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

Optimal. Leaf size=252 \[ \frac{5 (b c-a d)^3 (a d+7 b c) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x^n}}{\sqrt{b} \sqrt{c+d x^n}}\right )}{64 b^{3/2} d^{9/2} n}-\frac{5 (b c-a d)^2 (a d+7 b c) \sqrt{a+b x^n} \sqrt{c+d x^n}}{64 b d^4 n}+\frac{5 (b c-a d) (a d+7 b c) \left (a+b x^n\right )^{3/2} \sqrt{c+d x^n}}{96 b d^3 n}-\frac{(a d+7 b c) \left (a+b x^n\right )^{5/2} \sqrt{c+d x^n}}{24 b d^2 n}+\frac{\left (a+b x^n\right )^{7/2} \sqrt{c+d x^n}}{4 b d n} \]

[Out]

(-5*(b*c - a*d)^2*(7*b*c + a*d)*Sqrt[a + b*x^n]*Sqrt[c + d*x^n])/(64*b*d^4*n) +
(5*(b*c - a*d)*(7*b*c + a*d)*(a + b*x^n)^(3/2)*Sqrt[c + d*x^n])/(96*b*d^3*n) - (
(7*b*c + a*d)*(a + b*x^n)^(5/2)*Sqrt[c + d*x^n])/(24*b*d^2*n) + ((a + b*x^n)^(7/
2)*Sqrt[c + d*x^n])/(4*b*d*n) + (5*(b*c - a*d)^3*(7*b*c + a*d)*ArcTanh[(Sqrt[d]*
Sqrt[a + b*x^n])/(Sqrt[b]*Sqrt[c + d*x^n])])/(64*b^(3/2)*d^(9/2)*n)

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Rubi [A]  time = 0.601344, antiderivative size = 252, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167 \[ \frac{5 (b c-a d)^3 (a d+7 b c) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x^n}}{\sqrt{b} \sqrt{c+d x^n}}\right )}{64 b^{3/2} d^{9/2} n}-\frac{5 (b c-a d)^2 (a d+7 b c) \sqrt{a+b x^n} \sqrt{c+d x^n}}{64 b d^4 n}+\frac{5 (b c-a d) (a d+7 b c) \left (a+b x^n\right )^{3/2} \sqrt{c+d x^n}}{96 b d^3 n}-\frac{(a d+7 b c) \left (a+b x^n\right )^{5/2} \sqrt{c+d x^n}}{24 b d^2 n}+\frac{\left (a+b x^n\right )^{7/2} \sqrt{c+d x^n}}{4 b d n} \]

Antiderivative was successfully verified.

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

[Out]

(-5*(b*c - a*d)^2*(7*b*c + a*d)*Sqrt[a + b*x^n]*Sqrt[c + d*x^n])/(64*b*d^4*n) +
(5*(b*c - a*d)*(7*b*c + a*d)*(a + b*x^n)^(3/2)*Sqrt[c + d*x^n])/(96*b*d^3*n) - (
(7*b*c + a*d)*(a + b*x^n)^(5/2)*Sqrt[c + d*x^n])/(24*b*d^2*n) + ((a + b*x^n)^(7/
2)*Sqrt[c + d*x^n])/(4*b*d*n) + (5*(b*c - a*d)^3*(7*b*c + a*d)*ArcTanh[(Sqrt[d]*
Sqrt[a + b*x^n])/(Sqrt[b]*Sqrt[c + d*x^n])])/(64*b^(3/2)*d^(9/2)*n)

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Rubi in Sympy [A]  time = 51.3209, size = 221, normalized size = 0.88 \[ \frac{\left (a + b x^{n}\right )^{\frac{7}{2}} \sqrt{c + d x^{n}}}{4 b d n} - \frac{\left (a + b x^{n}\right )^{\frac{5}{2}} \sqrt{c + d x^{n}} \left (a d + 7 b c\right )}{24 b d^{2} n} - \frac{5 \left (a + b x^{n}\right )^{\frac{3}{2}} \sqrt{c + d x^{n}} \left (a d - b c\right ) \left (a d + 7 b c\right )}{96 b d^{3} n} - \frac{5 \sqrt{a + b x^{n}} \sqrt{c + d x^{n}} \left (a d - b c\right )^{2} \left (a d + 7 b c\right )}{64 b d^{4} n} - \frac{5 \left (a d - b c\right )^{3} \left (a d + 7 b c\right ) \operatorname{atanh}{\left (\frac{\sqrt{d} \sqrt{a + b x^{n}}}{\sqrt{b} \sqrt{c + d x^{n}}} \right )}}{64 b^{\frac{3}{2}} d^{\frac{9}{2}} n} \]

