3.903 \(\int \frac{x^{-1+3 n} \left (a+b x^n\right )^{5/2}}{\sqrt{c+d x^n}} \, dx\)

Optimal. Leaf size=358 \[ \frac{(b c-a d)^2 \left (3 a^2 d^2+14 a b c d+63 b^2 c^2\right ) \sqrt{a+b x^n} \sqrt{c+d x^n}}{128 b^2 d^5 n}-\frac{(b c-a d) \left (3 a^2 d^2+14 a b c d+63 b^2 c^2\right ) \left (a+b x^n\right )^{3/2} \sqrt{c+d x^n}}{192 b^2 d^4 n}+\frac{\left (3 a^2 d^2+14 a b c d+63 b^2 c^2\right ) \left (a+b x^n\right )^{5/2} \sqrt{c+d x^n}}{240 b^2 d^3 n}-\frac{(b c-a d)^3 \left (3 a^2 d^2+14 a b c d+63 b^2 c^2\right ) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x^n}}{\sqrt{b} \sqrt{c+d x^n}}\right )}{128 b^{5/2} d^{11/2} n}-\frac{3 (a d+3 b c) \left (a+b x^n\right )^{7/2} \sqrt{c+d x^n}}{40 b^2 d^2 n}+\frac{x^n \left (a+b x^n\right )^{7/2} \sqrt{c+d x^n}}{5 b d n} \]

[Out]

((b*c - a*d)^2*(63*b^2*c^2 + 14*a*b*c*d + 3*a^2*d^2)*Sqrt[a + b*x^n]*Sqrt[c + d*
x^n])/(128*b^2*d^5*n) - ((b*c - a*d)*(63*b^2*c^2 + 14*a*b*c*d + 3*a^2*d^2)*(a +
b*x^n)^(3/2)*Sqrt[c + d*x^n])/(192*b^2*d^4*n) + ((63*b^2*c^2 + 14*a*b*c*d + 3*a^
2*d^2)*(a + b*x^n)^(5/2)*Sqrt[c + d*x^n])/(240*b^2*d^3*n) - (3*(3*b*c + a*d)*(a
+ b*x^n)^(7/2)*Sqrt[c + d*x^n])/(40*b^2*d^2*n) + (x^n*(a + b*x^n)^(7/2)*Sqrt[c +
d*x^n])/(5*b*d*n) - ((b*c - a*d)^3*(63*b^2*c^2 + 14*a*b*c*d + 3*a^2*d^2)*ArcTan
h[(Sqrt[d]*Sqrt[a + b*x^n])/(Sqrt[b]*Sqrt[c + d*x^n])])/(128*b^(5/2)*d^(11/2)*n)

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Rubi [A]  time = 1.00002, antiderivative size = 358, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ \frac{(b c-a d)^2 \left (3 a^2 d^2+14 a b c d+63 b^2 c^2\right ) \sqrt{a+b x^n} \sqrt{c+d x^n}}{128 b^2 d^5 n}-\frac{(b c-a d) \left (3 a^2 d^2+14 a b c d+63 b^2 c^2\right ) \left (a+b x^n\right )^{3/2} \sqrt{c+d x^n}}{192 b^2 d^4 n}+\frac{\left (3 a^2 d^2+14 a b c d+63 b^2 c^2\right ) \left (a+b x^n\right )^{5/2} \sqrt{c+d x^n}}{240 b^2 d^3 n}-\frac{(b c-a d)^3 \left (3 a^2 d^2+14 a b c d+63 b^2 c^2\right ) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x^n}}{\sqrt{b} \sqrt{c+d x^n}}\right )}{128 b^{5/2} d^{11/2} n}-\frac{3 (a d+3 b c) \left (a+b x^n\right )^{7/2} \sqrt{c+d x^n}}{40 b^2 d^2 n}+\frac{x^n \left (a+b x^n\right )^{7/2} \sqrt{c+d x^n}}{5 b d n} \]

Antiderivative was successfully verified.

