3.1856 \(\int \frac{(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^{3/2}} \, dx\)

Optimal. Leaf size=446 \[ -\frac{4 b^2 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{5/2} (b d-a e)^2 (-a B e-A b e+2 b B d)}{e^7 (a+b x)}+\frac{10 b \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{3/2} (b d-a e)^3 (-a B e-2 A b e+3 b B d)}{3 e^7 (a+b x)}-\frac{2 \sqrt{a^2+2 a b x+b^2 x^2} \sqrt{d+e x} (b d-a e)^4 (-a B e-5 A b e+6 b B d)}{e^7 (a+b x)}-\frac{2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^5 (B d-A e)}{e^7 (a+b x) \sqrt{d+e x}}-\frac{2 b^4 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{9/2} (-5 a B e-A b e+6 b B d)}{9 e^7 (a+b x)}+\frac{10 b^3 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{7/2} (b d-a e) (-2 a B e-A b e+3 b B d)}{7 e^7 (a+b x)}+\frac{2 b^5 B \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{11/2}}{11 e^7 (a+b x)} \]

[Out]

(-2*(b*d - a*e)^5*(B*d - A*e)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(e^7*(a + b*x)*Sqrt
[d + e*x]) - (2*(b*d - a*e)^4*(6*b*B*d - 5*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*(b*d - a*e)^3*(3*b*B*d - 2*A*b*e
- 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)) - (4*b
^2*(b*d - a*e)^2*(2*b*B*d - A*b*e - a*B*e)*(d + e*x)^(5/2)*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)^(7/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(7*e^7*(a + b*x)) - (2*b^4*(6*b*B*d -
A*b*e - 5*a*B*e)*(d + e*x)^(9/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(9*e^7*(a + b*x
)) + (2*b^5*B*(d + e*x)^(11/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(11*e^7*(a + b*x))

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Rubi [A]  time = 0.706497, antiderivative size = 446, 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 \[ -\frac{4 b^2 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{5/2} (b d-a e)^2 (-a B e-A b e+2 b B d)}{e^7 (a+b x)}+\frac{10 b \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{3/2} (b d-a e)^3 (-a B e-2 A b e+3 b B d)}{3 e^7 (a+b x)}-\frac{2 \sqrt{a^2+2 a b x+b^2 x^2} \sqrt{d+e x} (b d-a e)^4 (-a B e-5 A b e+6 b B d)}{e^7 (a+b x)}-\frac{2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^5 (B d-A e)}{e^7 (a+b x) \sqrt{d+e x}}-\frac{2 b^4 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{9/2} (-5 a B e-A b e+6 b B d)}{9 e^7 (a+b x)}+\frac{10 b^3 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{7/2} (b d-a e) (-2 a B e-A b e+3 b B d)}{7 e^7 (a+b x)}+\frac{2 b^5 B \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{11/2}}{11 e^7 (a+b x)} \]

Antiderivative was successfully verified.

[In]  Int[((A + B*x)*(a^2 + 2*a*b*x + b^2*x^2)^(5/2))/(d + e*x)^(3/2),x]

[Out]

(-2*(b*d - a*e)^5*(B*d - A*e)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(e^7*(a + b*x)*Sqrt
[d + e*x]) - (2*(b*d - a*e)^4*(6*b*B*d - 5*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*(b*d - a*e)^3*(3*b*B*d - 2*A*b*e
- 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)) - (4*b
^2*(b*d - a*e)^2*(2*b*B*d - A*b*e - a*B*e)*(d + e*x)^(5/2)*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)^(7/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(7*e^7*(a + b*x)) - (2*b^4*(6*b*B*d -
A*b*e - 5*a*B*e)*(d + e*x)^(9/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(9*e^7*(a + b*x
)) + (2*b^5*B*(d + e*x)^(11/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(11*e^7*(a + b*x))