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]

(a + b*x**n)**(7/2)*sqrt(c + d*x**n)/(4*b*d*n) - (a + b*x**n)**(5/2)*sqrt(c + d*
x**n)*(a*d + 7*b*c)/(24*b*d**2*n) - 5*(a + b*x**n)**(3/2)*sqrt(c + d*x**n)*(a*d
- b*c)*(a*d + 7*b*c)/(96*b*d**3*n) - 5*sqrt(a + b*x**n)*sqrt(c + d*x**n)*(a*d -
b*c)**2*(a*d + 7*b*c)/(64*b*d**4*n) - 5*(a*d - b*c)**3*(a*d + 7*b*c)*atanh(sqrt(
d)*sqrt(a + b*x**n)/(sqrt(b)*sqrt(c + d*x**n)))/(64*b**(3/2)*d**(9/2)*n)

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Mathematica [A]  time = 0.535317, size = 218, normalized size = 0.87 \[ \frac{15 (b c-a d)^3 (a d+7 b c) \log \left (2 \sqrt{b} \sqrt{d} \sqrt{a+b x^n} \sqrt{c+d x^n}+a d+b c+2 b d x^n\right )-2 \sqrt{b} \sqrt{d} \sqrt{a+b x^n} \sqrt{c+d x^n} \left (-15 a^3 d^3+a^2 b d^2 \left (191 c-118 d x^n\right )-a b^2 d \left (265 c^2-172 c d x^n+136 d^2 x^{2 n}\right )+b^3 \left (105 c^3-70 c^2 d x^n+56 c d^2 x^{2 n}-48 d^3 x^{3 n}\right )\right )}{384 b^{3/2} d^{9/2} n} \]

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[b]*Sqrt[d]*Sqrt[a + b*x^n]*Sqrt[c + d*x^n]*(-15*a^3*d^3 + a^2*b*d^2*(19
1*c - 118*d*x^n) - a*b^2*d*(265*c^2 - 172*c*d*x^n + 136*d^2*x^(2*n)) + b^3*(105*
c^3 - 70*c^2*d*x^n + 56*c*d^2*x^(2*n) - 48*d^3*x^(3*n))) + 15*(b*c - a*d)^3*(7*b
*c + a*d)*Log[b*c + a*d + 2*b*d*x^n + 2*Sqrt[b]*Sqrt[d]*Sqrt[a + b*x^n]*Sqrt[c +
d*x^n]])/(384*b^(3/2)*d^(9/2)*n)