[In]  Int[(x^(-1 + 3*n)*(a + b*x^n)^(5/2))/Sqrt[c + d*x^n],x]

[Out]

((b*c - a*d)^2*(63*b^2*c^2 + 14*a*b*c*d + 3*a^2*d^2)*Sqrt[a + b*x^n]*Sqrt[c + d*
x^n])/(128*b^2*d^5*n) - ((b*c - a*d)*(63*b^2*c^2 + 14*a*b*c*d + 3*a^2*d^2)*(a +
b*x^n)^(3/2)*Sqrt[c + d*x^n])/(192*b^2*d^4*n) + ((63*b^2*c^2 + 14*a*b*c*d + 3*a^
2*d^2)*(a + b*x^n)^(5/2)*Sqrt[c + d*x^n])/(240*b^2*d^3*n) - (3*(3*b*c + a*d)*(a
+ b*x^n)^(7/2)*Sqrt[c + d*x^n])/(40*b^2*d^2*n) + (x^n*(a + b*x^n)^(7/2)*Sqrt[c +
d*x^n])/(5*b*d*n) - ((b*c - a*d)^3*(63*b^2*c^2 + 14*a*b*c*d + 3*a^2*d^2)*ArcTan
h[(Sqrt[d]*Sqrt[a + b*x^n])/(Sqrt[b]*Sqrt[c + d*x^n])])/(128*b^(5/2)*d^(11/2)*n)

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Rubi in Sympy [A]  time = 81.248, size = 333, normalized size = 0.93 \[ \frac{x^{n} \left (a + b x^{n}\right )^{\frac{7}{2}} \sqrt{c + d x^{n}}}{5 b d n} - \frac{3 \left (a + b x^{n}\right )^{\frac{7}{2}} \sqrt{c + d x^{n}} \left (a d + 3 b c\right )}{40 b^{2} d^{2} n} + \frac{\left (a + b x^{n}\right )^{\frac{5}{2}} \sqrt{c + d x^{n}} \left (3 a^{2} d^{2} + 14 a b c d + 63 b^{2} c^{2}\right )}{240 b^{2} d^{3} n} + \frac{\left (a + b x^{n}\right )^{\frac{3}{2}} \sqrt{c + d x^{n}} \left (a d - b c\right ) \left (3 a^{2} d^{2} + 14 a b c d + 63 b^{2} c^{2}\right )}{192 b^{2} d^{4} n} + \frac{\sqrt{a + b x^{n}} \sqrt{c + d x^{n}} \left (a d - b c\right )^{2} \left (3 a^{2} d^{2} + 14 a b c d + 63 b^{2} c^{2}\right )}{128 b^{2} d^{5} n} + \frac{\left (a d - b c\right )^{3} \left (3 a^{2} d^{2} + 14 a b c d + 63 b^{2} c^{2}\right ) \operatorname{atanh}{\left (\frac{\sqrt{d} \sqrt{a + b x^{n}}}{\sqrt{b} \sqrt{c + d x^{n}}} \right )}}{128 b^{\frac{5}{2}} d^{\frac{11}{2}} n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**(-1+3*n)*(a+b*x**n)**(5/2)/(c+d*x**n)**(1/2),x)

[Out]

x**n*(a + b*x**n)**(7/2)*sqrt(c + d*x**n)/(5*b*d*n) - 3*(a + b*x**n)**(7/2)*sqrt
(c + d*x**n)*(a*d + 3*b*c)/(40*b**2*d**2*n) + (a + b*x**n)**(5/2)*sqrt(c + d*x**
n)*(3*a**2*d**2 + 14*a*b*c*d + 63*b**2*c**2)/(240*b**2*d**3*n) + (a + b*x**n)**(
3/2)*sqrt(c + d*x**n)*(a*d - b*c)*(3*a**2*d**2 + 14*a*b*c*d + 63*b**2*c**2)/(192
*b**2*d**4*n) + sqrt(a + b*x**n)*sqrt(c + d*x**n)*(a*d - b*c)**2*(3*a**2*d**2 +
14*a*b*c*d + 63*b**2*c**2)/(128*b**2*d**5*n) + (a*d - b*c)**3*(3*a**2*d**2 + 14*
a*b*c*d + 63*b**2*c**2)*atanh(sqrt(d)*sqrt(a + b*x**n)/(sqrt(b)*sqrt(c + d*x**n)
))/(128*b**(5/2)*d**(11/2)*n)

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Mathematica [A]  time = 0.62574, size = 293, normalized size = 0.82 \[ \frac{2 \sqrt{b} \sqrt{d} \sqrt{a+b x^n} \sqrt{c+d x^n} \left (-45 a^4 d^4+30 a^3 b d^3 \left (d x^n-3 c\right )+2 a^2 b^2 d^2 \left (782 c^2-481 c d x^n+372 d^2 x^{2 n}\right )+2 a b^3 d \left (-1155 c^3+749 c^2 d x^n-592 c d^2 x^{2 n}+504 d^3 x^{3 n}\right )+b^4 \left (945 c^4-630 c^3 d x^n+504 c^2 d^2 x^{2 n}-432 c d^3 x^{3 n}+384 d^4 x^{4 n}\right )\right )-15 (b c-a d)^3 \left (3 a^2 d^2+14 a b c d+63 b^2 c^2\right ) \log \left (2 \sqrt{b} \sqrt{d} \sqrt{a+b x^n} \sqrt{c+d x^n}+a d+b c+2 b d x^n\right )}{3840 b^{5/2} d^{11/2} n} \]

Antiderivative was successfully verified.

