3.1691 \(\int \frac{\left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^{15/2}} \, dx\)

Optimal. Leaf size=318 \[ \frac{20 b^2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^3}{9 e^6 (a+b x) (d+e x)^{9/2}}-\frac{10 b \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^4}{11 e^6 (a+b x) (d+e x)^{11/2}}+\frac{2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^5}{13 e^6 (a+b x) (d+e x)^{13/2}}-\frac{2 b^5 \sqrt{a^2+2 a b x+b^2 x^2}}{3 e^6 (a+b x) (d+e x)^{3/2}}+\frac{2 b^4 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)}{e^6 (a+b x) (d+e x)^{5/2}}-\frac{20 b^3 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^2}{7 e^6 (a+b x) (d+e x)^{7/2}} \]

[Out]

(2*(b*d - a*e)^5*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(13*e^6*(a + b*x)*(d + e*x)^(13/
2)) - (10*b*(b*d - a*e)^4*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(11*e^6*(a + b*x)*(d +
e*x)^(11/2)) + (20*b^2*(b*d - a*e)^3*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(9*e^6*(a +
b*x)*(d + e*x)^(9/2)) - (20*b^3*(b*d - a*e)^2*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(7*
e^6*(a + b*x)*(d + e*x)^(7/2)) + (2*b^4*(b*d - a*e)*Sqrt[a^2 + 2*a*b*x + b^2*x^2
])/(e^6*(a + b*x)*(d + e*x)^(5/2)) - (2*b^5*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(3*e^
6*(a + b*x)*(d + e*x)^(3/2))

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Rubi [A]  time = 0.290422, antiderivative size = 318, 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 \[ \frac{20 b^2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^3}{9 e^6 (a+b x) (d+e x)^{9/2}}-\frac{10 b \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^4}{11 e^6 (a+b x) (d+e x)^{11/2}}+\frac{2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^5}{13 e^6 (a+b x) (d+e x)^{13/2}}-\frac{2 b^5 \sqrt{a^2+2 a b x+b^2 x^2}}{3 e^6 (a+b x) (d+e x)^{3/2}}+\frac{2 b^4 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)}{e^6 (a+b x) (d+e x)^{5/2}}-\frac{20 b^3 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^2}{7 e^6 (a+b x) (d+e x)^{7/2}} \]

Antiderivative was successfully verified.

[In]  Int[(a^2 + 2*a*b*x + b^2*x^2)^(5/2)/(d + e*x)^(15/2),x]

[Out]

(2*(b*d - a*e)^5*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(13*e^6*(a + b*x)*(d + e*x)^(13/
2)) - (10*b*(b*d - a*e)^4*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(11*e^6*(a + b*x)*(d +
e*x)^(11/2)) + (20*b^2*(b*d - a*e)^3*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(9*e^6*(a +
b*x)*(d + e*x)^(9/2)) - (20*b^3*(b*d - a*e)^2*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(7*
e^6*(a + b*x)*(d + e*x)^(7/2)) + (2*b^4*(b*d - a*e)*Sqrt[a^2 + 2*a*b*x + b^2*x^2
])/(e^6*(a + b*x)*(d + e*x)^(5/2)) - (2*b^5*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(3*e^
6*(a + b*x)*(d + e*x)^(3/2))

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Rubi in Sympy [A]  time = 36.7291, size = 255, normalized size = 0.8 \[ - \frac{1280 b^{4} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{9009 e^{5} \left (d + e x\right )^{\frac{5}{2}}} + \frac{512 b^{4} \left (a e - b d\right ) \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{9009 e^{6} \left (a + b x\right ) \left (d + e x\right )^{\frac{5}{2}}} - \frac{320 b^{3} \left (3 a + 3 b x\right ) \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{9009 e^{4} \left (d + e x\right )^{\frac{7}{2}}} - \frac{160 b^{2} \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{3}{2}}}{1287 e^{3} \left (d + e x\right )^{\frac{9}{2}}} - \frac{4 b \left (5 a + 5 b x\right ) \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{3}{2}}}{143 e^{2} \left (d + e x\right )^{\frac{11}{2}}} - \frac{2 \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{5}{2}}}{13 e \left (d + e x\right )^{\frac{13}{2}}} \]

