3.1687 \(\int \frac{\left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^{7/2}} \, dx\)

Optimal. Leaf size=316 \[ \frac{20 b^2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^3}{e^6 (a+b x) \sqrt{d+e x}}-\frac{10 b \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^4}{3 e^6 (a+b x) (d+e x)^{3/2}}+\frac{2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^5}{5 e^6 (a+b x) (d+e x)^{5/2}}+\frac{2 b^5 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{5/2}}{5 e^6 (a+b x)}-\frac{10 b^4 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{3/2} (b d-a e)}{3 e^6 (a+b x)}+\frac{20 b^3 \sqrt{a^2+2 a b x+b^2 x^2} \sqrt{d+e x} (b d-a e)^2}{e^6 (a+b x)} \]

[Out]

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

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Rubi [A]  time = 0.295326, antiderivative size = 316, 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 \[ \frac{20 b^2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^3}{e^6 (a+b x) \sqrt{d+e x}}-\frac{10 b \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^4}{3 e^6 (a+b x) (d+e x)^{3/2}}+\frac{2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^5}{5 e^6 (a+b x) (d+e x)^{5/2}}+\frac{2 b^5 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{5/2}}{5 e^6 (a+b x)}-\frac{10 b^4 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{3/2} (b d-a e)}{3 e^6 (a+b x)}+\frac{20 b^3 \sqrt{a^2+2 a b x+b^2 x^2} \sqrt{d+e x} (b d-a e)^2}{e^6 (a+b x)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 38.0758, size = 264, normalized size = 0.84 \[ \frac{64 b^{3} \left (3 a + 3 b x\right ) \sqrt{d + e x} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{15 e^{4}} + \frac{256 b^{3} \sqrt{d + e x} \left (a e - b d\right ) \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{15 e^{5}} + \frac{512 b^{3} \sqrt{d + e x} \left (a e - b d\right )^{2} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{15 e^{6} \left (a + b x\right )} - \frac{32 b^{2} \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{3}{2}}}{3 e^{3} \sqrt{d + e x}} - \frac{4 b \left (5 a + 5 b x\right ) \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{3}{2}}}{15 e^{2} \left (d + e x\right )^{\frac{3}{2}}} - \frac{2 \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{5}{2}}}{5 e \left (d + e x\right )^{\frac{5}{2}}} \]

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)**(7/2),x)

[Out]

64*b**3*(3*a + 3*b*x)*sqrt(d + e*x)*sqrt(a**2 + 2*a*b*x + b**2*x**2)/(15*e**4) +
256*b**3*sqrt(d + e*x)*(a*e - b*d)*sqrt(a**2 + 2*a*b*x + b**2*x**2)/(15*e**5) +
512*b**3*sqrt(d + e*x)*(a*e - b*d)**2*sqrt(a**2 + 2*a*b*x + b**2*x**2)/(15*e**6
*(a + b*x)) - 32*b**2*(a**2 + 2*a*b*x + b**2*x**2)**(3/2)/(3*e**3*sqrt(d + e*x))
- 4*b*(5*a + 5*b*x)*(a**2 + 2*a*b*x + b**2*x**2)**(3/2)/(15*e**2*(d + e*x)**(3/
2)) - 2*(a**2 + 2*a*b*x + b**2*x**2)**(5/2)/(5*e*(d + e*x)**(5/2))

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Mathematica [A]  time = 0.333837, size = 150, normalized size = 0.47 \[ \frac{2 \left ((a+b x)^2\right )^{5/2} \sqrt{d+e x} \left (b^3 \left (150 a^2 e^2-275 a b d e+128 b^2 d^2\right )-b^4 e x (19 b d-25 a e)+\frac{150 b^2 (b d-a e)^3}{d+e x}-\frac{25 b (b d-a e)^4}{(d+e x)^2}+\frac{3 (b d-a e)^5}{(d+e x)^3}+3 b^5 e^2 x^2\right )}{15 e^6 (a+b x)^5} \]

Antiderivative was successfully verified.

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

[Out]

(2*((a + b*x)^2)^(5/2)*Sqrt[d + e*x]*(b^3*(128*b^2*d^2 - 275*a*b*d*e + 150*a^2*e
^2) - b^4*e*(19*b*d - 25*a*e)*x + 3*b^5*e^2*x^2 + (3*(b*d - a*e)^5)/(d + e*x)^3
- (25*b*(b*d - a*e)^4)/(d + e*x)^2 + (150*b^2*(b*d - a*e)^3)/(d + e*x)))/(15*e^6
*(a + b*x)^5)

