{2} b^{3} d^{4} e^{2} + 3360 \, a^{3} b^{2} d^{3} e^{3} - 280 \, a^{4} b d^{2} e^{4} - 14 \, a^{5} d e^{5} - 3 \,{\left (12 \, b^{5} d e^{5} - 35 \, a b^{4} e^{6}\right )} x^{5} + 10 \,{\left (12 \, b^{5} d^{2} e^{4} - 35 \, a b^{4} d e^{5} + 35 \, a^{2} b^{3} e^{6}\right )} x^{4} - 10 \,{\left (96 \, b^{5} d^{3} e^{3} - 280 \, a b^{4} d^{2} e^{4} + 280 \, a^{2} b^{3} d e^{5} - 105 \, a^{3} b^{2} e^{6}\right )} x^{3} - 15 \,{\left (384 \, b^{5} d^{4} e^{2} - 1120 \, a b^{4} d^{3} e^{3} + 1120 \, a^{2} b^{3} d^{2} e^{4} - 420 \, a^{3} b^{2} d e^{5} + 35 \, a^{4} b e^{6}\right )} x^{2} - 5 \,{\left (1536 \, b^{5} d^{5} e - 4480 \, a b^{4} d^{4} e^{2} + 4480 \, a^{2} b^{3} d^{3} e^{3} - 1680 \, a^{3} b^{2} d^{2} e^{4} + 140 \, a^{4} b d e^{5} + 7 \, a^{5} e^{6}\right )} x\right )} B}{105 \,{\left (e^{9} x^{2} + 2 \, d e^{8} x + d^{2} e^{7}\right )} \sqrt{e x + d}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^2 + 2*a*b*x + a^2)^(5/2)*(B*x + A)/(e*x + d)^(7/2),x, algorithm="maxima")

[Out]

2/15*(3*b^5*e^5*x^5 + 256*b^5*d^5 - 640*a*b^4*d^4*e + 480*a^2*b^3*d^3*e^2 - 80*a
^3*b^2*d^2*e^3 - 10*a^4*b*d*e^4 - 3*a^5*e^5 - 5*(2*b^5*d*e^4 - 5*a*b^4*e^5)*x^4
+ 10*(8*b^5*d^2*e^3 - 20*a*b^4*d*e^4 + 15*a^2*b^3*e^5)*x^3 + 30*(16*b^5*d^3*e^2
- 40*a*b^4*d^2*e^3 + 30*a^2*b^3*d*e^4 - 5*a^3*b^2*e^5)*x^2 + 5*(128*b^5*d^4*e -
320*a*b^4*d^3*e^2 + 240*a^2*b^3*d^2*e^3 - 40*a^3*b^2*d*e^4 - 5*a^4*b*e^5)*x)*A/(
(e^8*x^2 + 2*d*e^7*x + d^2*e^6)*sqrt(e*x + d)) + 2/105*(15*b^5*e^6*x^6 - 3072*b^
5*d^6 + 8960*a*b^4*d^5*e - 8960*a^2*b^3*d^4*e^2 + 3360*a^3*b^2*d^3*e^3 - 280*a^4
*b*d^2*e^4 - 14*a^5*d*e^5 - 3*(12*b^5*d*e^5 - 35*a*b^4*e^6)*x^5 + 10*(12*b^5*d^2
*e^4 - 35*a*b^4*d*e^5 + 35*a^2*b^3*e^6)*x^4 - 10*(96*b^5*d^3*e^3 - 280*a*b^4*d^2
*e^4 + 280*a^2*b^3*d*e^5 - 105*a^3*b^2*e^6)*x^3 - 15*(384*b^5*d^4*e^2 - 1120*a*b
^4*d^3*e^3 + 1120*a^2*b^3*d^2*e^4 - 420*a^3*b^2*d*e^5 + 35*a^4*b*e^6)*x^2 - 5*(1
536*b^5*d^5*e - 4480*a*b^4*d^4*e^2 + 4480*a^2*b^3*d^3*e^3 - 1680*a^3*b^2*d^2*e^4
+ 140*a^4*b*d*e^5 + 7*a^5*e^6)*x)*B/((e^9*x^2 + 2*d*e^8*x + d^2*e^7)*sqrt(e*x +
d))

