3.1688 \(\int \frac{\left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^{9/2}} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.297843, 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 \[ \frac{20 b^2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^3}{3 e^6 (a+b x) (d+e x)^{3/2}}-\frac{2 b \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^4}{e^6 (a+b x) (d+e x)^{5/2}}+\frac{2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^5}{7 e^6 (a+b x) (d+e x)^{7/2}}+\frac{2 b^5 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{3/2}}{3 e^6 (a+b x)}-\frac{10 b^4 \sqrt{a^2+2 a b x+b^2 x^2} \sqrt{d+e x} (b d-a e)}{e^6 (a+b x)}-\frac{20 b^3 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^2}{e^6 (a+b x) \sqrt{d+e x}} \]

Antiderivative was successfully verified.

[In]  Int[(a^2 + 2*a*b*x + b^2*x^2)^(5/2)/(d + e*x)^(9/2),x]

[Out]

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

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Rubi in Sympy [A]  time = 35.6536, size = 255, normalized size = 0.81 \[ \frac{256 b^{4} \sqrt{d + e x} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{21 e^{5}} + \frac{512 b^{4} \sqrt{d + e x} \left (a e - b d\right ) \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{21 e^{6} \left (a + b x\right )} - \frac{64 b^{3} \left (3 a + 3 b x\right ) \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{21 e^{4} \sqrt{d + e x}} - \frac{32 b^{2} \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{3}{2}}}{21 e^{3} \left (d + e x\right )^{\frac{3}{2}}} - \frac{4 b \left (5 a + 5 b x\right ) \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{3}{2}}}{35 e^{2} \left (d + e x\right )^{\frac{5}{2}}} - \frac{2 \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{5}{2}}}{7 e \left (d + e x\right )^{\frac{7}{2}}} \]

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)**(9/2),x)

[Out]

256*b**4*sqrt(d + e*x)*sqrt(a**2 + 2*a*b*x + b**2*x**2)/(21*e**5) + 512*b**4*sqr
t(d + e*x)*(a*e - b*d)*sqrt(a**2 + 2*a*b*x + b**2*x**2)/(21*e**6*(a + b*x)) - 64
*b**3*(3*a + 3*b*x)*sqrt(a**2 + 2*a*b*x + b**2*x**2)/(21*e**4*sqrt(d + e*x)) - 3
2*b**2*(a**2 + 2*a*b*x + b**2*x**2)**(3/2)/(21*e**3*(d + e*x)**(3/2)) - 4*b*(5*a
+ 5*b*x)*(a**2 + 2*a*b*x + b**2*x**2)**(3/2)/(35*e**2*(d + e*x)**(5/2)) - 2*(a*
*2 + 2*a*b*x + b**2*x**2)**(5/2)/(7*e*(d + e*x)**(7/2))

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Mathematica [A]  time = 0.346399, size = 139, normalized size = 0.44 \[ \frac{2 \left ((a+b x)^2\right )^{5/2} \sqrt{d+e x} \left (-7 b^4 (14 b d-15 a e)-\frac{210 b^3 (b d-a e)^2}{d+e x}+\frac{70 b^2 (b d-a e)^3}{(d+e x)^2}-\frac{21 b (b d-a e)^4}{(d+e x)^3}+\frac{3 (b d-a e)^5}{(d+e x)^4}+7 b^5 e x\right )}{21 e^6 (a+b x)^5} \]

Antiderivative was successfully verified.

[In]  Integrate[(a^2 + 2*a*b*x + b^2*x^2)^(5/2)/(d + e*x)^(9/2),x]

[Out]