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)**(15/2),x)

[Out]

-1280*b**4*sqrt(a**2 + 2*a*b*x + b**2*x**2)/(9009*e**5*(d + e*x)**(5/2)) + 512*b
**4*(a*e - b*d)*sqrt(a**2 + 2*a*b*x + b**2*x**2)/(9009*e**6*(a + b*x)*(d + e*x)*
*(5/2)) - 320*b**3*(3*a + 3*b*x)*sqrt(a**2 + 2*a*b*x + b**2*x**2)/(9009*e**4*(d
+ e*x)**(7/2)) - 160*b**2*(a**2 + 2*a*b*x + b**2*x**2)**(3/2)/(1287*e**3*(d + e*
x)**(9/2)) - 4*b*(5*a + 5*b*x)*(a**2 + 2*a*b*x + b**2*x**2)**(3/2)/(143*e**2*(d
+ e*x)**(11/2)) - 2*(a**2 + 2*a*b*x + b**2*x**2)**(5/2)/(13*e*(d + e*x)**(13/2))

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Mathematica [A]  time = 0.352407, size = 141, normalized size = 0.44 \[ \frac{2 \left ((a+b x)^2\right )^{5/2} \left (9009 b^4 (d+e x)^4 (b d-a e)-12870 b^3 (d+e x)^3 (b d-a e)^2+10010 b^2 (d+e x)^2 (b d-a e)^3-4095 b (d+e x) (b d-a e)^4+693 (b d-a e)^5-3003 b^5 (d+e x)^5\right )}{9009 e^6 (a+b x)^5 (d+e x)^{13/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(a^2 + 2*a*b*x + b^2*x^2)^(5/2)/(d + e*x)^(15/2),x]

[Out]

(2*((a + b*x)^2)^(5/2)*(693*(b*d - a*e)^5 - 4095*b*(b*d - a*e)^4*(d + e*x) + 100
10*b^2*(b*d - a*e)^3*(d + e*x)^2 - 12870*b^3*(b*d - a*e)^2*(d + e*x)^3 + 9009*b^
4*(b*d - a*e)*(d + e*x)^4 - 3003*b^5*(d + e*x)^5))/(9009*e^6*(a + b*x)^5*(d + e*
x)^(13/2))

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Maple [A]  time = 0.012, size = 289, normalized size = 0.9 \[ -{\frac{6006\,{x}^{5}{b}^{5}{e}^{5}+18018\,{x}^{4}a{b}^{4}{e}^{5}+12012\,{x}^{4}{b}^{5}d{e}^{4}+25740\,{x}^{3}{a}^{2}{b}^{3}{e}^{5}+20592\,{x}^{3}a{b}^{4}d{e}^{4}+13728\,{x}^{3}{b}^{5}{d}^{2}{e}^{3}+20020\,{x}^{2}{a}^{3}{b}^{2}{e}^{5}+17160\,{x}^{2}{a}^{2}{b}^{3}d{e}^{4}+13728\,{x}^{2}a{b}^{4}{d}^{2}{e}^{3}+9152\,{x}^{2}{b}^{5}{d}^{3}{e}^{2}+8190\,x{a}^{4}b{e}^{5}+7280\,x{a}^{3}{b}^{2}d{e}^{4}+6240\,x{a}^{2}{b}^{3}{d}^{2}{e}^{3}+4992\,xa{b}^{4}{d}^{3}{e}^{2}+3328\,x{b}^{5}{d}^{4}e+1386\,{a}^{5}{e}^{5}+1260\,{a}^{4}bd{e}^{4}+1120\,{a}^{3}{b}^{2}{d}^{2}{e}^{3}+960\,{a}^{2}{b}^{3}{d}^{3}{e}^{2}+768\,a{b}^{4}{d}^{4}e+512\,{b}^{5}{d}^{5}}{9009\, \left ( bx+a \right ) ^{5}{e}^{6}} \left ( \left ( bx+a \right ) ^{2} \right ) ^{{\frac{5}{2}}} \left ( ex+d \right ) ^{-{\frac{13}{2}}}} \]

