3.1686 \(\int \frac{\left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^{5/2}} \, dx\)

Optimal. Leaf size=314 \[ -\frac{20 b^2 \sqrt{a^2+2 a b x+b^2 x^2} \sqrt{d+e x} (b d-a e)^3}{e^6 (a+b x)}-\frac{10 b \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^4}{e^6 (a+b x) \sqrt{d+e x}}+\frac{2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^5}{3 e^6 (a+b x) (d+e x)^{3/2}}+\frac{2 b^5 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{7/2}}{7 e^6 (a+b x)}-\frac{2 b^4 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{5/2} (b d-a e)}{e^6 (a+b x)}+\frac{20 b^3 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{3/2} (b d-a e)^2}{3 e^6 (a+b x)} \]

[Out]

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

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Rubi [A]  time = 0.300769, antiderivative size = 314, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067 \[ -\frac{20 b^2 \sqrt{a^2+2 a b x+b^2 x^2} \sqrt{d+e x} (b d-a e)^3}{e^6 (a+b x)}-\frac{10 b \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^4}{e^6 (a+b x) \sqrt{d+e x}}+\frac{2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^5}{3 e^6 (a+b x) (d+e x)^{3/2}}+\frac{2 b^5 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{7/2}}{7 e^6 (a+b x)}-\frac{2 b^4 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{5/2} (b d-a e)}{e^6 (a+b x)}+\frac{20 b^3 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{3/2} (b d-a e)^2}{3 e^6 (a+b x)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 42.1849, size = 269, normalized size = 0.86 \[ \frac{160 b^{2} \sqrt{d + e x} \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{3}{2}}}{21 e^{3}} + \frac{64 b^{2} \left (3 a + 3 b x\right ) \sqrt{d + e x} \left (a e - b d\right ) \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{21 e^{4}} + \frac{256 b^{2} \sqrt{d + e x} \left (a e - b d\right )^{2} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{21 e^{5}} + \frac{512 b^{2} \sqrt{d + e x} \left (a e - b d\right )^{3} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{21 e^{6} \left (a + b x\right )} - \frac{20 b \left (a + b x\right ) \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{3}{2}}}{3 e^{2} \sqrt{d + e x}} - \frac{2 \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{5}{2}}}{3 e \left (d + e x\right )^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

160*b**2*sqrt(d + e*x)*(a**2 + 2*a*b*x + b**2*x**2)**(3/2)/(21*e**3) + 64*b**2*(
3*a + 3*b*x)*sqrt(d + e*x)*(a*e - b*d)*sqrt(a**2 + 2*a*b*x + b**2*x**2)/(21*e**4
) + 256*b**2*sqrt(d + e*x)*(a*e - b*d)**2*sqrt(a**2 + 2*a*b*x + b**2*x**2)/(21*e
**5) + 512*b**2*sqrt(d + e*x)*(a*e - b*d)**3*sqrt(a**2 + 2*a*b*x + b**2*x**2)/(2
1*e**6*(a + b*x)) - 20*b*(a + b*x)*(a**2 + 2*a*b*x + b**2*x**2)**(3/2)/(3*e**2*s
qrt(d + e*x)) - 2*(a**2 + 2*a*b*x + b**2*x**2)**(5/2)/(3*e*(d + e*x)**(3/2))

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Mathematica [A]  time = 0.449169, size = 175, normalized size = 0.56 \[ \frac{2 \left ((a+b x)^2\right )^{5/2} \sqrt{d+e x} \left (b^3 e x \left (70 a^2 e^2-98 a b d e+37 b^2 d^2\right )+b^2 \left (210 a^3 e^3-560 a^2 b d e^2+511 a b^2 d^2 e-158 b^3 d^3\right )-3 b^4 e^2 x^2 (4 b d-7 a e)-\frac{105 b (b d-a e)^4}{d+e x}+\frac{7 (b d-a e)^5}{(d+e x)^2}+3 b^5 e^3 x^3\right )}{21 e^6 (a+b x)^5} \]

Antiderivative was successfully verified.

