3.904 \(\int \frac{x^{-1+3 n} \left (a+b x^n\right )^{3/2}}{\sqrt{c+d x^n}} \, dx\)

Optimal. Leaf size=291 \[ -\frac{(b c-a d) \left (3 a^2 d^2+10 a b c d+35 b^2 c^2\right ) \sqrt{a+b x^n} \sqrt{c+d x^n}}{64 b^2 d^4 n}+\frac{\left (3 a^2 d^2+10 a b c d+35 b^2 c^2\right ) \left (a+b x^n\right )^{3/2} \sqrt{c+d x^n}}{96 b^2 d^3 n}+\frac{(b c-a d)^2 \left (3 a^2 d^2+10 a b c d+35 b^2 c^2\right ) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x^n}}{\sqrt{b} \sqrt{c+d x^n}}\right )}{64 b^{5/2} d^{9/2} n}-\frac{(3 a d+7 b c) \left (a+b x^n\right )^{5/2} \sqrt{c+d x^n}}{24 b^2 d^2 n}+\frac{x^n \left (a+b x^n\right )^{5/2} \sqrt{c+d x^n}}{4 b d n} \]

[Out]

-((b*c - a*d)*(35*b^2*c^2 + 10*a*b*c*d + 3*a^2*d^2)*Sqrt[a + b*x^n]*Sqrt[c + d*x
^n])/(64*b^2*d^4*n) + ((35*b^2*c^2 + 10*a*b*c*d + 3*a^2*d^2)*(a + b*x^n)^(3/2)*S
qrt[c + d*x^n])/(96*b^2*d^3*n) - ((7*b*c + 3*a*d)*(a + b*x^n)^(5/2)*Sqrt[c + d*x
^n])/(24*b^2*d^2*n) + (x^n*(a + b*x^n)^(5/2)*Sqrt[c + d*x^n])/(4*b*d*n) + ((b*c
- a*d)^2*(35*b^2*c^2 + 10*a*b*c*d + 3*a^2*d^2)*ArcTanh[(Sqrt[d]*Sqrt[a + b*x^n])
/(Sqrt[b]*Sqrt[c + d*x^n])])/(64*b^(5/2)*d^(9/2)*n)

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Rubi [A]  time = 0.802315, antiderivative size = 291, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ -\frac{(b c-a d) \left (3 a^2 d^2+10 a b c d+35 b^2 c^2\right ) \sqrt{a+b x^n} \sqrt{c+d x^n}}{64 b^2 d^4 n}+\frac{\left (3 a^2 d^2+10 a b c d+35 b^2 c^2\right ) \left (a+b x^n\right )^{3/2} \sqrt{c+d x^n}}{96 b^2 d^3 n}+\frac{(b c-a d)^2 \left (3 a^2 d^2+10 a b c d+35 b^2 c^2\right ) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x^n}}{\sqrt{b} \sqrt{c+d x^n}}\right )}{64 b^{5/2} d^{9/2} n}-\frac{(3 a d+7 b c) \left (a+b x^n\right )^{5/2} \sqrt{c+d x^n}}{24 b^2 d^2 n}+\frac{x^n \left (a+b x^n\right )^{5/2} \sqrt{c+d x^n}}{4 b d n} \]

Antiderivative was successfully verified.

[In]  Int[(x^(-1 + 3*n)*(a + b*x^n)^(3/2))/Sqrt[c + d*x^n],x]

[Out]

-((b*c - a*d)*(35*b^2*c^2 + 10*a*b*c*d + 3*a^2*d^2)*Sqrt[a + b*x^n]*Sqrt[c + d*x
^n])/(64*b^2*d^4*n) + ((35*b^2*c^2 + 10*a*b*c*d + 3*a^2*d^2)*(a + b*x^n)^(3/2)*S
qrt[c + d*x^n])/(96*b^2*d^3*n) - ((7*b*c + 3*a*d)*(a + b*x^n)^(5/2)*Sqrt[c + d*x
^n])/(24*b^2*d^2*n) + (x^n*(a + b*x^n)^(5/2)*Sqrt[c + d*x^n])/(4*b*d*n) + ((b*c
- a*d)^2*(35*b^2*c^2 + 10*a*b*c*d + 3*a^2*d^2)*ArcTanh[(Sqrt[d]*Sqrt[a + b*x^n])
/(Sqrt[b]*Sqrt[c + d*x^n])])/(64*b^(5/2)*d^(9/2)*n)