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Maple [F]  time = 0.09, 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. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.369511, size = 1, normalized size = 0. \[ \left [\frac{4 \,{\left (48 \, \sqrt{b d} b^{3} d^{3} x^{3 \, n} - 8 \,{\left (7 \, b^{3} c d^{2} - 17 \, a b^{2} d^{3}\right )} \sqrt{b d} x^{2 \, n} + 2 \,{\left (35 \, b^{3} c^{2} d - 86 \, a b^{2} c d^{2} + 59 \, a^{2} b d^{3}\right )} \sqrt{b d} x^{n} -{\left (105 \, b^{3} c^{3} - 265 \, a b^{2} c^{2} d + 191 \, a^{2} b c d^{2} - 15 \, a^{3} d^{3}\right )} \sqrt{b d}\right )} \sqrt{b x^{n} + a} \sqrt{d x^{n} + c} - 15 \,{\left (7 \, b^{4} c^{4} - 20 \, a b^{3} c^{3} d + 18 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} - a^{4} d^{4}\right )} \log \left (8 \, \sqrt{b d} b^{2} d^{2} x^{2 \, n} + 8 \,{\left (b^{2} c d + a b d^{2}\right )} \sqrt{b d} x^{n} - 4 \,{\left (2 \, b^{2} d^{2} x^{n} + b^{2} c d + a b d^{2}\right )} \sqrt{b x^{n} + a} \sqrt{d x^{n} + c} +{\left (b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2}\right )} \sqrt{b d}\right )}{768 \, \sqrt{b d} b d^{4} n}, \frac{2 \,{\left (48 \, \sqrt{-b d} b^{3} d^{3} x^{3 \, n} - 8 \,{\left (7 \, b^{3} c d^{2} - 17 \, a b^{2} d^{3}\right )} \sqrt{-b d} x^{2 \, n} + 2 \,{\left (35 \, b^{3} c^{2} d - 86 \, a b^{2} c d^{2} + 59 \, a^{2} b d^{3}\right )} \sqrt{-b d} x^{n} -{\left (105 \, b^{3} c^{3} - 265 \, a b^{2} c^{2} d + 191 \, a^{2} b c d^{2} - 15 \, a^{3} d^{3}\right )} \sqrt{-b d}\right )} \sqrt{b x^{n} + a} \sqrt{d x^{n} + c} + 15 \,{\left (7 \, b^{4} c^{4} - 20 \, a b^{3} c^{3} d + 18 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} - a^{4} d^{4}\right )} \arctan \left (\frac{2 \, \sqrt{-b d} b d x^{n} +{\left (b c + a d\right )} \sqrt{-b d}}{2 \, \sqrt{b x^{n} + a} \sqrt{d x^{n} + c} b d}\right )}{384 \, \sqrt{-b d} b d^{4} n}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/768*(4*(48*sqrt(b*d)*b^3*d^3*x^(3*n) - 8*(7*b^3*c*d^2 - 17*a*b^2*d^3)*sqrt(b*
d)*x^(2*n) + 2*(35*b^3*c^2*d - 86*a*b^2*c*d^2 + 59*a^2*b*d^3)*sqrt(b*d)*x^n - (1
05*b^3*c^3 - 265*a*b^2*c^2*d + 191*a^2*b*c*d^2 - 15*a^3*d^3)*sqrt(b*d))*sqrt(b*x
^n + a)*sqrt(d*x^n + c) - 15*(7*b^4*c^4 - 20*a*b^3*c^3*d + 18*a^2*b^2*c^2*d^2 -
4*a^3*b*c*d^3 - a^4*d^4)*log(8*sqrt(b*d)*b^2*d^2*x^(2*n) + 8*(b^2*c*d + a*b*d^2)
*sqrt(b*d)*x^n - 4*(2*b^2*d^2*x^n + b^2*c*d + a*b*d^2)*sqrt(b*x^n + a)*sqrt(d*x^
n + c) + (b^2*c^2 + 6*a*b*c*d + a^2*d^2)*sqrt(b*d)))/(sqrt(b*d)*b*d^4*n), 1/384*
(2*(48*sqrt(-b*d)*b^3*d^3*x^(3*n) - 8*(7*b^3*c*d^2 - 17*a*b^2*d^3)*sqrt(-b*d)*x^
(2*n) + 2*(35*b^3*c^2*d - 86*a*b^2*c*d^2 + 59*a^2*b*d^3)*sqrt(-b*d)*x^n - (105*b
^3*c^3 - 265*a*b^2*c^2*d + 191*a^2*b*c*d^2 - 15*a^3*d^3)*sqrt(-b*d))*sqrt(b*x^n
+ a)*sqrt(d*x^n + c) + 15*(7*b^4*c^4 - 20*a*b^3*c^3*d + 18*a^2*b^2*c^2*d^2 - 4*a
^3*b*c*d^3 - a^4*d^4)*arctan(1/2*(2*sqrt(-b*d)*b*d*x^n + (b*c + a*d)*sqrt(-b*d))
/(sqrt(b*x^n + a)*sqrt(d*x^n + c)*b*d)))/(sqrt(-b*d)*b*d^4*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. \[ \int \frac{{\left (b x^{n} + a\right )}^{\frac{5}{2}} x^{2 \, n - 1}}{\sqrt{d x^{n} + c}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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