3.1575 \(\int \frac{\left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^{12}} \, dx\)

Optimal. Leaf size=308 \[ \frac{10 b^2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^3}{9 e^6 (a+b x) (d+e x)^9}-\frac{b \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^4}{2 e^6 (a+b x) (d+e x)^{10}}+\frac{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^5}{11 e^6 (a+b x) (d+e x)^{11}}-\frac{b^5 \sqrt{a^2+2 a b x+b^2 x^2}}{6 e^6 (a+b x) (d+e x)^6}+\frac{5 b^4 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)}{7 e^6 (a+b x) (d+e x)^7}-\frac{5 b^3 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^2}{4 e^6 (a+b x) (d+e x)^8} \]

[Out]

((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) -
(b*(b*d - a*e)^4*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(2*e^6*(a + b*x)*(d + e*x)^10) +
(10*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) - (5*b^3*(b*d - a*e)^2*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(4*e^6*(a + b*x)*(d +
e*x)^8) + (5*b^4*(b*d - a*e)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(7*e^6*(a + b*x)*(d
+ e*x)^7) - (b^5*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(6*e^6*(a + b*x)*(d + e*x)^6)

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Rubi [A]  time = 0.415113, antiderivative size = 308, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.071 \[ \frac{10 b^2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^3}{9 e^6 (a+b x) (d+e x)^9}-\frac{b \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^4}{2 e^6 (a+b x) (d+e x)^{10}}+\frac{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^5}{11 e^6 (a+b x) (d+e x)^{11}}-\frac{b^5 \sqrt{a^2+2 a b x+b^2 x^2}}{6 e^6 (a+b x) (d+e x)^6}+\frac{5 b^4 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)}{7 e^6 (a+b x) (d+e x)^7}-\frac{5 b^3 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^2}{4 e^6 (a+b x) (d+e x)^8} \]

Antiderivative was successfully verified.

[In]  Int[(a^2 + 2*a*b*x + b^2*x^2)^(5/2)/(d + e*x)^12,x]

[Out]

((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) -
(b*(b*d - a*e)^4*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(2*e^6*(a + b*x)*(d + e*x)^10) +
(10*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) - (5*b^3*(b*d - a*e)^2*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(4*e^6*(a + b*x)*(d +
e*x)^8) + (5*b^4*(b*d - a*e)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(7*e^6*(a + b*x)*(d
+ e*x)^7) - (b^5*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(6*e^6*(a + b*x)*(d + e*x)^6)

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Rubi in Sympy [A]  time = 35.1049, size = 236, normalized size = 0.77 \[ - \frac{b^{4} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{396 e^{5} \left (d + e x\right )^{7}} + \frac{b^{4} \left (a e - b d\right ) \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{2772 e^{6} \left (a + b x\right ) \left (d + e x\right )^{7}} - \frac{b^{3} \left (3 a + 3 b x\right ) \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{396 e^{4} \left (d + e x\right )^{8}} - \frac{2 b^{2} \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{3}{2}}}{99 e^{3} \left (d + e x\right )^{9}} - \frac{b \left (5 a + 5 b x\right ) \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{3}{2}}}{110 e^{2} \left (d + e x\right )^{10}} - \frac{\left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{5}{2}}}{11 e \left (d + e x\right )^{11}} \]

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)**12,x)

[Out]

-b**4*sqrt(a**2 + 2*a*b*x + b**2*x**2)/(396*e**5*(d + e*x)**7) + b**4*(a*e - b*d
)*sqrt(a**2 + 2*a*b*x + b**2*x**2)/(2772*e**6*(a + b*x)*(d + e*x)**7) - b**3*(3*
a + 3*b*x)*sqrt(a**2 + 2*a*b*x + b**2*x**2)/(396*e**4*(d + e*x)**8) - 2*b**2*(a*
*2 + 2*a*b*x + b**2*x**2)**(3/2)/(99*e**3*(d + e*x)**9) - b*(5*a + 5*b*x)*(a**2
+ 2*a*b*x + b**2*x**2)**(3/2)/(110*e**2*(d + e*x)**10) - (a**2 + 2*a*b*x + b**2*
x**2)**(5/2)/(11*e*(d + e*x)**11)

