3.1690 \(\int \frac{\left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^{13/2}} \, dx\)

Optimal. Leaf size=316 \[ \frac{20 b^2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^3}{7 e^6 (a+b x) (d+e x)^{7/2}}-\frac{10 b \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^4}{9 e^6 (a+b x) (d+e x)^{9/2}}+\frac{2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^5}{11 e^6 (a+b x) (d+e x)^{11/2}}-\frac{2 b^5 \sqrt{a^2+2 a b x+b^2 x^2}}{e^6 (a+b x) \sqrt{d+e x}}+\frac{10 b^4 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)}{3 e^6 (a+b x) (d+e x)^{3/2}}-\frac{4 b^3 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^2}{e^6 (a+b x) (d+e x)^{5/2}} \]

[Out]

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

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Rubi [A]  time = 0.294207, 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 \[ \frac{20 b^2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^3}{7 e^6 (a+b x) (d+e x)^{7/2}}-\frac{10 b \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^4}{9 e^6 (a+b x) (d+e x)^{9/2}}+\frac{2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^5}{11 e^6 (a+b x) (d+e x)^{11/2}}-\frac{2 b^5 \sqrt{a^2+2 a b x+b^2 x^2}}{e^6 (a+b x) \sqrt{d+e x}}+\frac{10 b^4 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)}{3 e^6 (a+b x) (d+e x)^{3/2}}-\frac{4 b^3 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^2}{e^6 (a+b x) (d+e x)^{5/2}} \]

Antiderivative was successfully verified.

[In]  Int[(a^2 + 2*a*b*x + b^2*x^2)^(5/2)/(d + e*x)^(13/2),x]

[Out]

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

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Rubi in Sympy [A]  time = 36.6337, size = 255, normalized size = 0.81 \[ - \frac{256 b^{4} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{231 e^{5} \left (d + e x\right )^{\frac{3}{2}}} + \frac{512 b^{4} \left (a e - b d\right ) \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{693 e^{6} \left (a + b x\right ) \left (d + e x\right )^{\frac{3}{2}}} - \frac{64 b^{3} \left (3 a + 3 b x\right ) \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{693 e^{4} \left (d + e x\right )^{\frac{5}{2}}} - \frac{160 b^{2} \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{3}{2}}}{693 e^{3} \left (d + e x\right )^{\frac{7}{2}}} - \frac{4 b \left (5 a + 5 b x\right ) \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{3}{2}}}{99 e^{2} \left (d + e x\right )^{\frac{9}{2}}} - \frac{2 \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{5}{2}}}{11 e \left (d + e x\right )^{\frac{11}{2}}} \]

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)**(13/2),x)

[Out]

-256*b**4*sqrt(a**2 + 2*a*b*x + b**2*x**2)/(231*e**5*(d + e*x)**(3/2)) + 512*b**
4*(a*e - b*d)*sqrt(a**2 + 2*a*b*x + b**2*x**2)/(693*e**6*(a + b*x)*(d + e*x)**(3
/2)) - 64*b**3*(3*a + 3*b*x)*sqrt(a**2 + 2*a*b*x + b**2*x**2)/(693*e**4*(d + e*x
)**(5/2)) - 160*b**2*(a**2 + 2*a*b*x + b**2*x**2)**(3/2)/(693*e**3*(d + e*x)**(7
/2)) - 4*b*(5*a + 5*b*x)*(a**2 + 2*a*b*x + b**2*x**2)**(3/2)/(99*e**2*(d + e*x)*
*(9/2)) - 2*(a**2 + 2*a*b*x + b**2*x**2)**(5/2)/(11*e*(d + e*x)**(11/2))

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Mathematica [A]  time = 0.343924, size = 141, normalized size = 0.45 \[ \frac{2 \left ((a+b x)^2\right )^{5/2} \left (1155 b^4 (d+e x)^4 (b d-a e)-1386 b^3 (d+e x)^3 (b d-a e)^2+990 b^2 (d+e x)^2 (b d-a e)^3-385 b (d+e x) (b d-a e)^4+63 (b d-a e)^5-693 b^5 (d+e x)^5\right )}{693 e^6 (a+b x)^5 (d+e x)^{11/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(a^2 + 2*a*b*x + b^2*x^2)^(5/2)/(d + e*x)^(13/2),x]

[Out]