[In]  Integrate[(x^(-1 + 3*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]*(-45*a^4*d^4 + 30*a^3*b*d^3*(
-3*c + d*x^n) + 2*a^2*b^2*d^2*(782*c^2 - 481*c*d*x^n + 372*d^2*x^(2*n)) + 2*a*b^
3*d*(-1155*c^3 + 749*c^2*d*x^n - 592*c*d^2*x^(2*n) + 504*d^3*x^(3*n)) + b^4*(945
*c^4 - 630*c^3*d*x^n + 504*c^2*d^2*x^(2*n) - 432*c*d^3*x^(3*n) + 384*d^4*x^(4*n)
)) - 15*(b*c - a*d)^3*(63*b^2*c^2 + 14*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]])/(3840*b^(5/2)*d^(11/
2)*n)

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Maple [F]  time = 0.084, size = 0, normalized size = 0. \[ \int{{x}^{-1+3\,n} \left ( a+b{x}^{n} \right ) ^{{\frac{5}{2}}}{\frac{1}{\sqrt{c+d{x}^{n}}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(-1+3*n)*(a+b*x^n)^(5/2)/(c+d*x^n)^(1/2),x)

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.38369, size = 1, normalized size = 0. \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^n + a)^(5/2)*x^(3*n - 1)/sqrt(d*x^n + c),x, algorithm="fricas")

[Out]

[1/7680*(4*(384*sqrt(b*d)*b^4*d^4*x^(4*n) - 144*(3*b^4*c*d^3 - 7*a*b^3*d^4)*sqrt
(b*d)*x^(3*n) + 8*(63*b^4*c^2*d^2 - 148*a*b^3*c*d^3 + 93*a^2*b^2*d^4)*sqrt(b*d)*
x^(2*n) - 2*(315*b^4*c^3*d - 749*a*b^3*c^2*d^2 + 481*a^2*b^2*c*d^3 - 15*a^3*b*d^
4)*sqrt(b*d)*x^n + (945*b^4*c^4 - 2310*a*b^3*c^3*d + 1564*a^2*b^2*c^2*d^2 - 90*a
^3*b*c*d^3 - 45*a^4*d^4)*sqrt(b*d))*sqrt(b*x^n + a)*sqrt(d*x^n + c) - 15*(63*b^5
*c^5 - 175*a*b^4*c^4*d + 150*a^2*b^3*c^3*d^2 - 30*a^3*b^2*c^2*d^3 - 5*a^4*b*c*d^
4 - 3*a^5*d^5)*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^2*d^5*n), 1/3840*(2*(384
*sqrt(-b*d)*b^4*d^4*x^(4*n) - 144*(3*b^4*c*d^3 - 7*a*b^3*d^4)*sqrt(-b*d)*x^(3*n)
+ 8*(63*b^4*c^2*d^2 - 148*a*b^3*c*d^3 + 93*a^2*b^2*d^4)*sqrt(-b*d)*x^(2*n) - 2*
(315*b^4*c^3*d - 749*a*b^3*c^2*d^2 + 481*a^2*b^2*c*d^3 - 15*a^3*b*d^4)*sqrt(-b*d
)*x^n + (945*b^4*c^4 - 2310*a*b^3*c^3*d + 1564*a^2*b^2*c^2*d^2 - 90*a^3*b*c*d^3
- 45*a^4*d^4)*sqrt(-b*d))*sqrt(b*x^n + a)*sqrt(d*x^n + c) - 15*(63*b^5*c^5 - 175
*a*b^4*c^4*d + 150*a^2*b^3*c^3*d^2 - 30*a^3*b^2*c^2*d^3 - 5*a^4*b*c*d^4 - 3*a^5*
d^5)*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^5*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)**(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. \[ \int \frac{{\left (b x^{n} + a\right )}^{\frac{5}{2}} x^{3 \, n - 1}}{\sqrt{d x^{n} + c}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^n + a)^(5/2)*x^(3*n - 1)/sqrt(d*x^n + c),x, algorithm="giac")

[Out]

integrate((b*x^n + a)^(5/2)*x^(3*n - 1)/sqrt(d*x^n + c), x)