3.987 \(\int \frac{x^3}{\left (a+b x^2\right )^{7/2} \sqrt{c+d x^2}} \, dx\)

Optimal. Leaf size=138 \[ \frac{a \sqrt{c+d x^2}}{5 b \left (a+b x^2\right )^{5/2} (b c-a d)}+\frac{2 d \sqrt{c+d x^2} (5 b c-a d)}{15 b \sqrt{a+b x^2} (b c-a d)^3}-\frac{\sqrt{c+d x^2} (5 b c-a d)}{15 b \left (a+b x^2\right )^{3/2} (b c-a d)^2} \]

[Out]

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

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Rubi [A]  time = 0.294932, antiderivative size = 138, 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 \[ \frac{a \sqrt{c+d x^2}}{5 b \left (a+b x^2\right )^{5/2} (b c-a d)}+\frac{2 d \sqrt{c+d x^2} (5 b c-a d)}{15 b \sqrt{a+b x^2} (b c-a d)^3}-\frac{\sqrt{c+d x^2} (5 b c-a d)}{15 b \left (a+b x^2\right )^{3/2} (b c-a d)^2} \]

Antiderivative was successfully verified.

[In]  Int[x^3/((a + b*x^2)^(7/2)*Sqrt[c + d*x^2]),x]

[Out]

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

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Rubi in Sympy [A]  time = 31.0271, size = 116, normalized size = 0.84 \[ - \frac{a \sqrt{c + d x^{2}}}{5 b \left (a + b x^{2}\right )^{\frac{5}{2}} \left (a d - b c\right )} + \frac{2 d \sqrt{c + d x^{2}} \left (a d - 5 b c\right )}{15 b \sqrt{a + b x^{2}} \left (a d - b c\right )^{3}} + \frac{\sqrt{c + d x^{2}} \left (a d - 5 b c\right )}{15 b \left (a + b x^{2}\right )^{\frac{3}{2}} \left (a d - b c\right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**3/(b*x**2+a)**(7/2)/(d*x**2+c)**(1/2),x)

[Out]

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

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Mathematica [A]  time = 0.119777, size = 91, normalized size = 0.66 \[ \frac{\sqrt{c+d x^2} \left (-5 a^2 d \left (d x^2-2 c\right )-2 a b \left (c^2-13 c d x^2+d^2 x^4\right )-5 b^2 c x^2 \left (c-2 d x^2\right )\right )}{15 \left (a+b x^2\right )^{5/2} (b c-a d)^3} \]

Antiderivative was successfully verified.

[In]  Integrate[x^3/((a + b*x^2)^(7/2)*Sqrt[c + d*x^2]),x]

[Out]

(Sqrt[c + d*x^2]*(-5*b^2*c*x^2*(c - 2*d*x^2) - 5*a^2*d*(-2*c + d*x^2) - 2*a*b*(c
^2 - 13*c*d*x^2 + d^2*x^4)))/(15*(b*c - a*d)^3*(a + b*x^2)^(5/2))

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Maple [A]  time = 0.013, size = 125, normalized size = 0.9 \[ -{\frac{-2\,ab{d}^{2}{x}^{4}+10\,{b}^{2}cd{x}^{4}-5\,{a}^{2}{d}^{2}{x}^{2}+26\,abcd{x}^{2}-5\,{b}^{2}{c}^{2}{x}^{2}+10\,{a}^{2}cd-2\,ab{c}^{2}}{15\,{a}^{3}{d}^{3}-45\,{a}^{2}c{d}^{2}b+45\,a{c}^{2}d{b}^{2}-15\,{c}^{3}{b}^{3}}\sqrt{d{x}^{2}+c} \left ( b{x}^{2}+a \right ) ^{-{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^3/(b*x^2+a)^(7/2)/(d*x^2+c)^(1/2),x)

[Out]