(2*((a + b*x)^2)^(5/2)*(63*(b*d - a*e)^5 - 385*b*(b*d - a*e)^4*(d + e*x) + 990*b
^2*(b*d - a*e)^3*(d + e*x)^2 - 1386*b^3*(b*d - a*e)^2*(d + e*x)^3 + 1155*b^4*(b*
d - a*e)*(d + e*x)^4 - 693*b^5*(d + e*x)^5))/(693*e^6*(a + b*x)^5*(d + e*x)^(11/
2))

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Maple [A]  time = 0.011, size = 289, normalized size = 0.9 \[ -{\frac{1386\,{x}^{5}{b}^{5}{e}^{5}+2310\,{x}^{4}a{b}^{4}{e}^{5}+4620\,{x}^{4}{b}^{5}d{e}^{4}+2772\,{x}^{3}{a}^{2}{b}^{3}{e}^{5}+3696\,{x}^{3}a{b}^{4}d{e}^{4}+7392\,{x}^{3}{b}^{5}{d}^{2}{e}^{3}+1980\,{x}^{2}{a}^{3}{b}^{2}{e}^{5}+2376\,{x}^{2}{a}^{2}{b}^{3}d{e}^{4}+3168\,{x}^{2}a{b}^{4}{d}^{2}{e}^{3}+6336\,{x}^{2}{b}^{5}{d}^{3}{e}^{2}+770\,x{a}^{4}b{e}^{5}+880\,x{a}^{3}{b}^{2}d{e}^{4}+1056\,x{a}^{2}{b}^{3}{d}^{2}{e}^{3}+1408\,xa{b}^{4}{d}^{3}{e}^{2}+2816\,x{b}^{5}{d}^{4}e+126\,{a}^{5}{e}^{5}+140\,{a}^{4}bd{e}^{4}+160\,{a}^{3}{b}^{2}{d}^{2}{e}^{3}+192\,{a}^{2}{b}^{3}{d}^{3}{e}^{2}+256\,a{b}^{4}{d}^{4}e+512\,{b}^{5}{d}^{5}}{693\, \left ( bx+a \right ) ^{5}{e}^{6}} \left ( \left ( bx+a \right ) ^{2} \right ) ^{{\frac{5}{2}}} \left ( ex+d \right ) ^{-{\frac{11}{2}}}} \]

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)^(13/2),x)

[Out]

-2/693/(e*x+d)^(11/2)*(693*b^5*e^5*x^5+1155*a*b^4*e^5*x^4+2310*b^5*d*e^4*x^4+138
6*a^2*b^3*e^5*x^3+1848*a*b^4*d*e^4*x^3+3696*b^5*d^2*e^3*x^3+990*a^3*b^2*e^5*x^2+
1188*a^2*b^3*d*e^4*x^2+1584*a*b^4*d^2*e^3*x^2+3168*b^5*d^3*e^2*x^2+385*a^4*b*e^5
*x+440*a^3*b^2*d*e^4*x+528*a^2*b^3*d^2*e^3*x+704*a*b^4*d^3*e^2*x+1408*b^5*d^4*e*
x+63*a^5*e^5+70*a^4*b*d*e^4+80*a^3*b^2*d^2*e^3+96*a^2*b^3*d^3*e^2+128*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 = 0.74423, size = 425, normalized size = 1.34 \[ -\frac{2 \,{\left (693 \, b^{5} e^{5} x^{5} + 256 \, b^{5} d^{5} + 128 \, a b^{4} d^{4} e + 96 \, a^{2} b^{3} d^{3} e^{2} + 80 \, a^{3} b^{2} d^{2} e^{3} + 70 \, a^{4} b d e^{4} + 63 \, a^{5} e^{5} + 1155 \,{\left (2 \, b^{5} d e^{4} + a b^{4} e^{5}\right )} x^{4} + 462 \,{\left (8 \, b^{5} d^{2} e^{3} + 4 \, a b^{4} d e^{4} + 3 \, a^{2} b^{3} e^{5}\right )} x^{3} + 198 \,{\left (16 \, b^{5} d^{3} e^{2} + 8 \, a b^{4} d^{2} e^{3} + 6 \, a^{2} b^{3} d e^{4} + 5 \, a^{3} b^{2} e^{5}\right )} x^{2} + 11 \,{\left (128 \, b^{5} d^{4} e + 64 \, a b^{4} d^{3} e^{2} + 48 \, a^{2} b^{3} d^{2} e^{3} + 40 \, a^{3} b^{2} d e^{4} + 35 \, a^{4} b e^{5}\right )} x\right )}}{693 \,{\left (e^{11} x^{5} + 5 \, d e^{10} x^{4} + 10 \, d^{2} e^{9} x^{3} + 10 \, d^{3} e^{8} x^{2} + 5 \, d^{4} e^{7} x + d^{5} e^{6}\right )} \sqrt{e x + d}} \]