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

[Out]

(2*((a + b*x)^2)^(5/2)*Sqrt[d + e*x]*(b^2*(-158*b^3*d^3 + 511*a*b^2*d^2*e - 560*
a^2*b*d*e^2 + 210*a^3*e^3) + b^3*e*(37*b^2*d^2 - 98*a*b*d*e + 70*a^2*e^2)*x - 3*
b^4*e^2*(4*b*d - 7*a*e)*x^2 + 3*b^5*e^3*x^3 + (7*(b*d - a*e)^5)/(d + e*x)^2 - (1
05*b*(b*d - a*e)^4)/(d + e*x)))/(21*e^6*(a + b*x)^5)

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Maple [A]  time = 0.01, size = 289, normalized size = 0.9 \[ -{\frac{-6\,{x}^{5}{b}^{5}{e}^{5}-42\,{x}^{4}a{b}^{4}{e}^{5}+12\,{x}^{4}{b}^{5}d{e}^{4}-140\,{x}^{3}{a}^{2}{b}^{3}{e}^{5}+112\,{x}^{3}a{b}^{4}d{e}^{4}-32\,{x}^{3}{b}^{5}{d}^{2}{e}^{3}-420\,{x}^{2}{a}^{3}{b}^{2}{e}^{5}+840\,{x}^{2}{a}^{2}{b}^{3}d{e}^{4}-672\,{x}^{2}a{b}^{4}{d}^{2}{e}^{3}+192\,{x}^{2}{b}^{5}{d}^{3}{e}^{2}+210\,x{a}^{4}b{e}^{5}-1680\,x{a}^{3}{b}^{2}d{e}^{4}+3360\,x{a}^{2}{b}^{3}{d}^{2}{e}^{3}-2688\,xa{b}^{4}{d}^{3}{e}^{2}+768\,x{b}^{5}{d}^{4}e+14\,{a}^{5}{e}^{5}+140\,{a}^{4}bd{e}^{4}-1120\,{a}^{3}{b}^{2}{d}^{2}{e}^{3}+2240\,{a}^{2}{b}^{3}{d}^{3}{e}^{2}-1792\,a{b}^{4}{d}^{4}e+512\,{b}^{5}{d}^{5}}{21\, \left ( bx+a \right ) ^{5}{e}^{6}} \left ( \left ( bx+a \right ) ^{2} \right ) ^{{\frac{5}{2}}} \left ( ex+d \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/21/(e*x+d)^(3/2)*(-3*b^5*e^5*x^5-21*a*b^4*e^5*x^4+6*b^5*d*e^4*x^4-70*a^2*b^3*
e^5*x^3+56*a*b^4*d*e^4*x^3-16*b^5*d^2*e^3*x^3-210*a^3*b^2*e^5*x^2+420*a^2*b^3*d*
e^4*x^2-336*a*b^4*d^2*e^3*x^2+96*b^5*d^3*e^2*x^2+105*a^4*b*e^5*x-840*a^3*b^2*d*e
^4*x+1680*a^2*b^3*d^2*e^3*x-1344*a*b^4*d^3*e^2*x+384*b^5*d^4*e*x+7*a^5*e^5+70*a^
4*b*d*e^4-560*a^3*b^2*d^2*e^3+1120*a^2*b^3*d^3*e^2-896*a*b^4*d^4*e+256*b^5*d^5)*
((b*x+a)^2)^(5/2)/e^6/(b*x+a)^5