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Mathematica [A]  time = 0.158017, size = 223, normalized size = 0.72 \[ -\frac{\sqrt{(a+b x)^2} \left (252 a^5 e^5+126 a^4 b e^4 (d+11 e x)+56 a^3 b^2 e^3 \left (d^2+11 d e x+55 e^2 x^2\right )+21 a^2 b^3 e^2 \left (d^3+11 d^2 e x+55 d e^2 x^2+165 e^3 x^3\right )+6 a b^4 e \left (d^4+11 d^3 e x+55 d^2 e^2 x^2+165 d e^3 x^3+330 e^4 x^4\right )+b^5 \left (d^5+11 d^4 e x+55 d^3 e^2 x^2+165 d^2 e^3 x^3+330 d e^4 x^4+462 e^5 x^5\right )\right )}{2772 e^6 (a+b x) (d+e x)^{11}} \]

Antiderivative was successfully verified.

[In]  Integrate[(a^2 + 2*a*b*x + b^2*x^2)^(5/2)/(d + e*x)^12,x]

[Out]

-(Sqrt[(a + b*x)^2]*(252*a^5*e^5 + 126*a^4*b*e^4*(d + 11*e*x) + 56*a^3*b^2*e^3*(
d^2 + 11*d*e*x + 55*e^2*x^2) + 21*a^2*b^3*e^2*(d^3 + 11*d^2*e*x + 55*d*e^2*x^2 +
165*e^3*x^3) + 6*a*b^4*e*(d^4 + 11*d^3*e*x + 55*d^2*e^2*x^2 + 165*d*e^3*x^3 + 3
30*e^4*x^4) + b^5*(d^5 + 11*d^4*e*x + 55*d^3*e^2*x^2 + 165*d^2*e^3*x^3 + 330*d*e
^4*x^4 + 462*e^5*x^5)))/(2772*e^6*(a + b*x)*(d + e*x)^11)

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Maple [A]  time = 0.015, size = 288, normalized size = 0.9 \[ -{\frac{462\,{x}^{5}{b}^{5}{e}^{5}+1980\,{x}^{4}a{b}^{4}{e}^{5}+330\,{x}^{4}{b}^{5}d{e}^{4}+3465\,{x}^{3}{a}^{2}{b}^{3}{e}^{5}+990\,{x}^{3}a{b}^{4}d{e}^{4}+165\,{x}^{3}{b}^{5}{d}^{2}{e}^{3}+3080\,{x}^{2}{a}^{3}{b}^{2}{e}^{5}+1155\,{x}^{2}{a}^{2}{b}^{3}d{e}^{4}+330\,{x}^{2}a{b}^{4}{d}^{2}{e}^{3}+55\,{x}^{2}{b}^{5}{d}^{3}{e}^{2}+1386\,x{a}^{4}b{e}^{5}+616\,x{a}^{3}{b}^{2}d{e}^{4}+231\,x{a}^{2}{b}^{3}{d}^{2}{e}^{3}+66\,xa{b}^{4}{d}^{3}{e}^{2}+11\,x{b}^{5}{d}^{4}e+252\,{a}^{5}{e}^{5}+126\,{a}^{4}bd{e}^{4}+56\,{a}^{3}{b}^{2}{d}^{2}{e}^{3}+21\,{a}^{2}{b}^{3}{d}^{3}{e}^{2}+6\,a{b}^{4}{d}^{4}e+{b}^{5}{d}^{5}}{2772\,{e}^{6} \left ( ex+d \right ) ^{11} \left ( bx+a \right ) ^{5}} \left ( \left ( bx+a \right ) ^{2} \right ) ^{{\frac{5}{2}}}} \]

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)^12,x)

[Out]

