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

Optimal. Leaf size=154 \[ -\frac{\sqrt{c+d x^2} \left (3 a^2 d^2-10 a b c d+15 b^2 c^2\right )}{15 b^2 \sqrt{a+b x^2} (b c-a d)^3}-\frac{a^2 \sqrt{c+d x^2}}{5 b^2 \left (a+b x^2\right )^{5/2} (b c-a d)}+\frac{2 a \sqrt{c+d x^2} (5 b c-3 a d)}{15 b^2 \left (a+b x^2\right )^{3/2} (b c-a d)^2} \]

[Out]

-(a^2*Sqrt[c + d*x^2])/(5*b^2*(b*c - a*d)*(a + b*x^2)^(5/2)) + (2*a*(5*b*c - 3*a
*d)*Sqrt[c + d*x^2])/(15*b^2*(b*c - a*d)^2*(a + b*x^2)^(3/2)) - ((15*b^2*c^2 - 1
0*a*b*c*d + 3*a^2*d^2)*Sqrt[c + d*x^2])/(15*b^2*(b*c - a*d)^3*Sqrt[a + b*x^2])

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Rubi [A]  time = 0.472144, antiderivative size = 154, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154 \[ -\frac{\sqrt{c+d x^2} \left (3 a^2 d^2-10 a b c d+15 b^2 c^2\right )}{15 b^2 \sqrt{a+b x^2} (b c-a d)^3}-\frac{a^2 \sqrt{c+d x^2}}{5 b^2 \left (a+b x^2\right )^{5/2} (b c-a d)}+\frac{2 a \sqrt{c+d x^2} (5 b c-3 a d)}{15 b^2 \left (a+b x^2\right )^{3/2} (b c-a d)^2} \]

Antiderivative was successfully verified.

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

[Out]

-(a^2*Sqrt[c + d*x^2])/(5*b^2*(b*c - a*d)*(a + b*x^2)^(5/2)) + (2*a*(5*b*c - 3*a
*d)*Sqrt[c + d*x^2])/(15*b^2*(b*c - a*d)^2*(a + b*x^2)^(3/2)) - ((15*b^2*c^2 - 1
0*a*b*c*d + 3*a^2*d^2)*Sqrt[c + d*x^2])/(15*b^2*(b*c - a*d)^3*Sqrt[a + b*x^2])

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Rubi in Sympy [A]  time = 43.9394, size = 141, normalized size = 0.92 \[ \frac{a^{2} \sqrt{c + d x^{2}}}{5 b^{2} \left (a + b x^{2}\right )^{\frac{5}{2}} \left (a d - b c\right )} - \frac{2 a \sqrt{c + d x^{2}} \left (3 a d - 5 b c\right )}{15 b^{2} \left (a + b x^{2}\right )^{\frac{3}{2}} \left (a d - b c\right )^{2}} + \frac{\sqrt{c + d x^{2}} \left (3 a^{2} d^{2} - 10 a b c d + 15 b^{2} c^{2}\right )}{15 b^{2} \sqrt{a + b x^{2}} \left (a d - b c\right )^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**5/(b*x**2+a)**(7/2)/(d*x**2+c)**(1/2),x)

[Out]

a**2*sqrt(c + d*x**2)/(5*b**2*(a + b*x**2)**(5/2)*(a*d - b*c)) - 2*a*sqrt(c + d*
x**2)*(3*a*d - 5*b*c)/(15*b**2*(a + b*x**2)**(3/2)*(a*d - b*c)**2) + sqrt(c + d*
x**2)*(3*a**2*d**2 - 10*a*b*c*d + 15*b**2*c**2)/(15*b**2*sqrt(a + b*x**2)*(a*d -
b*c)**3)

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Mathematica [A]  time = 0.15295, size = 91, normalized size = 0.59 \[ -\frac{\sqrt{c+d x^2} \left (a^2 \left (8 c^2-4 c d x^2+3 d^2 x^4\right )+10 a b c x^2 \left (2 c-d x^2\right )+15 b^2 c^2 x^4\right )}{15 \left (a+b x^2\right )^{5/2} (b c-a d)^3} \]

Antiderivative was successfully verified.