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Rubi in Sympy [A]  time = 84.8564, size = 439, normalized size = 0.98 \[ - \frac{\left (2 a + 2 b x\right ) \left (A e - B d\right ) \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{5}{2}}}{e \sqrt{d + e x} \left (a e - b d\right )} + \frac{2 \sqrt{d + e x} \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{5}{2}} \left (11 A b e + B a e - 12 B b d\right )}{11 e^{2} \left (a e - b d\right )} + \frac{4 \left (5 a + 5 b x\right ) \sqrt{d + e x} \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{3}{2}} \left (11 A b e + B a e - 12 B b d\right )}{99 e^{3}} + \frac{160 \sqrt{d + e x} \left (a e - b d\right ) \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{3}{2}} \left (11 A b e + B a e - 12 B b d\right )}{693 e^{4}} + \frac{64 \left (3 a + 3 b x\right ) \sqrt{d + e x} \left (a e - b d\right )^{2} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}} \left (11 A b e + B a e - 12 B b d\right )}{693 e^{5}} + \frac{256 \sqrt{d + e x} \left (a e - b d\right )^{3} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}} \left (11 A b e + B a e - 12 B b d\right )}{693 e^{6}} + \frac{512 \sqrt{d + e x} \left (a e - b d\right )^{4} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}} \left (11 A b e + B a e - 12 B b d\right )}{693 e^{7} \left (a + b x\right )} \]

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)**(3/2),x)

[Out]

-(2*a + 2*b*x)*(A*e - B*d)*(a**2 + 2*a*b*x + b**2*x**2)**(5/2)/(e*sqrt(d + e*x)*
(a*e - b*d)) + 2*sqrt(d + e*x)*(a**2 + 2*a*b*x + b**2*x**2)**(5/2)*(11*A*b*e + B
*a*e - 12*B*b*d)/(11*e**2*(a*e - b*d)) + 4*(5*a + 5*b*x)*sqrt(d + e*x)*(a**2 + 2
*a*b*x + b**2*x**2)**(3/2)*(11*A*b*e + B*a*e - 12*B*b*d)/(99*e**3) + 160*sqrt(d
+ e*x)*(a*e - b*d)*(a**2 + 2*a*b*x + b**2*x**2)**(3/2)*(11*A*b*e + B*a*e - 12*B*
b*d)/(693*e**4) + 64*(3*a + 3*b*x)*sqrt(d + e*x)*(a*e - b*d)**2*sqrt(a**2 + 2*a*
b*x + b**2*x**2)*(11*A*b*e + B*a*e - 12*B*b*d)/(693*e**5) + 256*sqrt(d + e*x)*(a
*e - b*d)**3*sqrt(a**2 + 2*a*b*x + b**2*x**2)*(11*A*b*e + B*a*e - 12*B*b*d)/(693
*e**6) + 512*sqrt(d + e*x)*(a*e - b*d)**4*sqrt(a**2 + 2*a*b*x + b**2*x**2)*(11*A
*b*e + B*a*e - 12*B*b*d)/(693*e**7*(a + b*x))

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Mathematica [A]  time = 0.66676, size = 486, normalized size = 1.09 \[ \frac{2 \sqrt{(a+b x)^2} \left (693 a^5 e^5 (-A e+2 B d+B e x)+1155 a^4 b e^4 \left (3 A e (2 d+e x)+B \left (-8 d^2-4 d e x+e^2 x^2\right )\right )+462 a^3 b^2 e^3 \left (5 A e \left (-8 d^2-4 d e x+e^2 x^2\right )+3 B \left (16 d^3+8 d^2 e x-2 d e^2 x^2+e^3 x^3\right )\right )-198 a^2 b^3 e^2 \left (B \left (128 d^4+64 d^3 e x-16 d^2 e^2 x^2+8 d e^3 x^3-5 e^4 x^4\right )-7 A e \left (16 d^3+8 d^2 e x-2 d e^2 x^2+e^3 x^3\right )\right )+11 a b^4 e \left (9 A e \left (-128 d^4-64 d^3 e x+16 d^2 e^2 x^2-8 d e^3 x^3+5 e^4 x^4\right )+5 B \left (256 d^5+128 d^4 e x-32 d^3 e^2 x^2+16 d^2 e^3 x^3-10 d e^4 x^4+7 e^5 x^5\right )\right )+b^5 \left (11 A e \left (256 d^5+128 d^4 e x-32 d^3 e^2 x^2+16 d^2 e^3 x^3-10 d e^4 x^4+7 e^5 x^5\right )-3 B \left (1024 d^6+512 d^5 e x-128 d^4 e^2 x^2+64 d^3 e^3 x^3-40 d^2 e^4 x^4+28 d e^5 x^5-21 e^6 x^6\right )\right )\right )}{693 e^7 (a+b x) \sqrt{d+e x}} \]

Antiderivative was successfully verified.