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Rubi in Sympy [A]  time = 59.3333, size = 269, normalized size = 0.92 \[ \frac{x^{n} \left (a + b x^{n}\right )^{\frac{5}{2}} \sqrt{c + d x^{n}}}{4 b d n} - \frac{\left (a + b x^{n}\right )^{\frac{5}{2}} \sqrt{c + d x^{n}} \left (3 a d + 7 b c\right )}{24 b^{2} d^{2} n} + \frac{\left (a + b x^{n}\right )^{\frac{3}{2}} \sqrt{c + d x^{n}} \left (3 a^{2} d^{2} + 10 a b c d + 35 b^{2} c^{2}\right )}{96 b^{2} d^{3} n} + \frac{\sqrt{a + b x^{n}} \sqrt{c + d x^{n}} \left (a d - b c\right ) \left (3 a^{2} d^{2} + 10 a b c d + 35 b^{2} c^{2}\right )}{64 b^{2} d^{4} n} + \frac{\left (a d - b c\right )^{2} \left (3 a^{2} d^{2} + 10 a b c d + 35 b^{2} c^{2}\right ) \operatorname{atanh}{\left (\frac{\sqrt{d} \sqrt{a + b x^{n}}}{\sqrt{b} \sqrt{c + d x^{n}}} \right )}}{64 b^{\frac{5}{2}} d^{\frac{9}{2}} n} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x**n*(a + b*x**n)**(5/2)*sqrt(c + d*x**n)/(4*b*d*n) - (a + b*x**n)**(5/2)*sqrt(c
+ d*x**n)*(3*a*d + 7*b*c)/(24*b**2*d**2*n) + (a + b*x**n)**(3/2)*sqrt(c + d*x**
n)*(3*a**2*d**2 + 10*a*b*c*d + 35*b**2*c**2)/(96*b**2*d**3*n) + sqrt(a + b*x**n)
*sqrt(c + d*x**n)*(a*d - b*c)*(3*a**2*d**2 + 10*a*b*c*d + 35*b**2*c**2)/(64*b**2
*d**4*n) + (a*d - b*c)**2*(3*a**2*d**2 + 10*a*b*c*d + 35*b**2*c**2)*atanh(sqrt(d
)*sqrt(a + b*x**n)/(sqrt(b)*sqrt(c + d*x**n)))/(64*b**(5/2)*d**(9/2)*n)

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Mathematica [A]  time = 0.455557, size = 234, normalized size = 0.8 \[ \frac{3 (b c-a d)^2 \left (3 a^2 d^2+10 a b c d+35 b^2 c^2\right ) \log \left (2 \sqrt{b} \sqrt{d} \sqrt{a+b x^n} \sqrt{c+d x^n}+a d+b c+2 b d x^n\right )-2 \sqrt{b} \sqrt{d} \sqrt{a+b x^n} \sqrt{c+d x^n} \left (9 a^3 d^3+3 a^2 b d^2 \left (5 c-2 d x^n\right )-a b^2 d \left (145 c^2-92 c d x^n+72 d^2 x^{2 n}\right )+b^3 \left (105 c^3-70 c^2 d x^n+56 c d^2 x^{2 n}-48 d^3 x^{3 n}\right )\right )}{384 b^{5/2} d^{9/2} n} \]

Antiderivative was successfully verified.

[In]  Integrate[(x^(-1 + 3*n)*(a + b*x^n)^(3/2))/Sqrt[c + d*x^n],x]

[Out]

(-2*Sqrt[b]*Sqrt[d]*Sqrt[a + b*x^n]*Sqrt[c + d*x^n]*(9*a^3*d^3 + 3*a^2*b*d^2*(5*
c - 2*d*x^n) - a*b^2*d*(145*c^2 - 92*c*d*x^n + 72*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))) + 3*(b*c - a*d)^2*(35*b^2*c^2
+ 10*a*b*c*d + 3*a^2*d^2)*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^(5/2)*d^(9/2)*n)

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Maple [F]  time = 0.086, size = 0, normalized size = 0. \[ \int{{x}^{-1+3\,n} \left ( a+b{x}^{n} \right ) ^{{\frac{3}{2}}}{\frac{1}{\sqrt{c+d{x}^{n}}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