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Maxima [A]  time = 0.734895, size = 367, normalized size = 1.17 \[ \frac{2 \,{\left (3 \, b^{5} e^{5} x^{5} - 256 \, b^{5} d^{5} + 896 \, a b^{4} d^{4} e - 1120 \, a^{2} b^{3} d^{3} e^{2} + 560 \, a^{3} b^{2} d^{2} e^{3} - 70 \, a^{4} b d e^{4} - 7 \, a^{5} e^{5} - 3 \,{\left (2 \, b^{5} d e^{4} - 7 \, a b^{4} e^{5}\right )} x^{4} + 2 \,{\left (8 \, b^{5} d^{2} e^{3} - 28 \, a b^{4} d e^{4} + 35 \, a^{2} b^{3} e^{5}\right )} x^{3} - 6 \,{\left (16 \, b^{5} d^{3} e^{2} - 56 \, a b^{4} d^{2} e^{3} + 70 \, a^{2} b^{3} d e^{4} - 35 \, a^{3} b^{2} e^{5}\right )} x^{2} - 3 \,{\left (128 \, b^{5} d^{4} e - 448 \, a b^{4} d^{3} e^{2} + 560 \, a^{2} b^{3} d^{2} e^{3} - 280 \, a^{3} b^{2} d e^{4} + 35 \, a^{4} b e^{5}\right )} x\right )}}{21 \,{\left (e^{7} x + d e^{6}\right )} \sqrt{e x + d}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/21*(3*b^5*e^5*x^5 - 256*b^5*d^5 + 896*a*b^4*d^4*e - 1120*a^2*b^3*d^3*e^2 + 560
*a^3*b^2*d^2*e^3 - 70*a^4*b*d*e^4 - 7*a^5*e^5 - 3*(2*b^5*d*e^4 - 7*a*b^4*e^5)*x^
4 + 2*(8*b^5*d^2*e^3 - 28*a*b^4*d*e^4 + 35*a^2*b^3*e^5)*x^3 - 6*(16*b^5*d^3*e^2
- 56*a*b^4*d^2*e^3 + 70*a^2*b^3*d*e^4 - 35*a^3*b^2*e^5)*x^2 - 3*(128*b^5*d^4*e -
448*a*b^4*d^3*e^2 + 560*a^2*b^3*d^2*e^3 - 280*a^3*b^2*d*e^4 + 35*a^4*b*e^5)*x)/
((e^7*x + d*e^6)*sqrt(e*x + d))

