3.426 \(\int \left (a x^j+b x^n\right )^{3/2} \, dx\)

Optimal. Leaf size=97 \[ \frac{2 b x^{n+1} \sqrt{a x^j+b x^n} \, _2F_1\left (-\frac{3}{2},\frac{\frac{3 n}{2}+1}{j-n};\frac{2 j+n+2}{2 (j-n)};-\frac{a x^{j-n}}{b}\right )}{(3 n+2) \sqrt{\frac{a x^{j-n}}{b}+1}} \]

[Out]

(2*b*x^(1 + n)*Sqrt[a*x^j + b*x^n]*Hypergeometric2F1[-3/2, (1 + (3*n)/2)/(j - n)
, (2 + 2*j + n)/(2*(j - n)), -((a*x^(j - n))/b)])/((2 + 3*n)*Sqrt[1 + (a*x^(j -
n))/b])

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Rubi [A]  time = 0.135562, antiderivative size = 97, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ \frac{2 b x^{n+1} \sqrt{a x^j+b x^n} \, _2F_1\left (-\frac{3}{2},\frac{\frac{3 n}{2}+1}{j-n};\frac{2 j+n+2}{2 (j-n)};-\frac{a x^{j-n}}{b}\right )}{(3 n+2) \sqrt{\frac{a x^{j-n}}{b}+1}} \]

Antiderivative was successfully verified.

[In]  Int[(a*x^j + b*x^n)^(3/2),x]

[Out]

(2*b*x^(1 + n)*Sqrt[a*x^j + b*x^n]*Hypergeometric2F1[-3/2, (1 + (3*n)/2)/(j - n)
, (2 + 2*j + n)/(2*(j - n)), -((a*x^(j - n))/b)])/((2 + 3*n)*Sqrt[1 + (a*x^(j -
n))/b])

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Rubi in Sympy [A]  time = 12.4465, size = 80, normalized size = 0.82 \[ \frac{2 b x^{- \frac{n}{2}} x^{\frac{3 n}{2} + 1} \sqrt{a x^{j} + b x^{n}}{{}_{2}F_{1}\left (\begin{matrix} - \frac{3}{2}, \frac{\frac{3 n}{2} + 1}{j - n} \\ \frac{j + \frac{n}{2} + 1}{j - n} \end{matrix}\middle |{- \frac{a x^{j - n}}{b}} \right )}}{\left (3 n + 2\right ) \sqrt{\frac{a x^{j - n}}{b} + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((a*x**j+b*x**n)**(3/2),x)

[Out]

2*b*x**(-n/2)*x**(3*n/2 + 1)*sqrt(a*x**j + b*x**n)*hyper((-3/2, (3*n/2 + 1)/(j -
n)), ((j + n/2 + 1)/(j - n),), -a*x**(j - n)/b)/((3*n + 2)*sqrt(a*x**(j - n)/b
+ 1))

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Mathematica [A]  time = 0.316899, size = 177, normalized size = 1.82 \[ \frac{2 x \left (3 a^2 (j-n)^2 x^{2 j} \sqrt{\frac{a x^{j-n}}{b}+1} \, _2F_1\left (\frac{1}{2},\frac{4 j-n+2}{2 j-2 n};\frac{6 j-3 n+2}{2 j-2 n};-\frac{a x^{j-n}}{b}\right )+(4 j-n+2) \left (a x^j+b x^n\right ) \left (a (-j+4 n+2) x^j+b (2 j+n+2) x^n\right )\right )}{(3 n+2) (4 j-n+2) (2 j+n+2) \sqrt{a x^j+b x^n}} \]

Antiderivative was successfully verified.

[In]  Integrate[(a*x^j + b*x^n)^(3/2),x]

[Out]

(2*x*((2 + 4*j - n)*(a*x^j + b*x^n)*(a*(2 - j + 4*n)*x^j + b*(2 + 2*j + n)*x^n)
+ 3*a^2*(j - n)^2*x^(2*j)*Sqrt[1 + (a*x^(j - n))/b]*Hypergeometric2F1[1/2, (2 +
4*j - n)/(2*j - 2*n), (2 + 6*j - 3*n)/(2*j - 2*n), -((a*x^(j - n))/b)]))/((2 + 4
*j - n)*(2 + 2*j + n)*(2 + 3*n)*Sqrt[a*x^j + b*x^n])

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Maple [F]  time = 0.118, size = 0, normalized size = 0. \[ \int \left ( a{x}^{j}+b{x}^{n} \right ) ^{{\frac{3}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a*x^j+b*x^n)^(3/2),x)

[Out]

int((a*x^j+b*x^n)^(3/2),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int{\left (a x^{j} + b x^{n}\right )}^{\frac{3}{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a*x^j + b*x^n)^(3/2),x, algorithm="maxima")

[Out]

integrate((a*x^j + b*x^n)^(3/2), x)

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Fricas [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a*x^j + b*x^n)^(3/2),x, algorithm="fricas")

[Out]

Exception raised: TypeError

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \left (a x^{j} + b x^{n}\right )^{\frac{3}{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a*x**j+b*x**n)**(3/2),x)

[Out]

Integral((a*x**j + b*x**n)**(3/2), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a*x^j + b*x^n)^(3/2),x, algorithm="giac")

[Out]

integrate((a*x^j + b*x^n)^(3/2), x)