-1/15*(d*x^2+c)^(1/2)*(-2*a*b*d^2*x^4+10*b^2*c*d*x^4-5*a^2*d^2*x^2+26*a*b*c*d*x^
2-5*b^2*c^2*x^2+10*a^2*c*d-2*a*b*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^3/((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 = 0.334607, size = 363, normalized size = 2.63 \[ \frac{{\left (2 \,{\left (5 \, b^{2} c d - a b d^{2}\right )} x^{4} - 2 \, a b c^{2} + 10 \, a^{2} c d -{\left (5 \, b^{2} c^{2} - 26 \, a b c d + 5 \, a^{2} d^{2}\right )} x^{2}\right )} \sqrt{b x^{2} + a} \sqrt{d x^{2} + c}}{15 \,{\left (a^{3} b^{3} c^{3} - 3 \, a^{4} b^{2} c^{2} d + 3 \, a^{5} b c d^{2} - a^{6} d^{3} +{\left (b^{6} c^{3} - 3 \, a b^{5} c^{2} d + 3 \, a^{2} b^{4} c d^{2} - a^{3} b^{3} d^{3}\right )} x^{6} + 3 \,{\left (a b^{5} c^{3} - 3 \, a^{2} b^{4} c^{2} d + 3 \, a^{3} b^{3} c d^{2} - a^{4} b^{2} d^{3}\right )} x^{4} + 3 \,{\left (a^{2} b^{4} c^{3} - 3 \, a^{3} b^{3} c^{2} d + 3 \, a^{4} b^{2} c d^{2} - a^{5} b d^{3}\right )} x^{2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^3/((b*x^2 + a)^(7/2)*sqrt(d*x^2 + c)),x, algorithm="fricas")

[Out]

1/15*(2*(5*b^2*c*d - a*b*d^2)*x^4 - 2*a*b*c^2 + 10*a^2*c*d - (5*b^2*c^2 - 26*a*b
*c*d + 5*a^2*d^2)*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. \[ \int \frac{x^{3}}{\left (a + b x^{2}\right )^{\frac{7}{2}} \sqrt{c + d x^{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**3/(b*x**2+a)**(7/2)/(d*x**2+c)**(1/2),x)

[Out]

Integral(x**3/((a + b*x**2)**(7/2)*sqrt(c + d*x**2)), x)

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GIAC/XCAS [A]  time = 0.284648, size = 637, normalized size = 4.62 \[ \frac{4 \,{\left (5 \, \sqrt{b d} b^{8} c^{3} d - 11 \, \sqrt{b d} a b^{7} c^{2} d^{2} + 7 \, \sqrt{b d} a^{2} b^{6} c d^{3} - \sqrt{b d} a^{3} b^{5} d^{4} - 25 \, \sqrt{b d}{\left (\sqrt{b x^{2} + a} \sqrt{b d} - \sqrt{b^{2} c +{\left (b x^{2} + a\right )} b d - a b d}\right )}^{2} b^{6} c^{2} d + 30 \, \sqrt{b d}{\left (\sqrt{b x^{2} + a} \sqrt{b d} - \sqrt{b^{2} c +{\left (b x^{2} + a\right )} b d - a b d}\right )}^{2} a b^{5} c d^{2} - 5 \, \sqrt{b d}{\left (\sqrt{b x^{2} + a} \sqrt{b d} - \sqrt{b^{2} c +{\left (b x^{2} + a\right )} b d - a b d}\right )}^{2} a^{2} b^{4} d^{3} + 35 \, \sqrt{b d}{\left (\sqrt{b x^{2} + a} \sqrt{b d} - \sqrt{b^{2} c +{\left (b x^{2} + a\right )} b d - a b d}\right )}^{4} b^{4} c d + 5 \, \sqrt{b d}{\left (\sqrt{b x^{2} + a} \sqrt{b d} - \sqrt{b^{2} c +{\left (b x^{2} + a\right )} b d - a b d}\right )}^{4} a b^{3} d^{2} - 15 \, \sqrt{b d}{\left (\sqrt{b x^{2} + a} \sqrt{b d} - \sqrt{b^{2} c +{\left (b x^{2} + a\right )} b d - a b d}\right )}^{6} b^{2} d\right )}}{15 \,{\left (b^{2} c - a b d -{\left (\sqrt{b x^{2} + a} \sqrt{b d} - \sqrt{b^{2} c +{\left (b x^{2} + a\right )} b d - a b d}\right )}^{2}\right )}^{5} b{\left | b \right |}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^3/((b*x^2 + a)^(7/2)*sqrt(d*x^2 + c)),x, algorithm="giac")

[Out]

4/15*(5*sqrt(b*d)*b^8*c^3*d - 11*sqrt(b*d)*a*b^7*c^2*d^2 + 7*sqrt(b*d)*a^2*b^6*c
*d^3 - sqrt(b*d)*a^3*b^5*d^4 - 25*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*b^6*c^2*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))^2*a*b^5*c*d^2 - 5*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*d^3 + 3
5*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*d + 5*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*d^2 - 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))^6*b^2*d)/((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))