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

[Out]

-(Sqrt[c + d*x^2]*(15*b^2*c^2*x^4 + 10*a*b*c*x^2*(2*c - d*x^2) + a^2*(8*c^2 - 4*
c*d*x^2 + 3*d^2*x^4)))/(15*(b*c - a*d)^3*(a + b*x^2)^(5/2))

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Maple [A]  time = 0.012, size = 119, normalized size = 0.8 \[{\frac{3\,{x}^{4}{a}^{2}{d}^{2}-10\,{x}^{4}abcd+15\,{x}^{4}{b}^{2}{c}^{2}-4\,{x}^{2}{a}^{2}cd+20\,a{c}^{2}b{x}^{2}+8\,{a}^{2}{c}^{2}}{15\,{a}^{3}{d}^{3}-45\,{a}^{2}c{d}^{2}b+45\,a{c}^{2}d{b}^{2}-15\,{c}^{3}{b}^{3}}\sqrt{d{x}^{2}+c} \left ( b{x}^{2}+a \right ) ^{-{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^5/(b*x^2+a)^(7/2)/(d*x^2+c)^(1/2),x)

[Out]

1/15*(d*x^2+c)^(1/2)*(3*a^2*d^2*x^4-10*a*b*c*d*x^4+15*b^2*c^2*x^4-4*a^2*c*d*x^2+
20*a*b*c^2*x^2+8*a^2*c^2)/(b*x^2+a)^(5/2)/(a^3*d^3-3*a^2*b*c*d^2+3*a*b^2*c^2*d-b
^3*c^3)

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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(x^5/((b*x^2 + a)^(7/2)*sqrt(d*x^2 + c)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.326913, size = 354, normalized size = 2.3 \[ -\frac{{\left ({\left (15 \, b^{2} c^{2} - 10 \, a b c d + 3 \, a^{2} d^{2}\right )} x^{4} + 8 \, a^{2} c^{2} + 4 \,{\left (5 \, a b c^{2} - a^{2} c d\right )} x^{2}\right )} \sqrt{b x^{2} + a} \sqrt{d x^{2} + c}}{15 \,{\left (a^{3} b^{3} c^{3} - 3 \, a^{4} b^{2} c^{2} d + 3 \, a^{5} b c d^{2} - a^{6} d^{3} +{\left (b^{6} c^{3} - 3 \, a b^{5} c^{2} d + 3 \, a^{2} b^{4} c d^{2} - a^{3} b^{3} d^{3}\right )} x^{6} + 3 \,{\left (a b^{5} c^{3} - 3 \, a^{2} b^{4} c^{2} d + 3 \, a^{3} b^{3} c d^{2} - a^{4} b^{2} d^{3}\right )} x^{4} + 3 \,{\left (a^{2} b^{4} c^{3} - 3 \, a^{3} b^{3} c^{2} d + 3 \, a^{4} b^{2} c d^{2} - a^{5} b d^{3}\right )} x^{2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/15*((15*b^2*c^2 - 10*a*b*c*d + 3*a^2*d^2)*x^4 + 8*a^2*c^2 + 4*(5*a*b*c^2 - a^
2*c*d)*x^2)*sqrt(b*x^2 + a)*sqrt(d*x^2 + c)/(a^3*b^3*c^3 - 3*a^4*b^2*c^2*d + 3*a
^5*b*c*d^2 - a^6*d^3 + (b^6*c^3 - 3*a*b^5*c^2*d + 3*a^2*b^4*c*d^2 - a^3*b^3*d^3)
*x^6 + 3*(a*b^5*c^3 - 3*a^2*b^4*c^2*d + 3*a^3*b^3*c*d^2 - a^4*b^2*d^3)*x^4 + 3*(
a^2*b^4*c^3 - 3*a^3*b^3*c^2*d + 3*a^4*b^2*c*d^2 - a^5*b*d^3)*x^2)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{5}}{\left (a + b x^{2}\right )^{\frac{7}{2}} \sqrt{c + d x^{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**5/(b*x**2+a)**(7/2)/(d*x**2+c)**(1/2),x)

[Out]

Integral(x**5/((a + b*x**2)**(7/2)*sqrt(c + d*x**2)), x)