(2*((a + b*x)^2)^(5/2)*Sqrt[d + e*x]*(-7*b^4*(14*b*d - 15*a*e) + 7*b^5*e*x + (3*
(b*d - a*e)^5)/(d + e*x)^4 - (21*b*(b*d - a*e)^4)/(d + e*x)^3 + (70*b^2*(b*d - a
*e)^3)/(d + e*x)^2 - (210*b^3*(b*d - a*e)^2)/(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 \[ -{\frac{-14\,{x}^{5}{b}^{5}{e}^{5}-210\,{x}^{4}a{b}^{4}{e}^{5}+140\,{x}^{4}{b}^{5}d{e}^{4}+420\,{x}^{3}{a}^{2}{b}^{3}{e}^{5}-1680\,{x}^{3}a{b}^{4}d{e}^{4}+1120\,{x}^{3}{b}^{5}{d}^{2}{e}^{3}+140\,{x}^{2}{a}^{3}{b}^{2}{e}^{5}+840\,{x}^{2}{a}^{2}{b}^{3}d{e}^{4}-3360\,{x}^{2}a{b}^{4}{d}^{2}{e}^{3}+2240\,{x}^{2}{b}^{5}{d}^{3}{e}^{2}+42\,x{a}^{4}b{e}^{5}+112\,x{a}^{3}{b}^{2}d{e}^{4}+672\,x{a}^{2}{b}^{3}{d}^{2}{e}^{3}-2688\,xa{b}^{4}{d}^{3}{e}^{2}+1792\,x{b}^{5}{d}^{4}e+6\,{a}^{5}{e}^{5}+12\,{a}^{4}bd{e}^{4}+32\,{a}^{3}{b}^{2}{d}^{2}{e}^{3}+192\,{a}^{2}{b}^{3}{d}^{3}{e}^{2}-768\,a{b}^{4}{d}^{4}e+512\,{b}^{5}{d}^{5}}{21\, \left ( bx+a \right ) ^{5}{e}^{6}} \left ( \left ( bx+a \right ) ^{2} \right ) ^{{\frac{5}{2}}} \left ( ex+d \right ) ^{-{\frac{7}{2}}}} \]

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)^(9/2),x)

[Out]

-2/21/(e*x+d)^(7/2)*(-7*b^5*e^5*x^5-105*a*b^4*e^5*x^4+70*b^5*d*e^4*x^4+210*a^2*b
^3*e^5*x^3-840*a*b^4*d*e^4*x^3+560*b^5*d^2*e^3*x^3+70*a^3*b^2*e^5*x^2+420*a^2*b^
3*d*e^4*x^2-1680*a*b^4*d^2*e^3*x^2+1120*b^5*d^3*e^2*x^2+21*a^4*b*e^5*x+56*a^3*b^
2*d*e^4*x+336*a^2*b^3*d^2*e^3*x-1344*a*b^4*d^3*e^2*x+896*b^5*d^4*e*x+3*a^5*e^5+6
*a^4*b*d*e^4+16*a^3*b^2*d^2*e^3+96*a^2*b^3*d^3*e^2-384*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 = 0.751033, size = 396, normalized size = 1.26 \[ \frac{2 \,{\left (7 \, b^{5} e^{5} x^{5} - 256 \, b^{5} d^{5} + 384 \, a b^{4} d^{4} e - 96 \, a^{2} b^{3} d^{3} e^{2} - 16 \, a^{3} b^{2} d^{2} e^{3} - 6 \, a^{4} b d e^{4} - 3 \, a^{5} e^{5} - 35 \,{\left (2 \, b^{5} d e^{4} - 3 \, a b^{4} e^{5}\right )} x^{4} - 70 \,{\left (8 \, b^{5} d^{2} e^{3} - 12 \, a b^{4} d e^{4} + 3 \, a^{2} b^{3} e^{5}\right )} x^{3} - 70 \,{\left (16 \, b^{5} d^{3} e^{2} - 24 \, a b^{4} d^{2} e^{3} + 6 \, a^{2} b^{3} d e^{4} + a^{3} b^{2} e^{5}\right )} x^{2} - 7 \,{\left (128 \, b^{5} d^{4} e - 192 \, a b^{4} d^{3} e^{2} + 48 \, a^{2} b^{3} d^{2} e^{3} + 8 \, a^{3} b^{2} d e^{4} + 3 \, a^{4} b e^{5}\right )} x\right )}}{21 \,{\left (e^{9} x^{3} + 3 \, d e^{8} x^{2} + 3 \, d^{2} e^{7} x + d^{3} e^{6}\right )} \sqrt{e x + d}} \]