int(x^(-1+3*n)*(a+b*x^n)^(3/2)/(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)^(3/2)*x^(3*n - 1)/sqrt(d*x^n + c),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.357617, size = 1, normalized size = 0. \[ \left [\frac{4 \,{\left (48 \, \sqrt{b d} b^{3} d^{3} x^{3 \, n} - 8 \,{\left (7 \, b^{3} c d^{2} - 9 \, a b^{2} d^{3}\right )} \sqrt{b d} x^{2 \, n} + 2 \,{\left (35 \, b^{3} c^{2} d - 46 \, a b^{2} c d^{2} + 3 \, a^{2} b d^{3}\right )} \sqrt{b d} x^{n} -{\left (105 \, b^{3} c^{3} - 145 \, a b^{2} c^{2} d + 15 \, a^{2} b c d^{2} + 9 \, a^{3} d^{3}\right )} \sqrt{b d}\right )} \sqrt{b x^{n} + a} \sqrt{d x^{n} + c} + 3 \,{\left (35 \, b^{4} c^{4} - 60 \, a b^{3} c^{3} d + 18 \, a^{2} b^{2} c^{2} d^{2} + 4 \, a^{3} b c d^{3} + 3 \, a^{4} d^{4}\right )} \log \left (8 \, \sqrt{b d} b^{2} d^{2} x^{2 \, n} + 8 \,{\left (b^{2} c d + a b d^{2}\right )} \sqrt{b d} x^{n} + 4 \,{\left (2 \, b^{2} d^{2} x^{n} + b^{2} c d + a b d^{2}\right )} \sqrt{b x^{n} + a} \sqrt{d x^{n} + c} +{\left (b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2}\right )} \sqrt{b d}\right )}{768 \, \sqrt{b d} b^{2} d^{4} n}, \frac{2 \,{\left (48 \, \sqrt{-b d} b^{3} d^{3} x^{3 \, n} - 8 \,{\left (7 \, b^{3} c d^{2} - 9 \, a b^{2} d^{3}\right )} \sqrt{-b d} x^{2 \, n} + 2 \,{\left (35 \, b^{3} c^{2} d - 46 \, a b^{2} c d^{2} + 3 \, a^{2} b d^{3}\right )} \sqrt{-b d} x^{n} -{\left (105 \, b^{3} c^{3} - 145 \, a b^{2} c^{2} d + 15 \, a^{2} b c d^{2} + 9 \, a^{3} d^{3}\right )} \sqrt{-b d}\right )} \sqrt{b x^{n} + a} \sqrt{d x^{n} + c} + 3 \,{\left (35 \, b^{4} c^{4} - 60 \, a b^{3} c^{3} d + 18 \, a^{2} b^{2} c^{2} d^{2} + 4 \, a^{3} b c d^{3} + 3 \, a^{4} d^{4}\right )} \arctan \left (\frac{2 \, \sqrt{-b d} b d x^{n} +{\left (b c + a d\right )} \sqrt{-b d}}{2 \, \sqrt{b x^{n} + a} \sqrt{d x^{n} + c} b d}\right )}{384 \, \sqrt{-b d} b^{2} d^{4} n}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^n + a)^(3/2)*x^(3*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 - 9*a*b^2*d^3)*sqrt(b*d
)*x^(2*n) + 2*(35*b^3*c^2*d - 46*a*b^2*c*d^2 + 3*a^2*b*d^3)*sqrt(b*d)*x^n - (105
*b^3*c^3 - 145*a*b^2*c^2*d + 15*a^2*b*c*d^2 + 9*a^3*d^3)*sqrt(b*d))*sqrt(b*x^n +
a)*sqrt(d*x^n + c) + 3*(35*b^4*c^4 - 60*a*b^3*c^3*d + 18*a^2*b^2*c^2*d^2 + 4*a^
3*b*c*d^3 + 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)*s
qrt(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^2*d^4*n), 1/384*
(2*(48*sqrt(-b*d)*b^3*d^3*x^(3*n) - 8*(7*b^3*c*d^2 - 9*a*b^2*d^3)*sqrt(-b*d)*x^(
2*n) + 2*(35*b^3*c^2*d - 46*a*b^2*c*d^2 + 3*a^2*b*d^3)*sqrt(-b*d)*x^n - (105*b^3
*c^3 - 145*a*b^2*c^2*d + 15*a^2*b*c*d^2 + 9*a^3*d^3)*sqrt(-b*d))*sqrt(b*x^n + a)
*sqrt(d*x^n + c) + 3*(35*b^4*c^4 - 60*a*b^3*c^3*d + 18*a^2*b^2*c^2*d^2 + 4*a^3*b
*c*d^3 + 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^2*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+3*n)*(a+b*x**n)**(3/2)/(c+d*x**n)**(1/2),x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (b x^{n} + a\right )}^{\frac{3}{2}} x^{3 \, n - 1}}{\sqrt{d x^{n} + c}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^n + a)^(3/2)*x^(3*n - 1)/sqrt(d*x^n + c),x, algorithm="giac")

[Out]

integrate((b*x^n + a)^(3/2)*x^(3*n - 1)/sqrt(d*x^n + c), x)