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)^(15/2),x)

[Out]

-2/9009/(e*x+d)^(13/2)*(3003*b^5*e^5*x^5+9009*a*b^4*e^5*x^4+6006*b^5*d*e^4*x^4+1
2870*a^2*b^3*e^5*x^3+10296*a*b^4*d*e^4*x^3+6864*b^5*d^2*e^3*x^3+10010*a^3*b^2*e^
5*x^2+8580*a^2*b^3*d*e^4*x^2+6864*a*b^4*d^2*e^3*x^2+4576*b^5*d^3*e^2*x^2+4095*a^
4*b*e^5*x+3640*a^3*b^2*d*e^4*x+3120*a^2*b^3*d^2*e^3*x+2496*a*b^4*d^3*e^2*x+1664*
b^5*d^4*e*x+693*a^5*e^5+630*a^4*b*d*e^4+560*a^3*b^2*d^2*e^3+480*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 = 0.749511, size = 441, normalized size = 1.39 \[ -\frac{2 \,{\left (3003 \, b^{5} e^{5} x^{5} + 256 \, b^{5} d^{5} + 384 \, a b^{4} d^{4} e + 480 \, a^{2} b^{3} d^{3} e^{2} + 560 \, a^{3} b^{2} d^{2} e^{3} + 630 \, a^{4} b d e^{4} + 693 \, a^{5} e^{5} + 3003 \,{\left (2 \, b^{5} d e^{4} + 3 \, a b^{4} e^{5}\right )} x^{4} + 858 \,{\left (8 \, b^{5} d^{2} e^{3} + 12 \, a b^{4} d e^{4} + 15 \, a^{2} b^{3} e^{5}\right )} x^{3} + 286 \,{\left (16 \, b^{5} d^{3} e^{2} + 24 \, a b^{4} d^{2} e^{3} + 30 \, a^{2} b^{3} d e^{4} + 35 \, a^{3} b^{2} e^{5}\right )} x^{2} + 13 \,{\left (128 \, b^{5} d^{4} e + 192 \, a b^{4} d^{3} e^{2} + 240 \, a^{2} b^{3} d^{2} e^{3} + 280 \, a^{3} b^{2} d e^{4} + 315 \, a^{4} b e^{5}\right )} x\right )}}{9009 \,{\left (e^{12} x^{6} + 6 \, d e^{11} x^{5} + 15 \, d^{2} e^{10} x^{4} + 20 \, d^{3} e^{9} x^{3} + 15 \, d^{4} e^{8} x^{2} + 6 \, d^{5} e^{7} x + d^{6} e^{6}\right )} \sqrt{e x + d}} \]

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)^(15/2),x, algorithm="maxima")

[Out]

-2/9009*(3003*b^5*e^5*x^5 + 256*b^5*d^5 + 384*a*b^4*d^4*e + 480*a^2*b^3*d^3*e^2
+ 560*a^3*b^2*d^2*e^3 + 630*a^4*b*d*e^4 + 693*a^5*e^5 + 3003*(2*b^5*d*e^4 + 3*a*
b^4*e^5)*x^4 + 858*(8*b^5*d^2*e^3 + 12*a*b^4*d*e^4 + 15*a^2*b^3*e^5)*x^3 + 286*(
16*b^5*d^3*e^2 + 24*a*b^4*d^2*e^3 + 30*a^2*b^3*d*e^4 + 35*a^3*b^2*e^5)*x^2 + 13*
(128*b^5*d^4*e + 192*a*b^4*d^3*e^2 + 240*a^2*b^3*d^2*e^3 + 280*a^3*b^2*d*e^4 + 3
15*a^4*b*e^5)*x)/((e^12*x^6 + 6*d*e^11*x^5 + 15*d^2*e^10*x^4 + 20*d^3*e^9*x^3 +
15*d^4*e^8*x^2 + 6*d^5*e^7*x + d^6*e^6)*sqrt(e*x + d))