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)^(9/2),x, algorithm="maxima")

[Out]

2/21*(7*b^5*e^5*x^5 - 256*b^5*d^5 + 384*a*b^4*d^4*e - 96*a^2*b^3*d^3*e^2 - 16*a^
3*b^2*d^2*e^3 - 6*a^4*b*d*e^4 - 3*a^5*e^5 - 35*(2*b^5*d*e^4 - 3*a*b^4*e^5)*x^4 -
70*(8*b^5*d^2*e^3 - 12*a*b^4*d*e^4 + 3*a^2*b^3*e^5)*x^3 - 70*(16*b^5*d^3*e^2 -
24*a*b^4*d^2*e^3 + 6*a^2*b^3*d*e^4 + a^3*b^2*e^5)*x^2 - 7*(128*b^5*d^4*e - 192*a
*b^4*d^3*e^2 + 48*a^2*b^3*d^2*e^3 + 8*a^3*b^2*d*e^4 + 3*a^4*b*e^5)*x)/((e^9*x^3
+ 3*d*e^8*x^2 + 3*d^2*e^7*x + d^3*e^6)*sqrt(e*x + d))

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Fricas [A]  time = 0.208651, size = 396, normalized size = 1.26 \[ \frac{2 \,{\left (7 \, b^{5} e^{5} x^{5} - 256 \, b^{5} d^{5} + 384 \, a b^{4} d^{4} e - 96 \, a^{2} b^{3} d^{3} e^{2} - 16 \, a^{3} b^{2} d^{2} e^{3} - 6 \, a^{4} b d e^{4} - 3 \, a^{5} e^{5} - 35 \,{\left (2 \, b^{5} d e^{4} - 3 \, a b^{4} e^{5}\right )} x^{4} - 70 \,{\left (8 \, b^{5} d^{2} e^{3} - 12 \, a b^{4} d e^{4} + 3 \, a^{2} b^{3} e^{5}\right )} x^{3} - 70 \,{\left (16 \, b^{5} d^{3} e^{2} - 24 \, a b^{4} d^{2} e^{3} + 6 \, a^{2} b^{3} d e^{4} + a^{3} b^{2} e^{5}\right )} x^{2} - 7 \,{\left (128 \, b^{5} d^{4} e - 192 \, a b^{4} d^{3} e^{2} + 48 \, a^{2} b^{3} d^{2} e^{3} + 8 \, a^{3} b^{2} d e^{4} + 3 \, a^{4} b e^{5}\right )} x\right )}}{21 \,{\left (e^{9} x^{3} + 3 \, d e^{8} x^{2} + 3 \, d^{2} e^{7} x + d^{3} e^{6}\right )} \sqrt{e x + d}} \]

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)^(9/2),x, algorithm="fricas")

[Out]