-1/2772/e^6*(462*b^5*e^5*x^5+1980*a*b^4*e^5*x^4+330*b^5*d*e^4*x^4+3465*a^2*b^3*e
^5*x^3+990*a*b^4*d*e^4*x^3+165*b^5*d^2*e^3*x^3+3080*a^3*b^2*e^5*x^2+1155*a^2*b^3
*d*e^4*x^2+330*a*b^4*d^2*e^3*x^2+55*b^5*d^3*e^2*x^2+1386*a^4*b*e^5*x+616*a^3*b^2
*d*e^4*x+231*a^2*b^3*d^2*e^3*x+66*a*b^4*d^3*e^2*x+11*b^5*d^4*e*x+252*a^5*e^5+126
*a^4*b*d*e^4+56*a^3*b^2*d^2*e^3+21*a^2*b^3*d^3*e^2+6*a*b^4*d^4*e+b^5*d^5)*((b*x+
a)^2)^(5/2)/(e*x+d)^11/(b*x+a)^5

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

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)^12,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.213478, size = 500, normalized size = 1.62 \[ -\frac{462 \, b^{5} e^{5} x^{5} + b^{5} d^{5} + 6 \, a b^{4} d^{4} e + 21 \, a^{2} b^{3} d^{3} e^{2} + 56 \, a^{3} b^{2} d^{2} e^{3} + 126 \, a^{4} b d e^{4} + 252 \, a^{5} e^{5} + 330 \,{\left (b^{5} d e^{4} + 6 \, a b^{4} e^{5}\right )} x^{4} + 165 \,{\left (b^{5} d^{2} e^{3} + 6 \, a b^{4} d e^{4} + 21 \, a^{2} b^{3} e^{5}\right )} x^{3} + 55 \,{\left (b^{5} d^{3} e^{2} + 6 \, a b^{4} d^{2} e^{3} + 21 \, a^{2} b^{3} d e^{4} + 56 \, a^{3} b^{2} e^{5}\right )} x^{2} + 11 \,{\left (b^{5} d^{4} e + 6 \, a b^{4} d^{3} e^{2} + 21 \, a^{2} b^{3} d^{2} e^{3} + 56 \, a^{3} b^{2} d e^{4} + 126 \, a^{4} b e^{5}\right )} x}{2772 \,{\left (e^{17} x^{11} + 11 \, d e^{16} x^{10} + 55 \, d^{2} e^{15} x^{9} + 165 \, d^{3} e^{14} x^{8} + 330 \, d^{4} e^{13} x^{7} + 462 \, d^{5} e^{12} x^{6} + 462 \, d^{6} e^{11} x^{5} + 330 \, d^{7} e^{10} x^{4} + 165 \, d^{8} e^{9} x^{3} + 55 \, d^{9} e^{8} x^{2} + 11 \, d^{10} e^{7} x + d^{11} e^{6}\right )}} \]

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)^12,x, algorithm="fricas")

[Out]

-1/2772*(462*b^5*e^5*x^5 + b^5*d^5 + 6*a*b^4*d^4*e + 21*a^2*b^3*d^3*e^2 + 56*a^3
*b^2*d^2*e^3 + 126*a^4*b*d*e^4 + 252*a^5*e^5 + 330*(b^5*d*e^4 + 6*a*b^4*e^5)*x^4
+ 165*(b^5*d^2*e^3 + 6*a*b^4*d*e^4 + 21*a^2*b^3*e^5)*x^3 + 55*(b^5*d^3*e^2 + 6*
a*b^4*d^2*e^3 + 21*a^2*b^3*d*e^4 + 56*a^3*b^2*e^5)*x^2 + 11*(b^5*d^4*e + 6*a*b^4
*d^3*e^2 + 21*a^2*b^3*d^2*e^3 + 56*a^3*b^2*d*e^4 + 126*a^4*b*e^5)*x)/(e^17*x^11
+ 11*d*e^16*x^10 + 55*d^2*e^15*x^9 + 165*d^3*e^14*x^8 + 330*d^4*e^13*x^7 + 462*d
^5*e^12*x^6 + 462*d^6*e^11*x^5 + 330*d^7*e^10*x^4 + 165*d^8*e^9*x^3 + 55*d^9*e^8
*x^2 + 11*d^10*e^7*x + d^11*e^6)