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Fricas [A]  time = 0.210609, size = 441, normalized size = 1.39 \[ -\frac{2 \,{\left (3003 \, b^{5} e^{5} x^{5} + 256 \, b^{5} d^{5} + 384 \, a b^{4} d^{4} e + 480 \, a^{2} b^{3} d^{3} e^{2} + 560 \, a^{3} b^{2} d^{2} e^{3} + 630 \, a^{4} b d e^{4} + 693 \, a^{5} e^{5} + 3003 \,{\left (2 \, b^{5} d e^{4} + 3 \, a b^{4} e^{5}\right )} x^{4} + 858 \,{\left (8 \, b^{5} d^{2} e^{3} + 12 \, a b^{4} d e^{4} + 15 \, a^{2} b^{3} e^{5}\right )} x^{3} + 286 \,{\left (16 \, b^{5} d^{3} e^{2} + 24 \, a b^{4} d^{2} e^{3} + 30 \, a^{2} b^{3} d e^{4} + 35 \, a^{3} b^{2} e^{5}\right )} x^{2} + 13 \,{\left (128 \, b^{5} d^{4} e + 192 \, a b^{4} d^{3} e^{2} + 240 \, a^{2} b^{3} d^{2} e^{3} + 280 \, a^{3} b^{2} d e^{4} + 315 \, a^{4} b e^{5}\right )} x\right )}}{9009 \,{\left (e^{12} x^{6} + 6 \, d e^{11} x^{5} + 15 \, d^{2} e^{10} x^{4} + 20 \, d^{3} e^{9} x^{3} + 15 \, d^{4} e^{8} x^{2} + 6 \, d^{5} e^{7} x + d^{6} e^{6}\right )} \sqrt{e x + d}} \]

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)^(15/2),x, algorithm="fricas")

[Out]