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GIAC/XCAS [A]  time = 0.280999, size = 806, normalized size = 5.23 \[ -\frac{2 \,{\left (15 \, \sqrt{b d} b^{8} c^{4} - 40 \, \sqrt{b d} a b^{7} c^{3} d + 38 \, \sqrt{b d} a^{2} b^{6} c^{2} d^{2} - 16 \, \sqrt{b d} a^{3} b^{5} c d^{3} + 3 \, \sqrt{b d} a^{4} b^{4} d^{4} - 60 \, \sqrt{b d}{\left (\sqrt{b x^{2} + a} \sqrt{b d} - \sqrt{b^{2} c +{\left (b x^{2} + a\right )} b d - a b d}\right )}^{2} b^{6} c^{3} + 80 \, \sqrt{b d}{\left (\sqrt{b x^{2} + a} \sqrt{b d} - \sqrt{b^{2} c +{\left (b x^{2} + a\right )} b d - a b d}\right )}^{2} a b^{5} c^{2} d - 20 \, \sqrt{b d}{\left (\sqrt{b x^{2} + a} \sqrt{b d} - \sqrt{b^{2} c +{\left (b x^{2} + a\right )} b d - a b d}\right )}^{2} a^{2} b^{4} c d^{2} + 90 \, \sqrt{b d}{\left (\sqrt{b x^{2} + a} \sqrt{b d} - \sqrt{b^{2} c +{\left (b x^{2} + a\right )} b d - a b d}\right )}^{4} b^{4} c^{2} - 40 \, \sqrt{b d}{\left (\sqrt{b x^{2} + a} \sqrt{b d} - \sqrt{b^{2} c +{\left (b x^{2} + a\right )} b d - a b d}\right )}^{4} a b^{3} c d + 30 \, \sqrt{b d}{\left (\sqrt{b x^{2} + a} \sqrt{b d} - \sqrt{b^{2} c +{\left (b x^{2} + a\right )} b d - a b d}\right )}^{4} a^{2} b^{2} d^{2} - 60 \, \sqrt{b d}{\left (\sqrt{b x^{2} + a} \sqrt{b d} - \sqrt{b^{2} c +{\left (b x^{2} + a\right )} b d - a b d}\right )}^{6} b^{2} c + 15 \, \sqrt{b d}{\left (\sqrt{b x^{2} + a} \sqrt{b d} - \sqrt{b^{2} c +{\left (b x^{2} + a\right )} b d - a b d}\right )}^{8}\right )}}{15 \,{\left (b^{2} c - a b d -{\left (\sqrt{b x^{2} + a} \sqrt{b d} - \sqrt{b^{2} c +{\left (b x^{2} + a\right )} b d - a b d}\right )}^{2}\right )}^{5} b{\left | b \right |}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/15*(15*sqrt(b*d)*b^8*c^4 - 40*sqrt(b*d)*a*b^7*c^3*d + 38*sqrt(b*d)*a^2*b^6*c^
2*d^2 - 16*sqrt(b*d)*a^3*b^5*c*d^3 + 3*sqrt(b*d)*a^4*b^4*d^4 - 60*sqrt(b*d)*(sqr
t(b*x^2 + a)*sqrt(b*d) - sqrt(b^2*c + (b*x^2 + a)*b*d - a*b*d))^2*b^6*c^3 + 80*s
qrt(b*d)*(sqrt(b*x^2 + a)*sqrt(b*d) - sqrt(b^2*c + (b*x^2 + a)*b*d - a*b*d))^2*a
*b^5*c^2*d - 20*sqrt(b*d)*(sqrt(b*x^2 + a)*sqrt(b*d) - sqrt(b^2*c + (b*x^2 + a)*
b*d - a*b*d))^2*a^2*b^4*c*d^2 + 90*sqrt(b*d)*(sqrt(b*x^2 + a)*sqrt(b*d) - sqrt(b
^2*c + (b*x^2 + a)*b*d - a*b*d))^4*b^4*c^2 - 40*sqrt(b*d)*(sqrt(b*x^2 + a)*sqrt(
b*d) - sqrt(b^2*c + (b*x^2 + a)*b*d - a*b*d))^4*a*b^3*c*d + 30*sqrt(b*d)*(sqrt(b
*x^2 + a)*sqrt(b*d) - sqrt(b^2*c + (b*x^2 + a)*b*d - a*b*d))^4*a^2*b^2*d^2 - 60*
sqrt(b*d)*(sqrt(b*x^2 + a)*sqrt(b*d) - sqrt(b^2*c + (b*x^2 + a)*b*d - a*b*d))^6*
b^2*c + 15*sqrt(b*d)*(sqrt(b*x^2 + a)*sqrt(b*d) - sqrt(b^2*c + (b*x^2 + a)*b*d -
a*b*d))^8)/((b^2*c - a*b*d - (sqrt(b*x^2 + a)*sqrt(b*d) - sqrt(b^2*c + (b*x^2 +
a)*b*d - a*b*d))^2)^5*b*abs(b))