3.902 \(\int \frac{x^{-1+2 n}}{\left (a+b x^n\right )^{5/2} \sqrt{c+d x^n}} \, dx\)

Optimal. Leaf size=95 \[ \frac{2 a \sqrt{c+d x^n}}{3 b n (b c-a d) \left (a+b x^n\right )^{3/2}}-\frac{2 (3 b c-a d) \sqrt{c+d x^n}}{3 b n (b c-a d)^2 \sqrt{a+b x^n}} \]

[Out]

(2*a*Sqrt[c + d*x^n])/(3*b*(b*c - a*d)*n*(a + b*x^n)^(3/2)) - (2*(3*b*c - a*d)*S
qrt[c + d*x^n])/(3*b*(b*c - a*d)^2*n*Sqrt[a + b*x^n])

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Rubi [A]  time = 0.22541, antiderivative size = 95, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1 \[ \frac{2 a \sqrt{c+d x^n}}{3 b n (b c-a d) \left (a+b x^n\right )^{3/2}}-\frac{2 (3 b c-a d) \sqrt{c+d x^n}}{3 b n (b c-a d)^2 \sqrt{a+b x^n}} \]

Antiderivative was successfully verified.

[In]  Int[x^(-1 + 2*n)/((a + b*x^n)^(5/2)*Sqrt[c + d*x^n]),x]

[Out]

(2*a*Sqrt[c + d*x^n])/(3*b*(b*c - a*d)*n*(a + b*x^n)^(3/2)) - (2*(3*b*c - a*d)*S
qrt[c + d*x^n])/(3*b*(b*c - a*d)^2*n*Sqrt[a + b*x^n])

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Rubi in Sympy [A]  time = 19.8986, size = 78, normalized size = 0.82 \[ - \frac{2 a \sqrt{c + d x^{n}}}{3 b n \left (a + b x^{n}\right )^{\frac{3}{2}} \left (a d - b c\right )} + \frac{2 \sqrt{c + d x^{n}} \left (a d - 3 b c\right )}{3 b n \sqrt{a + b x^{n}} \left (a d - b c\right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.154746, size = 57, normalized size = 0.6 \[ \frac{2 \sqrt{c+d x^n} \left (-2 a c+a d x^n-3 b c x^n\right )}{3 n (b c-a d)^2 \left (a+b x^n\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[x^(-1 + 2*n)/((a + b*x^n)^(5/2)*Sqrt[c + d*x^n]),x]

[Out]

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

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Maple [F]  time = 0.087, size = 0, normalized size = 0. \[ \int{{x}^{-1+2\,n} \left ( a+b{x}^{n} \right ) ^{-{\frac{5}{2}}}{\frac{1}{\sqrt{c+d{x}^{n}}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

int(x^(-1+2*n)/(a+b*x^n)^(5/2)/(c+d*x^n)^(1/2),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{2 \, n - 1}}{{\left (b x^{n} + a\right )}^{\frac{5}{2}} \sqrt{d x^{n} + c}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^(2*n - 1)/((b*x^n + a)^(5/2)*sqrt(d*x^n + c)),x, algorithm="maxima")

[Out]

integrate(x^(2*n - 1)/((b*x^n + a)^(5/2)*sqrt(d*x^n + c)), x)

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Fricas [A]  time = 0.31698, size = 182, normalized size = 1.92 \[ -\frac{2 \,{\left (2 \, a c +{\left (3 \, b c - a d\right )} x^{n}\right )} \sqrt{b x^{n} + a} \sqrt{d x^{n} + c}}{3 \,{\left ({\left (b^{4} c^{2} - 2 \, a b^{3} c d + a^{2} b^{2} d^{2}\right )} n x^{2 \, n} + 2 \,{\left (a b^{3} c^{2} - 2 \, a^{2} b^{2} c d + a^{3} b d^{2}\right )} n x^{n} +{\left (a^{2} b^{2} c^{2} - 2 \, a^{3} b c d + a^{4} d^{2}\right )} n\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^(2*n - 1)/((b*x^n + a)^(5/2)*sqrt(d*x^n + c)),x, algorithm="fricas")

[Out]

-2/3*(2*a*c + (3*b*c - a*d)*x^n)*sqrt(b*x^n + a)*sqrt(d*x^n + c)/((b^4*c^2 - 2*a
*b^3*c*d + a^2*b^2*d^2)*n*x^(2*n) + 2*(a*b^3*c^2 - 2*a^2*b^2*c*d + a^3*b*d^2)*n*
x^n + (a^2*b^2*c^2 - 2*a^3*b*c*d + a^4*d^2)*n)

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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(x**(-1+2*n)/(a+b*x**n)**(5/2)/(c+d*x**n)**(1/2),x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{2 \, n - 1}}{{\left (b x^{n} + a\right )}^{\frac{5}{2}} \sqrt{d x^{n} + c}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^(2*n - 1)/((b*x^n + a)^(5/2)*sqrt(d*x^n + c)),x, algorithm="giac")

[Out]

integrate(x^(2*n - 1)/((b*x^n + a)^(5/2)*sqrt(d*x^n + c)), x)