2/21*(7*b^5*e^5*x^5 - 256*b^5*d^5 + 384*a*b^4*d^4*e - 96*a^2*b^3*d^3*e^2 - 16*a^
3*b^2*d^2*e^3 - 6*a^4*b*d*e^4 - 3*a^5*e^5 - 35*(2*b^5*d*e^4 - 3*a*b^4*e^5)*x^4 -
70*(8*b^5*d^2*e^3 - 12*a*b^4*d*e^4 + 3*a^2*b^3*e^5)*x^3 - 70*(16*b^5*d^3*e^2 -
24*a*b^4*d^2*e^3 + 6*a^2*b^3*d*e^4 + a^3*b^2*e^5)*x^2 - 7*(128*b^5*d^4*e - 192*a
*b^4*d^3*e^2 + 48*a^2*b^3*d^2*e^3 + 8*a^3*b^2*d*e^4 + 3*a^4*b*e^5)*x)/((e^9*x^3
+ 3*d*e^8*x^2 + 3*d^2*e^7*x + d^3*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)**(9/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.232532, size = 616, normalized size = 1.96 \[ \frac{2}{3} \,{\left ({\left (x e + d\right )}^{\frac{3}{2}} b^{5} e^{12}{\rm sign}\left (b x + a\right ) - 15 \, \sqrt{x e + d} b^{5} d e^{12}{\rm sign}\left (b x + a\right ) + 15 \, \sqrt{x e + d} a b^{4} e^{13}{\rm sign}\left (b x + a\right )\right )} e^{\left (-18\right )} - \frac{2 \,{\left (210 \,{\left (x e + d\right )}^{3} b^{5} d^{2}{\rm sign}\left (b x + a\right ) - 70 \,{\left (x e + d\right )}^{2} b^{5} d^{3}{\rm sign}\left (b x + a\right ) + 21 \,{\left (x e + d\right )} b^{5} d^{4}{\rm sign}\left (b x + a\right ) - 3 \, b^{5} d^{5}{\rm sign}\left (b x + a\right ) - 420 \,{\left (x e + d\right )}^{3} a b^{4} d e{\rm sign}\left (b x + a\right ) + 210 \,{\left (x e + d\right )}^{2} a b^{4} d^{2} e{\rm sign}\left (b x + a\right ) - 84 \,{\left (x e + d\right )} a b^{4} d^{3} e{\rm sign}\left (b x + a\right ) + 15 \, a b^{4} d^{4} e{\rm sign}\left (b x + a\right ) + 210 \,{\left (x e + d\right )}^{3} a^{2} b^{3} e^{2}{\rm sign}\left (b x + a\right ) - 210 \,{\left (x e + d\right )}^{2} a^{2} b^{3} d e^{2}{\rm sign}\left (b x + a\right ) + 126 \,{\left (x e + d\right )} a^{2} b^{3} d^{2} e^{2}{\rm sign}\left (b x + a\right ) - 30 \, a^{2} b^{3} d^{3} e^{2}{\rm sign}\left (b x + a\right ) + 70 \,{\left (x e + d\right )}^{2} a^{3} b^{2} e^{3}{\rm sign}\left (b x + a\right ) - 84 \,{\left (x e + d\right )} a^{3} b^{2} d e^{3}{\rm sign}\left (b x + a\right ) + 30 \, a^{3} b^{2} d^{2} e^{3}{\rm sign}\left (b x + a\right ) + 21 \,{\left (x e + d\right )} a^{4} b e^{4}{\rm sign}\left (b x + a\right ) - 15 \, a^{4} b d e^{4}{\rm sign}\left (b x + a\right ) + 3 \, a^{5} e^{5}{\rm sign}\left (b x + a\right )\right )} e^{\left (-6\right )}}{21 \,{\left (x e + d\right )}^{\frac{7}{2}}} \]

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)^(9/2),x, algorithm="giac")

[Out]

2/3*((x*e + d)^(3/2)*b^5*e^12*sign(b*x + a) - 15*sqrt(x*e + d)*b^5*d*e^12*sign(b
*x + a) + 15*sqrt(x*e + d)*a*b^4*e^13*sign(b*x + a))*e^(-18) - 2/21*(210*(x*e +
d)^3*b^5*d^2*sign(b*x + a) - 70*(x*e + d)^2*b^5*d^3*sign(b*x + a) + 21*(x*e + d)
*b^5*d^4*sign(b*x + a) - 3*b^5*d^5*sign(b*x + a) - 420*(x*e + d)^3*a*b^4*d*e*sig
n(b*x + a) + 210*(x*e + d)^2*a*b^4*d^2*e*sign(b*x + a) - 84*(x*e + d)*a*b^4*d^3*
e*sign(b*x + a) + 15*a*b^4*d^4*e*sign(b*x + a) + 210*(x*e + d)^3*a^2*b^3*e^2*sig
n(b*x + a) - 210*(x*e + d)^2*a^2*b^3*d*e^2*sign(b*x + a) + 126*(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) + 70*(x*e + d)^2*a^3*b
^2*e^3*sign(b*x + a) - 84*(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) + 21*(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)^(7/2)