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

Optimal. Leaf size=107 \[ \frac{2 b x^{n+1} (c x)^m \sqrt{a x^j+b x^n} \, _2F_1\left (-\frac{3}{2},\frac{m+\frac{3 n}{2}+1}{j-n};\frac{m+\frac{3 n}{2}+1}{j-n}+1;-\frac{a x^{j-n}}{b}\right )}{(2 m+3 n+2) \sqrt{\frac{a x^{j-n}}{b}+1}} \]

[Out]

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

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Rubi [A]  time = 0.20198, antiderivative size = 107, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143 \[ \frac{2 b x^{n+1} (c x)^m \sqrt{a x^j+b x^n} \, _2F_1\left (-\frac{3}{2},\frac{m+\frac{3 n}{2}+1}{j-n};\frac{m+\frac{3 n}{2}+1}{j-n}+1;-\frac{a x^{j-n}}{b}\right )}{(2 m+3 n+2) \sqrt{\frac{a x^{j-n}}{b}+1}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 24.1581, size = 97, normalized size = 0.91 \[ \frac{2 b x^{- m - \frac{n}{2}} x^{m + \frac{3 n}{2} + 1} \left (c x\right )^{m} \sqrt{a x^{j} + b x^{n}}{{}_{2}F_{1}\left (\begin{matrix} - \frac{3}{2}, \frac{m + \frac{3 n}{2} + 1}{j - n} \\ \frac{j + m + \frac{n}{2} + 1}{j - n} \end{matrix}\middle |{- \frac{a x^{j - n}}{b}} \right )}}{\sqrt{\frac{a x^{j - n}}{b} + 1} \left (2 m + 3 n + 2\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [B]  time = 0.490255, size = 218, normalized size = 2.04 \[ \frac{2 (c x)^m \left (3 a^2 (j-n)^2 x^{2 j+1} \sqrt{\frac{a x^{j-n}}{b}+1} \, _2F_1\left (\frac{1}{2},\frac{4 j+2 m-n+2}{2 j-2 n};\frac{6 j+2 m-3 n+2}{2 j-2 n};-\frac{a x^{j-n}}{b}\right )+x^{-m} (4 j+2 m-n+2) \left (a x^j+b x^n\right ) \left (a (-j+2 m+4 n+2) x^{j+m+1}+b (2 j+2 m+n+2) x^{m+n+1}\right )\right )}{(2 m+3 n+2) (4 j+2 m-n+2) (2 j+2 m+n+2) \sqrt{a x^j+b x^n}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [F]  time = 0.428, size = 0, normalized size = 0. \[ \int \left ( cx \right ) ^{m} \left ( a{x}^{j}+b{x}^{n} \right ) ^{{\frac{3}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((a*x^j + b*x^n)^(3/2)*(c*x)^m, 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)*(c*x)^m,x, algorithm="fricas")

[Out]

Exception raised: TypeError

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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((c*x)**m*(a*x**j+b*x**n)**(3/2),x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((a*x^j + b*x^n)^(3/2)*(c*x)^m, x)