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Fricas [A]  time = 0.289187, size = 784, normalized size = 1.75 \[ \frac{2 \,{\left (15 \, B b^{5} e^{6} x^{6} - 3072 \, B b^{5} d^{6} - 21 \, A a^{5} e^{6} + 1792 \,{\left (5 \, B a b^{4} + A b^{5}\right )} d^{5} e - 4480 \,{\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} d^{4} e^{2} + 3360 \,{\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} d^{3} e^{3} - 280 \,{\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} d^{2} e^{4} - 14 \,{\left (B a^{5} + 5 \, A a^{4} b\right )} d e^{5} - 3 \,{\left (12 \, B b^{5} d e^{5} - 7 \,{\left (5 \, B a b^{4} + A b^{5}\right )} e^{6}\right )} x^{5} + 5 \,{\left (24 \, B b^{5} d^{2} e^{4} - 14 \,{\left (5 \, B a b^{4} + A b^{5}\right )} d e^{5} + 35 \,{\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} e^{6}\right )} x^{4} - 10 \,{\left (96 \, B b^{5} d^{3} e^{3} - 56 \,{\left (5 \, B a b^{4} + A b^{5}\right )} d^{2} e^{4} + 140 \,{\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} d e^{5} - 105 \,{\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} e^{6}\right )} x^{3} - 15 \,{\left (384 \, B b^{5} d^{4} e^{2} - 224 \,{\left (5 \, B a b^{4} + A b^{5}\right )} d^{3} e^{3} + 560 \,{\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} d^{2} e^{4} - 420 \,{\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} d e^{5} + 35 \,{\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} e^{6}\right )} x^{2} - 5 \,{\left (1536 \, B b^{5} d^{5} e - 896 \,{\left (5 \, B a b^{4} + A b^{5}\right )} d^{4} e^{2} + 2240 \,{\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} d^{3} e^{3} - 1680 \,{\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} d^{2} e^{4} + 140 \,{\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} d e^{5} + 7 \,{\left (B a^{5} + 5 \, A a^{4} b\right )} e^{6}\right )} x\right )}}{105 \,{\left (e^{9} x^{2} + 2 \, d e^{8} x + d^{2} e^{7}\right )} \sqrt{e x + d}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^2 + 2*a*b*x + a^2)^(5/2)*(B*x + A)/(e*x + d)^(7/2),x, algorithm="fricas")

[Out]

2/105*(15*B*b^5*e^6*x^6 - 3072*B*b^5*d^6 - 21*A*a^5*e^6 + 1792*(5*B*a*b^4 + A*b^
5)*d^5*e - 4480*(2*B*a^2*b^3 + A*a*b^4)*d^4*e^2 + 3360*(B*a^3*b^2 + A*a^2*b^3)*d
^3*e^3 - 280*(B*a^4*b + 2*A*a^3*b^2)*d^2*e^4 - 14*(B*a^5 + 5*A*a^4*b)*d*e^5 - 3*
(12*B*b^5*d*e^5 - 7*(5*B*a*b^4 + A*b^5)*e^6)*x^5 + 5*(24*B*b^5*d^2*e^4 - 14*(5*B
*a*b^4 + A*b^5)*d*e^5 + 35*(2*B*a^2*b^3 + A*a*b^4)*e^6)*x^4 - 10*(96*B*b^5*d^3*e
^3 - 56*(5*B*a*b^4 + A*b^5)*d^2*e^4 + 140*(2*B*a^2*b^3 + A*a*b^4)*d*e^5 - 105*(B
*a^3*b^2 + A*a^2*b^3)*e^6)*x^3 - 15*(384*B*b^5*d^4*e^2 - 224*(5*B*a*b^4 + A*b^5)
*d^3*e^3 + 560*(2*B*a^2*b^3 + A*a*b^4)*d^2*e^4 - 420*(B*a^3*b^2 + A*a^2*b^3)*d*e
^5 + 35*(B*a^4*b + 2*A*a^3*b^2)*e^6)*x^2 - 5*(1536*B*b^5*d^5*e - 896*(5*B*a*b^4
+ A*b^5)*d^4*e^2 + 2240*(2*B*a^2*b^3 + A*a*b^4)*d^3*e^3 - 1680*(B*a^3*b^2 + A*a^
2*b^3)*d^2*e^4 + 140*(B*a^4*b + 2*A*a^3*b^2)*d*e^5 + 7*(B*a^5 + 5*A*a^4*b)*e^6)*
x)/((e^9*x^2 + 2*d*e^8*x + d^2*e^7)*sqrt(e*x + d))

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x+A)*(b**2*x**2+2*a*b*x+a**2)**(5/2)/(e*x+d)**(7/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.317808, size = 1, normalized size = 0. \[ \mathit{Done} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^2 + 2*a*b*x + a^2)^(5/2)*(B*x + A)/(e*x + d)^(7/2),x, algorithm="giac")

[Out]

Done