[In]  Integrate[((A + B*x)*(a^2 + 2*a*b*x + b^2*x^2)^(5/2))/(d + e*x)^(3/2),x]

[Out]

(2*Sqrt[(a + b*x)^2]*(693*a^5*e^5*(2*B*d - A*e + B*e*x) + 1155*a^4*b*e^4*(3*A*e*
(2*d + e*x) + B*(-8*d^2 - 4*d*e*x + e^2*x^2)) + 462*a^3*b^2*e^3*(5*A*e*(-8*d^2 -
4*d*e*x + e^2*x^2) + 3*B*(16*d^3 + 8*d^2*e*x - 2*d*e^2*x^2 + e^3*x^3)) - 198*a^
2*b^3*e^2*(-7*A*e*(16*d^3 + 8*d^2*e*x - 2*d*e^2*x^2 + e^3*x^3) + B*(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)) + 11*a*b^4*e*(9*A*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*B*(256*d^5 + 12
8*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)) + b^5*(
11*A*e*(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) - 3*B*(1024*d^6 + 512*d^5*e*x - 128*d^4*e^2*x^2 + 64*d^3*e^3*x^3 - 4
0*d^2*e^4*x^4 + 28*d*e^5*x^5 - 21*e^6*x^6))))/(693*e^7*(a + b*x)*Sqrt[d + e*x])

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Maple [A]  time = 0.014, size = 689, normalized size = 1.5 \[ -{\frac{-126\,B{x}^{6}{b}^{5}{e}^{6}-154\,A{x}^{5}{b}^{5}{e}^{6}-770\,B{x}^{5}a{b}^{4}{e}^{6}+168\,B{x}^{5}{b}^{5}d{e}^{5}-990\,A{x}^{4}a{b}^{4}{e}^{6}+220\,A{x}^{4}{b}^{5}d{e}^{5}-1980\,B{x}^{4}{a}^{2}{b}^{3}{e}^{6}+1100\,B{x}^{4}a{b}^{4}d{e}^{5}-240\,B{x}^{4}{b}^{5}{d}^{2}{e}^{4}-2772\,A{x}^{3}{a}^{2}{b}^{3}{e}^{6}+1584\,A{x}^{3}a{b}^{4}d{e}^{5}-352\,A{x}^{3}{b}^{5}{d}^{2}{e}^{4}-2772\,B{x}^{3}{a}^{3}{b}^{2}{e}^{6}+3168\,B{x}^{3}{a}^{2}{b}^{3}d{e}^{5}-1760\,B{x}^{3}a{b}^{4}{d}^{2}{e}^{4}+384\,B{x}^{3}{b}^{5}{d}^{3}{e}^{3}-4620\,A{x}^{2}{a}^{3}{b}^{2}{e}^{6}+5544\,A{x}^{2}{a}^{2}{b}^{3}d{e}^{5}-3168\,A{x}^{2}a{b}^{4}{d}^{2}{e}^{4}+704\,A{x}^{2}{b}^{5}{d}^{3}{e}^{3}-2310\,B{x}^{2}{a}^{4}b{e}^{6}+5544\,B{x}^{2}{a}^{3}{b}^{2}d{e}^{5}-6336\,B{x}^{2}{a}^{2}{b}^{3}{d}^{2}{e}^{4}+3520\,B{x}^{2}a{b}^{4}{d}^{3}{e}^{3}-768\,B{x}^{2}{b}^{5}{d}^{4}{e}^{2}-6930\,Ax{a}^{4}b{e}^{6}+18480\,Ax{a}^{3}{b}^{2}d{e}^{5}-22176\,Ax{a}^{2}{b}^{3}{d}^{2}{e}^{4}+12672\,Axa{b}^{4}{d}^{3}{e}^{3}-2816\,Ax{b}^{5}{d}^{4}{e}^{2}-1386\,Bx{a}^{5}{e}^{6}+9240\,Bx{a}^{4}bd{e}^{5}-22176\,Bx{a}^{3}{b}^{2}{d}^{2}{e}^{4}+25344\,Bx{a}^{2}{b}^{3}{d}^{3}{e}^{3}-14080\,Bxa{b}^{4}{d}^{4}{e}^{2}+3072\,Bx{b}^{5}{d}^{5}e+1386\,A{a}^{5}{e}^{6}-13860\,Ad{e}^{5}{a}^{4}b+36960\,A{a}^{3}{b}^{2}{d}^{2}{e}^{4}-44352\,A{a}^{2}{b}^{3}{d}^{3}{e}^{3}+25344\,Aa{b}^{4}{d}^{4}{e}^{2}-5632\,A{b}^{5}{d}^{5}e-2772\,Bd{e}^{5}{a}^{5}+18480\,B{a}^{4}b{d}^{2}{e}^{4}-44352\,B{a}^{3}{b}^{2}{d}^{3}{e}^{3}+50688\,B{a}^{2}{b}^{3}{d}^{4}{e}^{2}-28160\,Ba{b}^{4}{d}^{5}e+6144\,B{b}^{5}{d}^{6}}{693\, \left ( bx+a \right ) ^{5}{e}^{7}} \left ( \left ( bx+a \right ) ^{2} \right ) ^{{\frac{5}{2}}}{\frac{1}{\sqrt{ex+d}}}} \]