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)^(13/2),x, algorithm="maxima")

[Out]

-2/693*(693*b^5*e^5*x^5 + 256*b^5*d^5 + 128*a*b^4*d^4*e + 96*a^2*b^3*d^3*e^2 + 8
0*a^3*b^2*d^2*e^3 + 70*a^4*b*d*e^4 + 63*a^5*e^5 + 1155*(2*b^5*d*e^4 + a*b^4*e^5)
*x^4 + 462*(8*b^5*d^2*e^3 + 4*a*b^4*d*e^4 + 3*a^2*b^3*e^5)*x^3 + 198*(16*b^5*d^3
*e^2 + 8*a*b^4*d^2*e^3 + 6*a^2*b^3*d*e^4 + 5*a^3*b^2*e^5)*x^2 + 11*(128*b^5*d^4*
e + 64*a*b^4*d^3*e^2 + 48*a^2*b^3*d^2*e^3 + 40*a^3*b^2*d*e^4 + 35*a^4*b*e^5)*x)/
((e^11*x^5 + 5*d*e^10*x^4 + 10*d^2*e^9*x^3 + 10*d^3*e^8*x^2 + 5*d^4*e^7*x + d^5*
e^6)*sqrt(e*x + d))

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Fricas [A]  time = 0.208429, size = 425, normalized size = 1.34 \[ -\frac{2 \,{\left (693 \, b^{5} e^{5} x^{5} + 256 \, b^{5} d^{5} + 128 \, a b^{4} d^{4} e + 96 \, a^{2} b^{3} d^{3} e^{2} + 80 \, a^{3} b^{2} d^{2} e^{3} + 70 \, a^{4} b d e^{4} + 63 \, a^{5} e^{5} + 1155 \,{\left (2 \, b^{5} d e^{4} + a b^{4} e^{5}\right )} x^{4} + 462 \,{\left (8 \, b^{5} d^{2} e^{3} + 4 \, a b^{4} d e^{4} + 3 \, a^{2} b^{3} e^{5}\right )} x^{3} + 198 \,{\left (16 \, b^{5} d^{3} e^{2} + 8 \, a b^{4} d^{2} e^{3} + 6 \, a^{2} b^{3} d e^{4} + 5 \, a^{3} b^{2} e^{5}\right )} x^{2} + 11 \,{\left (128 \, b^{5} d^{4} e + 64 \, a b^{4} d^{3} e^{2} + 48 \, a^{2} b^{3} d^{2} e^{3} + 40 \, a^{3} b^{2} d e^{4} + 35 \, a^{4} b e^{5}\right )} x\right )}}{693 \,{\left (e^{11} x^{5} + 5 \, d e^{10} x^{4} + 10 \, d^{2} e^{9} x^{3} + 10 \, d^{3} e^{8} x^{2} + 5 \, d^{4} e^{7} x + d^{5} e^{6}\right )} \sqrt{e x + d}} \]

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)^(13/2),x, algorithm="fricas")

[Out]