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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)**12,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.223062, size = 514, normalized size = 1.67 \[ -\frac{{\left (462 \, b^{5} x^{5} e^{5}{\rm sign}\left (b x + a\right ) + 330 \, b^{5} d x^{4} e^{4}{\rm sign}\left (b x + a\right ) + 165 \, b^{5} d^{2} x^{3} e^{3}{\rm sign}\left (b x + a\right ) + 55 \, b^{5} d^{3} x^{2} e^{2}{\rm sign}\left (b x + a\right ) + 11 \, b^{5} d^{4} x e{\rm sign}\left (b x + a\right ) + b^{5} d^{5}{\rm sign}\left (b x + a\right ) + 1980 \, a b^{4} x^{4} e^{5}{\rm sign}\left (b x + a\right ) + 990 \, a b^{4} d x^{3} e^{4}{\rm sign}\left (b x + a\right ) + 330 \, a b^{4} d^{2} x^{2} e^{3}{\rm sign}\left (b x + a\right ) + 66 \, a b^{4} d^{3} x e^{2}{\rm sign}\left (b x + a\right ) + 6 \, a b^{4} d^{4} e{\rm sign}\left (b x + a\right ) + 3465 \, a^{2} b^{3} x^{3} e^{5}{\rm sign}\left (b x + a\right ) + 1155 \, a^{2} b^{3} d x^{2} e^{4}{\rm sign}\left (b x + a\right ) + 231 \, a^{2} b^{3} d^{2} x e^{3}{\rm sign}\left (b x + a\right ) + 21 \, a^{2} b^{3} d^{3} e^{2}{\rm sign}\left (b x + a\right ) + 3080 \, a^{3} b^{2} x^{2} e^{5}{\rm sign}\left (b x + a\right ) + 616 \, a^{3} b^{2} d x e^{4}{\rm sign}\left (b x + a\right ) + 56 \, a^{3} b^{2} d^{2} e^{3}{\rm sign}\left (b x + a\right ) + 1386 \, a^{4} b x e^{5}{\rm sign}\left (b x + a\right ) + 126 \, a^{4} b d e^{4}{\rm sign}\left (b x + a\right ) + 252 \, a^{5} e^{5}{\rm sign}\left (b x + a\right )\right )} e^{\left (-6\right )}}{2772 \,{\left (x e + d\right )}^{11}} \]

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)^12,x, algorithm="giac")

[Out]

-1/2772*(462*b^5*x^5*e^5*sign(b*x + a) + 330*b^5*d*x^4*e^4*sign(b*x + a) + 165*b
^5*d^2*x^3*e^3*sign(b*x + a) + 55*b^5*d^3*x^2*e^2*sign(b*x + a) + 11*b^5*d^4*x*e
*sign(b*x + a) + b^5*d^5*sign(b*x + a) + 1980*a*b^4*x^4*e^5*sign(b*x + a) + 990*
a*b^4*d*x^3*e^4*sign(b*x + a) + 330*a*b^4*d^2*x^2*e^3*sign(b*x + a) + 66*a*b^4*d
^3*x*e^2*sign(b*x + a) + 6*a*b^4*d^4*e*sign(b*x + a) + 3465*a^2*b^3*x^3*e^5*sign
(b*x + a) + 1155*a^2*b^3*d*x^2*e^4*sign(b*x + a) + 231*a^2*b^3*d^2*x*e^3*sign(b*
x + a) + 21*a^2*b^3*d^3*e^2*sign(b*x + a) + 3080*a^3*b^2*x^2*e^5*sign(b*x + a) +
616*a^3*b^2*d*x*e^4*sign(b*x + a) + 56*a^3*b^2*d^2*e^3*sign(b*x + a) + 1386*a^4
*b*x*e^5*sign(b*x + a) + 126*a^4*b*d*e^4*sign(b*x + a) + 252*a^5*e^5*sign(b*x +
a))*e^(-6)/(x*e + d)^11