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)^(3/2),x)

[Out]

-2/693/(e*x+d)^(1/2)*(-63*B*b^5*e^6*x^6-77*A*b^5*e^6*x^5-385*B*a*b^4*e^6*x^5+84*
B*b^5*d*e^5*x^5-495*A*a*b^4*e^6*x^4+110*A*b^5*d*e^5*x^4-990*B*a^2*b^3*e^6*x^4+55
0*B*a*b^4*d*e^5*x^4-120*B*b^5*d^2*e^4*x^4-1386*A*a^2*b^3*e^6*x^3+792*A*a*b^4*d*e
^5*x^3-176*A*b^5*d^2*e^4*x^3-1386*B*a^3*b^2*e^6*x^3+1584*B*a^2*b^3*d*e^5*x^3-880
*B*a*b^4*d^2*e^4*x^3+192*B*b^5*d^3*e^3*x^3-2310*A*a^3*b^2*e^6*x^2+2772*A*a^2*b^3
*d*e^5*x^2-1584*A*a*b^4*d^2*e^4*x^2+352*A*b^5*d^3*e^3*x^2-1155*B*a^4*b*e^6*x^2+2
772*B*a^3*b^2*d*e^5*x^2-3168*B*a^2*b^3*d^2*e^4*x^2+1760*B*a*b^4*d^3*e^3*x^2-384*
B*b^5*d^4*e^2*x^2-3465*A*a^4*b*e^6*x+9240*A*a^3*b^2*d*e^5*x-11088*A*a^2*b^3*d^2*
e^4*x+6336*A*a*b^4*d^3*e^3*x-1408*A*b^5*d^4*e^2*x-693*B*a^5*e^6*x+4620*B*a^4*b*d
*e^5*x-11088*B*a^3*b^2*d^2*e^4*x+12672*B*a^2*b^3*d^3*e^3*x-7040*B*a*b^4*d^4*e^2*
x+1536*B*b^5*d^5*e*x+693*A*a^5*e^6-6930*A*a^4*b*d*e^5+18480*A*a^3*b^2*d^2*e^4-22
176*A*a^2*b^3*d^3*e^3+12672*A*a*b^4*d^4*e^2-2816*A*b^5*d^5*e-1386*B*a^5*d*e^5+92
40*B*a^4*b*d^2*e^4-22176*B*a^3*b^2*d^3*e^3+25344*B*a^2*b^3*d^4*e^2-14080*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 = 0.744579, size = 814, normalized size = 1.83 \[ \frac{2 \,{\left (7 \, b^{5} e^{5} x^{5} + 256 \, b^{5} d^{5} - 1152 \, a b^{4} d^{4} e + 2016 \, a^{2} b^{3} d^{3} e^{2} - 1680 \, a^{3} b^{2} d^{2} e^{3} + 630 \, a^{4} b d e^{4} - 63 \, a^{5} e^{5} - 5 \,{\left (2 \, b^{5} d e^{4} - 9 \, a b^{4} e^{5}\right )} x^{4} + 2 \,{\left (8 \, b^{5} d^{2} e^{3} - 36 \, a b^{4} d e^{4} + 63 \, a^{2} b^{3} e^{5}\right )} x^{3} - 2 \,{\left (16 \, b^{5} d^{3} e^{2} - 72 \, a b^{4} d^{2} e^{3} + 126 \, a^{2} b^{3} d e^{4} - 105 \, a^{3} b^{2} e^{5}\right )} x^{2} +{\left (128 \, b^{5} d^{4} e - 576 \, a b^{4} d^{3} e^{2} + 1008 \, a^{2} b^{3} d^{2} e^{3} - 840 \, a^{3} b^{2} d e^{4} + 315 \, a^{4} b e^{5}\right )} x\right )} A}{63 \, \sqrt{e x + d} e^{6}} + \frac{2 \,{\left (63 \, b^{5} e^{6} x^{6} - 3072 \, b^{5} d^{6} + 14080 \, a b^{4} d^{5} e - 25344 \, a^{2} b^{3} d^{4} e^{2} + 22176 \, a^{3} b^{2} d^{3} e^{3} - 9240 \, a^{4} b d^{2} e^{4} + 1386 \, a^{5} d e^{5} - 7 \,{\left (12 \, b^{5} d e^{5} - 55 \, a b^{4} e^{6}\right )} x^{5} + 10 \,{\left (12 \, b^{5} d^{2} e^{4} - 55 \, a b^{4} d e^{5} + 99 \, a^{2} b^{3} e^{6}\right )} x^{4} - 2 \,{\left (96 \, b^{5} d^{3} e^{3} - 440 \, a b^{4} d^{2} e^{4} + 792 \, a^{2} b^{3} d e^{5} - 693 \, a^{3} b^{2} e^{6}\right )} x^{3} +{\left (384 \, b^{5} d^{4} e^{2} - 1760 \, a b^{4} d^{3} e^{3} + 3168 \, a^{2} b^{3} d^{2} e^{4} - 2772 \, a^{3} b^{2} d e^{5} + 1155 \, a^{4} b e^{6}\right )} x^{2} -{\left (1536 \, b^{5} d^{5} e - 7040 \, a b^{4} d^{4} e^{2} + 12672 \, a^{2} b^{3} d^{3} e^{3} - 11088 \, a^{3} b^{2} d^{2} e^{4} + 4620 \, a^{4} b d e^{5} - 693 \, a^{5} e^{6}\right )} x\right )} B}{693 \, \sqrt{e x + d} e^{7}} \]