-2/693*(693*b^5*e^5*x^5 + 256*b^5*d^5 + 128*a*b^4*d^4*e + 96*a^2*b^3*d^3*e^2 + 8
0*a^3*b^2*d^2*e^3 + 70*a^4*b*d*e^4 + 63*a^5*e^5 + 1155*(2*b^5*d*e^4 + a*b^4*e^5)
*x^4 + 462*(8*b^5*d^2*e^3 + 4*a*b^4*d*e^4 + 3*a^2*b^3*e^5)*x^3 + 198*(16*b^5*d^3
*e^2 + 8*a*b^4*d^2*e^3 + 6*a^2*b^3*d*e^4 + 5*a^3*b^2*e^5)*x^2 + 11*(128*b^5*d^4*
e + 64*a*b^4*d^3*e^2 + 48*a^2*b^3*d^2*e^3 + 40*a^3*b^2*d*e^4 + 35*a^4*b*e^5)*x)/
((e^11*x^5 + 5*d*e^10*x^4 + 10*d^2*e^9*x^3 + 10*d^3*e^8*x^2 + 5*d^4*e^7*x + d^5*
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)**(13/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.23275, size = 603, normalized size = 1.91 \[ -\frac{2 \,{\left (693 \,{\left (x e + d\right )}^{5} b^{5}{\rm sign}\left (b x + a\right ) - 1155 \,{\left (x e + d\right )}^{4} b^{5} d{\rm sign}\left (b x + a\right ) + 1386 \,{\left (x e + d\right )}^{3} b^{5} d^{2}{\rm sign}\left (b x + a\right ) - 990 \,{\left (x e + d\right )}^{2} b^{5} d^{3}{\rm sign}\left (b x + a\right ) + 385 \,{\left (x e + d\right )} b^{5} d^{4}{\rm sign}\left (b x + a\right ) - 63 \, b^{5} d^{5}{\rm sign}\left (b x + a\right ) + 1155 \,{\left (x e + d\right )}^{4} a b^{4} e{\rm sign}\left (b x + a\right ) - 2772 \,{\left (x e + d\right )}^{3} a b^{4} d e{\rm sign}\left (b x + a\right ) + 2970 \,{\left (x e + d\right )}^{2} a b^{4} d^{2} e{\rm sign}\left (b x + a\right ) - 1540 \,{\left (x e + d\right )} a b^{4} d^{3} e{\rm sign}\left (b x + a\right ) + 315 \, a b^{4} d^{4} e{\rm sign}\left (b x + a\right ) + 1386 \,{\left (x e + d\right )}^{3} a^{2} b^{3} e^{2}{\rm sign}\left (b x + a\right ) - 2970 \,{\left (x e + d\right )}^{2} a^{2} b^{3} d e^{2}{\rm sign}\left (b x + a\right ) + 2310 \,{\left (x e + d\right )} a^{2} b^{3} d^{2} e^{2}{\rm sign}\left (b x + a\right ) - 630 \, a^{2} b^{3} d^{3} e^{2}{\rm sign}\left (b x + a\right ) + 990 \,{\left (x e + d\right )}^{2} a^{3} b^{2} e^{3}{\rm sign}\left (b x + a\right ) - 1540 \,{\left (x e + d\right )} a^{3} b^{2} d e^{3}{\rm sign}\left (b x + a\right ) + 630 \, a^{3} b^{2} d^{2} e^{3}{\rm sign}\left (b x + a\right ) + 385 \,{\left (x e + d\right )} a^{4} b e^{4}{\rm sign}\left (b x + a\right ) - 315 \, a^{4} b d e^{4}{\rm sign}\left (b x + a\right ) + 63 \, a^{5} e^{5}{\rm sign}\left (b x + a\right )\right )} e^{\left (-6\right )}}{693 \,{\left (x e + d\right )}^{\frac{11}{2}}} \]

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)^(13/2),x, algorithm="giac")

[Out]

-2/693*(693*(x*e + d)^5*b^5*sign(b*x + a) - 1155*(x*e + d)^4*b^5*d*sign(b*x + a)
+ 1386*(x*e + d)^3*b^5*d^2*sign(b*x + a) - 990*(x*e + d)^2*b^5*d^3*sign(b*x + a
) + 385*(x*e + d)*b^5*d^4*sign(b*x + a) - 63*b^5*d^5*sign(b*x + a) + 1155*(x*e +
d)^4*a*b^4*e*sign(b*x + a) - 2772*(x*e + d)^3*a*b^4*d*e*sign(b*x + a) + 2970*(x
*e + d)^2*a*b^4*d^2*e*sign(b*x + a) - 1540*(x*e + d)*a*b^4*d^3*e*sign(b*x + a) +
315*a*b^4*d^4*e*sign(b*x + a) + 1386*(x*e + d)^3*a^2*b^3*e^2*sign(b*x + a) - 29
70*(x*e + d)^2*a^2*b^3*d*e^2*sign(b*x + a) + 2310*(x*e + d)*a^2*b^3*d^2*e^2*sign
(b*x + a) - 630*a^2*b^3*d^3*e^2*sign(b*x + a) + 990*(x*e + d)^2*a^3*b^2*e^3*sign
(b*x + a) - 1540*(x*e + d)*a^3*b^2*d*e^3*sign(b*x + a) + 630*a^3*b^2*d^2*e^3*sig
n(b*x + a) + 385*(x*e + d)*a^4*b*e^4*sign(b*x + a) - 315*a^4*b*d*e^4*sign(b*x +
a) + 63*a^5*e^5*sign(b*x + a))*e^(-6)/(x*e + d)^(11/2)