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Maple [A]  time = 0.01, size = 289, normalized size = 0.9 \[ -{\frac{-6\,{x}^{5}{b}^{5}{e}^{5}-50\,{x}^{4}a{b}^{4}{e}^{5}+20\,{x}^{4}{b}^{5}d{e}^{4}-300\,{x}^{3}{a}^{2}{b}^{3}{e}^{5}+400\,{x}^{3}a{b}^{4}d{e}^{4}-160\,{x}^{3}{b}^{5}{d}^{2}{e}^{3}+300\,{x}^{2}{a}^{3}{b}^{2}{e}^{5}-1800\,{x}^{2}{a}^{2}{b}^{3}d{e}^{4}+2400\,{x}^{2}a{b}^{4}{d}^{2}{e}^{3}-960\,{x}^{2}{b}^{5}{d}^{3}{e}^{2}+50\,x{a}^{4}b{e}^{5}+400\,x{a}^{3}{b}^{2}d{e}^{4}-2400\,x{a}^{2}{b}^{3}{d}^{2}{e}^{3}+3200\,xa{b}^{4}{d}^{3}{e}^{2}-1280\,x{b}^{5}{d}^{4}e+6\,{a}^{5}{e}^{5}+20\,{a}^{4}bd{e}^{4}+160\,{a}^{3}{b}^{2}{d}^{2}{e}^{3}-960\,{a}^{2}{b}^{3}{d}^{3}{e}^{2}+1280\,a{b}^{4}{d}^{4}e-512\,{b}^{5}{d}^{5}}{15\, \left ( bx+a \right ) ^{5}{e}^{6}} \left ( \left ( bx+a \right ) ^{2} \right ) ^{{\frac{5}{2}}} \left ( ex+d \right ) ^{-{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/15/(e*x+d)^(5/2)*(-3*b^5*e^5*x^5-25*a*b^4*e^5*x^4+10*b^5*d*e^4*x^4-150*a^2*b^
3*e^5*x^3+200*a*b^4*d*e^4*x^3-80*b^5*d^2*e^3*x^3+150*a^3*b^2*e^5*x^2-900*a^2*b^3
*d*e^4*x^2+1200*a*b^4*d^2*e^3*x^2-480*b^5*d^3*e^2*x^2+25*a^4*b*e^5*x+200*a^3*b^2
*d*e^4*x-1200*a^2*b^3*d^2*e^3*x+1600*a*b^4*d^3*e^2*x-640*b^5*d^4*e*x+3*a^5*e^5+1
0*a^4*b*d*e^4+80*a^3*b^2*d^2*e^3-480*a^2*b^3*d^3*e^2+640*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 = 0.735125, size = 382, normalized size = 1.21 \[ \frac{2 \,{\left (3 \, b^{5} e^{5} x^{5} + 256 \, b^{5} d^{5} - 640 \, a b^{4} d^{4} e + 480 \, a^{2} b^{3} d^{3} e^{2} - 80 \, a^{3} b^{2} d^{2} e^{3} - 10 \, a^{4} b d e^{4} - 3 \, a^{5} e^{5} - 5 \,{\left (2 \, b^{5} d e^{4} - 5 \, a b^{4} e^{5}\right )} x^{4} + 10 \,{\left (8 \, b^{5} d^{2} e^{3} - 20 \, a b^{4} d e^{4} + 15 \, a^{2} b^{3} e^{5}\right )} x^{3} + 30 \,{\left (16 \, b^{5} d^{3} e^{2} - 40 \, a b^{4} d^{2} e^{3} + 30 \, a^{2} b^{3} d e^{4} - 5 \, a^{3} b^{2} e^{5}\right )} x^{2} + 5 \,{\left (128 \, b^{5} d^{4} e - 320 \, a b^{4} d^{3} e^{2} + 240 \, a^{2} b^{3} d^{2} e^{3} - 40 \, a^{3} b^{2} d e^{4} - 5 \, a^{4} b e^{5}\right )} x\right )}}{15 \,{\left (e^{8} x^{2} + 2 \, d e^{7} x + d^{2} e^{6}\right )} \sqrt{e x + d}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^2 + 2*a*b*x + a^2)^(5/2)/(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)/((e
^8*x^2 + 2*d*e^7*x + d^2*e^6)*sqrt(e*x + d))