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)^(3/2),x, algorithm="maxima")

[Out]

2/63*(7*b^5*e^5*x^5 + 256*b^5*d^5 - 1152*a*b^4*d^4*e + 2016*a^2*b^3*d^3*e^2 - 16
80*a^3*b^2*d^2*e^3 + 630*a^4*b*d*e^4 - 63*a^5*e^5 - 5*(2*b^5*d*e^4 - 9*a*b^4*e^5
)*x^4 + 2*(8*b^5*d^2*e^3 - 36*a*b^4*d*e^4 + 63*a^2*b^3*e^5)*x^3 - 2*(16*b^5*d^3*
e^2 - 72*a*b^4*d^2*e^3 + 126*a^2*b^3*d*e^4 - 105*a^3*b^2*e^5)*x^2 + (128*b^5*d^4
*e - 576*a*b^4*d^3*e^2 + 1008*a^2*b^3*d^2*e^3 - 840*a^3*b^2*d*e^4 + 315*a^4*b*e^
5)*x)*A/(sqrt(e*x + d)*e^6) + 2/693*(63*b^5*e^6*x^6 - 3072*b^5*d^6 + 14080*a*b^4
*d^5*e - 25344*a^2*b^3*d^4*e^2 + 22176*a^3*b^2*d^3*e^3 - 9240*a^4*b*d^2*e^4 + 13
86*a^5*d*e^5 - 7*(12*b^5*d*e^5 - 55*a*b^4*e^6)*x^5 + 10*(12*b^5*d^2*e^4 - 55*a*b
^4*d*e^5 + 99*a^2*b^3*e^6)*x^4 - 2*(96*b^5*d^3*e^3 - 440*a*b^4*d^2*e^4 + 792*a^2
*b^3*d*e^5 - 693*a^3*b^2*e^6)*x^3 + (384*b^5*d^4*e^2 - 1760*a*b^4*d^3*e^3 + 3168
*a^2*b^3*d^2*e^4 - 2772*a^3*b^2*d*e^5 + 1155*a^4*b*e^6)*x^2 - (1536*b^5*d^5*e -
7040*a*b^4*d^4*e^2 + 12672*a^2*b^3*d^3*e^3 - 11088*a^3*b^2*d^2*e^4 + 4620*a^4*b*
d*e^5 - 693*a^5*e^6)*x)*B/(sqrt(e*x + d)*e^7)