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Fricas [A]  time = 0.210266, size = 367, normalized size = 1.17 \[ \frac{2 \,{\left (3 \, b^{5} e^{5} x^{5} - 256 \, b^{5} d^{5} + 896 \, a b^{4} d^{4} e - 1120 \, a^{2} b^{3} d^{3} e^{2} + 560 \, a^{3} b^{2} d^{2} e^{3} - 70 \, a^{4} b d e^{4} - 7 \, a^{5} e^{5} - 3 \,{\left (2 \, b^{5} d e^{4} - 7 \, a b^{4} e^{5}\right )} x^{4} + 2 \,{\left (8 \, b^{5} d^{2} e^{3} - 28 \, a b^{4} d e^{4} + 35 \, a^{2} b^{3} e^{5}\right )} x^{3} - 6 \,{\left (16 \, b^{5} d^{3} e^{2} - 56 \, a b^{4} d^{2} e^{3} + 70 \, a^{2} b^{3} d e^{4} - 35 \, a^{3} b^{2} e^{5}\right )} x^{2} - 3 \,{\left (128 \, b^{5} d^{4} e - 448 \, a b^{4} d^{3} e^{2} + 560 \, a^{2} b^{3} d^{2} e^{3} - 280 \, a^{3} b^{2} d e^{4} + 35 \, a^{4} b e^{5}\right )} x\right )}}{21 \,{\left (e^{7} x + d e^{6}\right )} \sqrt{e x + d}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/21*(3*b^5*e^5*x^5 - 256*b^5*d^5 + 896*a*b^4*d^4*e - 1120*a^2*b^3*d^3*e^2 + 560
*a^3*b^2*d^2*e^3 - 70*a^4*b*d*e^4 - 7*a^5*e^5 - 3*(2*b^5*d*e^4 - 7*a*b^4*e^5)*x^
4 + 2*(8*b^5*d^2*e^3 - 28*a*b^4*d*e^4 + 35*a^2*b^3*e^5)*x^3 - 6*(16*b^5*d^3*e^2
- 56*a*b^4*d^2*e^3 + 70*a^2*b^3*d*e^4 - 35*a^3*b^2*e^5)*x^2 - 3*(128*b^5*d^4*e -
448*a*b^4*d^3*e^2 + 560*a^2*b^3*d^2*e^3 - 280*a^3*b^2*d*e^4 + 35*a^4*b*e^5)*x)/
((e^7*x + d*e^6)*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**2*x**2+2*a*b*x+a**2)**(5/2)/(e*x+d)**(5/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.231147, size = 621, normalized size = 1.98 \[ \frac{2}{21} \,{\left (3 \,{\left (x e + d\right )}^{\frac{7}{2}} b^{5} e^{36}{\rm sign}\left (b x + a\right ) - 21 \,{\left (x e + d\right )}^{\frac{5}{2}} b^{5} d e^{36}{\rm sign}\left (b x + a\right ) + 70 \,{\left (x e + d\right )}^{\frac{3}{2}} b^{5} d^{2} e^{36}{\rm sign}\left (b x + a\right ) - 210 \, \sqrt{x e + d} b^{5} d^{3} e^{36}{\rm sign}\left (b x + a\right ) + 21 \,{\left (x e + d\right )}^{\frac{5}{2}} a b^{4} e^{37}{\rm sign}\left (b x + a\right ) - 140 \,{\left (x e + d\right )}^{\frac{3}{2}} a b^{4} d e^{37}{\rm sign}\left (b x + a\right ) + 630 \, \sqrt{x e + d} a b^{4} d^{2} e^{37}{\rm sign}\left (b x + a\right ) + 70 \,{\left (x e + d\right )}^{\frac{3}{2}} a^{2} b^{3} e^{38}{\rm sign}\left (b x + a\right ) - 630 \, \sqrt{x e + d} a^{2} b^{3} d e^{38}{\rm sign}\left (b x + a\right ) + 210 \, \sqrt{x e + d} a^{3} b^{2} e^{39}{\rm sign}\left (b x + a\right )\right )} e^{\left (-42\right )} - \frac{2 \,{\left (15 \,{\left (x e + d\right )} b^{5} d^{4}{\rm sign}\left (b x + a\right ) - b^{5} d^{5}{\rm sign}\left (b x + a\right ) - 60 \,{\left (x e + d\right )} a b^{4} d^{3} e{\rm sign}\left (b x + a\right ) + 5 \, a b^{4} d^{4} e{\rm sign}\left (b x + a\right ) + 90 \,{\left (x e + d\right )} a^{2} b^{3} d^{2} e^{2}{\rm sign}\left (b x + a\right ) - 10 \, a^{2} b^{3} d^{3} e^{2}{\rm sign}\left (b x + a\right ) - 60 \,{\left (x e + d\right )} a^{3} b^{2} d e^{3}{\rm sign}\left (b x + a\right ) + 10 \, a^{3} b^{2} d^{2} e^{3}{\rm sign}\left (b x + a\right ) + 15 \,{\left (x e + d\right )} a^{4} b e^{4}{\rm sign}\left (b x + a\right ) - 5 \, a^{4} b d e^{4}{\rm sign}\left (b x + a\right ) + a^{5} e^{5}{\rm sign}\left (b x + a\right )\right )} e^{\left (-6\right )}}{3 \,{\left (x e + d\right )}^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/21*(3*(x*e + d)^(7/2)*b^5*e^36*sign(b*x + a) - 21*(x*e + d)^(5/2)*b^5*d*e^36*s
ign(b*x + a) + 70*(x*e + d)^(3/2)*b^5*d^2*e^36*sign(b*x + a) - 210*sqrt(x*e + d)
*b^5*d^3*e^36*sign(b*x + a) + 21*(x*e + d)^(5/2)*a*b^4*e^37*sign(b*x + a) - 140*
(x*e + d)^(3/2)*a*b^4*d*e^37*sign(b*x + a) + 630*sqrt(x*e + d)*a*b^4*d^2*e^37*si
gn(b*x + a) + 70*(x*e + d)^(3/2)*a^2*b^3*e^38*sign(b*x + a) - 630*sqrt(x*e + d)*
a^2*b^3*d*e^38*sign(b*x + a) + 210*sqrt(x*e + d)*a^3*b^2*e^39*sign(b*x + a))*e^(
-42) - 2/3*(15*(x*e + d)*b^5*d^4*sign(b*x + a) - b^5*d^5*sign(b*x + a) - 60*(x*e
+ d)*a*b^4*d^3*e*sign(b*x + a) + 5*a*b^4*d^4*e*sign(b*x + a) + 90*(x*e + d)*a^2
*b^3*d^2*e^2*sign(b*x + a) - 10*a^2*b^3*d^3*e^2*sign(b*x + a) - 60*(x*e + d)*a^3
*b^2*d*e^3*sign(b*x + a) + 10*a^3*b^2*d^2*e^3*sign(b*x + a) + 15*(x*e + d)*a^4*b
*e^4*sign(b*x + a) - 5*a^4*b*d*e^4*sign(b*x + a) + a^5*e^5*sign(b*x + a))*e^(-6)
/(x*e + d)^(3/2)