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Fricas [A]  time = 0.208973, size = 382, normalized size = 1.21 \[ \frac{2 \,{\left (3 \, b^{5} e^{5} x^{5} + 256 \, b^{5} d^{5} - 640 \, a b^{4} d^{4} e + 480 \, a^{2} b^{3} d^{3} e^{2} - 80 \, a^{3} b^{2} d^{2} e^{3} - 10 \, a^{4} b d e^{4} - 3 \, a^{5} e^{5} - 5 \,{\left (2 \, b^{5} d e^{4} - 5 \, a b^{4} e^{5}\right )} x^{4} + 10 \,{\left (8 \, b^{5} d^{2} e^{3} - 20 \, a b^{4} d e^{4} + 15 \, a^{2} b^{3} e^{5}\right )} x^{3} + 30 \,{\left (16 \, b^{5} d^{3} e^{2} - 40 \, a b^{4} d^{2} e^{3} + 30 \, a^{2} b^{3} d e^{4} - 5 \, a^{3} b^{2} e^{5}\right )} x^{2} + 5 \,{\left (128 \, b^{5} d^{4} e - 320 \, a b^{4} d^{3} e^{2} + 240 \, a^{2} b^{3} d^{2} e^{3} - 40 \, a^{3} b^{2} d e^{4} - 5 \, a^{4} b e^{5}\right )} x\right )}}{15 \,{\left (e^{8} x^{2} + 2 \, d e^{7} x + d^{2} e^{6}\right )} \sqrt{e x + d}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[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)/((e
^8*x^2 + 2*d*e^7*x + d^2*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)**(7/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.233203, size = 620, normalized size = 1.96 \[ \frac{2}{15} \,{\left (3 \,{\left (x e + d\right )}^{\frac{5}{2}} b^{5} e^{24}{\rm sign}\left (b x + a\right ) - 25 \,{\left (x e + d\right )}^{\frac{3}{2}} b^{5} d e^{24}{\rm sign}\left (b x + a\right ) + 150 \, \sqrt{x e + d} b^{5} d^{2} e^{24}{\rm sign}\left (b x + a\right ) + 25 \,{\left (x e + d\right )}^{\frac{3}{2}} a b^{4} e^{25}{\rm sign}\left (b x + a\right ) - 300 \, \sqrt{x e + d} a b^{4} d e^{25}{\rm sign}\left (b x + a\right ) + 150 \, \sqrt{x e + d} a^{2} b^{3} e^{26}{\rm sign}\left (b x + a\right )\right )} e^{\left (-30\right )} + \frac{2 \,{\left (150 \,{\left (x e + d\right )}^{2} b^{5} d^{3}{\rm sign}\left (b x + a\right ) - 25 \,{\left (x e + d\right )} b^{5} d^{4}{\rm sign}\left (b x + a\right ) + 3 \, b^{5} d^{5}{\rm sign}\left (b x + a\right ) - 450 \,{\left (x e + d\right )}^{2} a b^{4} d^{2} e{\rm sign}\left (b x + a\right ) + 100 \,{\left (x e + d\right )} a b^{4} d^{3} e{\rm sign}\left (b x + a\right ) - 15 \, a b^{4} d^{4} e{\rm sign}\left (b x + a\right ) + 450 \,{\left (x e + d\right )}^{2} a^{2} b^{3} d e^{2}{\rm sign}\left (b x + a\right ) - 150 \,{\left (x e + d\right )} a^{2} b^{3} d^{2} e^{2}{\rm sign}\left (b x + a\right ) + 30 \, a^{2} b^{3} d^{3} e^{2}{\rm sign}\left (b x + a\right ) - 150 \,{\left (x e + d\right )}^{2} a^{3} b^{2} e^{3}{\rm sign}\left (b x + a\right ) + 100 \,{\left (x e + d\right )} a^{3} b^{2} d e^{3}{\rm sign}\left (b x + a\right ) - 30 \, a^{3} b^{2} d^{2} e^{3}{\rm sign}\left (b x + a\right ) - 25 \,{\left (x e + d\right )} a^{4} b e^{4}{\rm sign}\left (b x + a\right ) + 15 \, a^{4} b d e^{4}{\rm sign}\left (b x + a\right ) - 3 \, a^{5} e^{5}{\rm sign}\left (b x + a\right )\right )} e^{\left (-6\right )}}{15 \,{\left (x e + d\right )}^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/15*(3*(x*e + d)^(5/2)*b^5*e^24*sign(b*x + a) - 25*(x*e + d)^(3/2)*b^5*d*e^24*s
ign(b*x + a) + 150*sqrt(x*e + d)*b^5*d^2*e^24*sign(b*x + a) + 25*(x*e + d)^(3/2)
*a*b^4*e^25*sign(b*x + a) - 300*sqrt(x*e + d)*a*b^4*d*e^25*sign(b*x + a) + 150*s
qrt(x*e + d)*a^2*b^3*e^26*sign(b*x + a))*e^(-30) + 2/15*(150*(x*e + d)^2*b^5*d^3
*sign(b*x + a) - 25*(x*e + d)*b^5*d^4*sign(b*x + a) + 3*b^5*d^5*sign(b*x + a) -
450*(x*e + d)^2*a*b^4*d^2*e*sign(b*x + a) + 100*(x*e + d)*a*b^4*d^3*e*sign(b*x +
a) - 15*a*b^4*d^4*e*sign(b*x + a) + 450*(x*e + d)^2*a^2*b^3*d*e^2*sign(b*x + a)
- 150*(x*e + d)*a^2*b^3*d^2*e^2*sign(b*x + a) + 30*a^2*b^3*d^3*e^2*sign(b*x + a
) - 150*(x*e + d)^2*a^3*b^2*e^3*sign(b*x + a) + 100*(x*e + d)*a^3*b^2*d*e^3*sign
(b*x + a) - 30*a^3*b^2*d^2*e^3*sign(b*x + a) - 25*(x*e + d)*a^4*b*e^4*sign(b*x +
a) + 15*a^4*b*d*e^4*sign(b*x + a) - 3*a^5*e^5*sign(b*x + a))*e^(-6)/(x*e + d)^(
5/2)