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Fricas [A]  time = 0.285568, size = 755, normalized size = 1.69 \[ \frac{2 \,{\left (63 \, B b^{5} e^{6} x^{6} - 3072 \, B b^{5} d^{6} - 693 \, A a^{5} e^{6} + 2816 \,{\left (5 \, B a b^{4} + A b^{5}\right )} d^{5} e - 12672 \,{\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} d^{4} e^{2} + 22176 \,{\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} d^{3} e^{3} - 9240 \,{\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} d^{2} e^{4} + 1386 \,{\left (B a^{5} + 5 \, A a^{4} b\right )} d e^{5} - 7 \,{\left (12 \, B b^{5} d e^{5} - 11 \,{\left (5 \, B a b^{4} + A b^{5}\right )} e^{6}\right )} x^{5} + 5 \,{\left (24 \, B b^{5} d^{2} e^{4} - 22 \,{\left (5 \, B a b^{4} + A b^{5}\right )} d e^{5} + 99 \,{\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} e^{6}\right )} x^{4} - 2 \,{\left (96 \, B b^{5} d^{3} e^{3} - 88 \,{\left (5 \, B a b^{4} + A b^{5}\right )} d^{2} e^{4} + 396 \,{\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} d e^{5} - 693 \,{\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} e^{6}\right )} x^{3} +{\left (384 \, B b^{5} d^{4} e^{2} - 352 \,{\left (5 \, B a b^{4} + A b^{5}\right )} d^{3} e^{3} + 1584 \,{\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} d^{2} e^{4} - 2772 \,{\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} d e^{5} + 1155 \,{\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} e^{6}\right )} x^{2} -{\left (1536 \, B b^{5} d^{5} e - 1408 \,{\left (5 \, B a b^{4} + A b^{5}\right )} d^{4} e^{2} + 6336 \,{\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} d^{3} e^{3} - 11088 \,{\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} d^{2} e^{4} + 4620 \,{\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} d e^{5} - 693 \,{\left (B a^{5} + 5 \, A a^{4} b\right )} e^{6}\right )} x\right )}}{693 \, \sqrt{e x + d} e^{7}} \]

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)^(3/2),x, algorithm="fricas")

[Out]

2/693*(63*B*b^5*e^6*x^6 - 3072*B*b^5*d^6 - 693*A*a^5*e^6 + 2816*(5*B*a*b^4 + A*b
^5)*d^5*e - 12672*(2*B*a^2*b^3 + A*a*b^4)*d^4*e^2 + 22176*(B*a^3*b^2 + A*a^2*b^3
)*d^3*e^3 - 9240*(B*a^4*b + 2*A*a^3*b^2)*d^2*e^4 + 1386*(B*a^5 + 5*A*a^4*b)*d*e^
5 - 7*(12*B*b^5*d*e^5 - 11*(5*B*a*b^4 + A*b^5)*e^6)*x^5 + 5*(24*B*b^5*d^2*e^4 -
22*(5*B*a*b^4 + A*b^5)*d*e^5 + 99*(2*B*a^2*b^3 + A*a*b^4)*e^6)*x^4 - 2*(96*B*b^5
*d^3*e^3 - 88*(5*B*a*b^4 + A*b^5)*d^2*e^4 + 396*(2*B*a^2*b^3 + A*a*b^4)*d*e^5 -
693*(B*a^3*b^2 + A*a^2*b^3)*e^6)*x^3 + (384*B*b^5*d^4*e^2 - 352*(5*B*a*b^4 + A*b
^5)*d^3*e^3 + 1584*(2*B*a^2*b^3 + A*a*b^4)*d^2*e^4 - 2772*(B*a^3*b^2 + A*a^2*b^3
)*d*e^5 + 1155*(B*a^4*b + 2*A*a^3*b^2)*e^6)*x^2 - (1536*B*b^5*d^5*e - 1408*(5*B*
a*b^4 + A*b^5)*d^4*e^2 + 6336*(2*B*a^2*b^3 + A*a*b^4)*d^3*e^3 - 11088*(B*a^3*b^2
+ A*a^2*b^3)*d^2*e^4 + 4620*(B*a^4*b + 2*A*a^3*b^2)*d*e^5 - 693*(B*a^5 + 5*A*a^
4*b)*e^6)*x)/(sqrt(e*x + d)*e^7)

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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)**(3/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.314067, 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)^(3/2),x, algorithm="giac")

[Out]

Done