-2/9009*(3003*b^5*e^5*x^5 + 256*b^5*d^5 + 384*a*b^4*d^4*e + 480*a^2*b^3*d^3*e^2
+ 560*a^3*b^2*d^2*e^3 + 630*a^4*b*d*e^4 + 693*a^5*e^5 + 3003*(2*b^5*d*e^4 + 3*a*
b^4*e^5)*x^4 + 858*(8*b^5*d^2*e^3 + 12*a*b^4*d*e^4 + 15*a^2*b^3*e^5)*x^3 + 286*(
16*b^5*d^3*e^2 + 24*a*b^4*d^2*e^3 + 30*a^2*b^3*d*e^4 + 35*a^3*b^2*e^5)*x^2 + 13*
(128*b^5*d^4*e + 192*a*b^4*d^3*e^2 + 240*a^2*b^3*d^2*e^3 + 280*a^3*b^2*d*e^4 + 3
15*a^4*b*e^5)*x)/((e^12*x^6 + 6*d*e^11*x^5 + 15*d^2*e^10*x^4 + 20*d^3*e^9*x^3 +
15*d^4*e^8*x^2 + 6*d^5*e^7*x + d^6*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)**(15/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.23419, size = 603, normalized size = 1.9 \[ -\frac{2 \,{\left (3003 \,{\left (x e + d\right )}^{5} b^{5}{\rm sign}\left (b x + a\right ) - 9009 \,{\left (x e + d\right )}^{4} b^{5} d{\rm sign}\left (b x + a\right ) + 12870 \,{\left (x e + d\right )}^{3} b^{5} d^{2}{\rm sign}\left (b x + a\right ) - 10010 \,{\left (x e + d\right )}^{2} b^{5} d^{3}{\rm sign}\left (b x + a\right ) + 4095 \,{\left (x e + d\right )} b^{5} d^{4}{\rm sign}\left (b x + a\right ) - 693 \, b^{5} d^{5}{\rm sign}\left (b x + a\right ) + 9009 \,{\left (x e + d\right )}^{4} a b^{4} e{\rm sign}\left (b x + a\right ) - 25740 \,{\left (x e + d\right )}^{3} a b^{4} d e{\rm sign}\left (b x + a\right ) + 30030 \,{\left (x e + d\right )}^{2} a b^{4} d^{2} e{\rm sign}\left (b x + a\right ) - 16380 \,{\left (x e + d\right )} a b^{4} d^{3} e{\rm sign}\left (b x + a\right ) + 3465 \, a b^{4} d^{4} e{\rm sign}\left (b x + a\right ) + 12870 \,{\left (x e + d\right )}^{3} a^{2} b^{3} e^{2}{\rm sign}\left (b x + a\right ) - 30030 \,{\left (x e + d\right )}^{2} a^{2} b^{3} d e^{2}{\rm sign}\left (b x + a\right ) + 24570 \,{\left (x e + d\right )} a^{2} b^{3} d^{2} e^{2}{\rm sign}\left (b x + a\right ) - 6930 \, a^{2} b^{3} d^{3} e^{2}{\rm sign}\left (b x + a\right ) + 10010 \,{\left (x e + d\right )}^{2} a^{3} b^{2} e^{3}{\rm sign}\left (b x + a\right ) - 16380 \,{\left (x e + d\right )} a^{3} b^{2} d e^{3}{\rm sign}\left (b x + a\right ) + 6930 \, a^{3} b^{2} d^{2} e^{3}{\rm sign}\left (b x + a\right ) + 4095 \,{\left (x e + d\right )} a^{4} b e^{4}{\rm sign}\left (b x + a\right ) - 3465 \, a^{4} b d e^{4}{\rm sign}\left (b x + a\right ) + 693 \, a^{5} e^{5}{\rm sign}\left (b x + a\right )\right )} e^{\left (-6\right )}}{9009 \,{\left (x e + d\right )}^{\frac{13}{2}}} \]

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)^(15/2),x, algorithm="giac")

[Out]

-2/9009*(3003*(x*e + d)^5*b^5*sign(b*x + a) - 9009*(x*e + d)^4*b^5*d*sign(b*x +
a) + 12870*(x*e + d)^3*b^5*d^2*sign(b*x + a) - 10010*(x*e + d)^2*b^5*d^3*sign(b*
x + a) + 4095*(x*e + d)*b^5*d^4*sign(b*x + a) - 693*b^5*d^5*sign(b*x + a) + 9009
*(x*e + d)^4*a*b^4*e*sign(b*x + a) - 25740*(x*e + d)^3*a*b^4*d*e*sign(b*x + a) +
30030*(x*e + d)^2*a*b^4*d^2*e*sign(b*x + a) - 16380*(x*e + d)*a*b^4*d^3*e*sign(
b*x + a) + 3465*a*b^4*d^4*e*sign(b*x + a) + 12870*(x*e + d)^3*a^2*b^3*e^2*sign(b
*x + a) - 30030*(x*e + d)^2*a^2*b^3*d*e^2*sign(b*x + a) + 24570*(x*e + d)*a^2*b^
3*d^2*e^2*sign(b*x + a) - 6930*a^2*b^3*d^3*e^2*sign(b*x + a) + 10010*(x*e + d)^2
*a^3*b^2*e^3*sign(b*x + a) - 16380*(x*e + d)*a^3*b^2*d*e^3*sign(b*x + a) + 6930*
a^3*b^2*d^2*e^3*sign(b*x + a) + 4095*(x*e + d)*a^4*b*e^4*sign(b*x + a) - 3465*a^
4*b*d*e^4*sign(b*x + a) + 693*a^5*e^5*sign(b*x + a))*e^(-6)/(x*e + d)^(13/2)