3.708 \(\int x^m (a+b x)^{3/2} \, dx\)

Optimal. Leaf size=48 \[ \frac{2 x^m (a+b x)^{5/2} \left (-\frac{b x}{a}\right )^{-m} \, _2F_1\left (\frac{5}{2},-m;\frac{7}{2};\frac{b x}{a}+1\right )}{5 b} \]

[Out]

(2*x^m*(a + b*x)^(5/2)*Hypergeometric2F1[5/2, -m, 7/2, 1 + (b*x)/a])/(5*b*(-((b*
x)/a))^m)

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Rubi [A]  time = 0.0388495, antiderivative size = 48, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154 \[ \frac{2 x^m (a+b x)^{5/2} \left (-\frac{b x}{a}\right )^{-m} \, _2F_1\left (\frac{5}{2},-m;\frac{7}{2};\frac{b x}{a}+1\right )}{5 b} \]

Antiderivative was successfully verified.

[In]  Int[x^m*(a + b*x)^(3/2),x]

[Out]

(2*x^m*(a + b*x)^(5/2)*Hypergeometric2F1[5/2, -m, 7/2, 1 + (b*x)/a])/(5*b*(-((b*
x)/a))^m)

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Rubi in Sympy [A]  time = 6.11254, size = 37, normalized size = 0.77 \[ \frac{2 x^{m} \left (- \frac{b x}{a}\right )^{- m} \left (a + b x\right )^{\frac{5}{2}}{{}_{2}F_{1}\left (\begin{matrix} - m, \frac{5}{2} \\ \frac{7}{2} \end{matrix}\middle |{1 + \frac{b x}{a}} \right )}}{5 b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*x**m*(-b*x/a)**(-m)*(a + b*x)**(5/2)*hyper((-m, 5/2), (7/2,), 1 + b*x/a)/(5*b)

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Mathematica [A]  time = 0.0717578, size = 83, normalized size = 1.73 \[ \frac{x^{m+1} \sqrt{a+b x} \left (a (m+2) \, _2F_1\left (-\frac{1}{2},m+1;m+2;-\frac{b x}{a}\right )+b (m+1) x \, _2F_1\left (-\frac{1}{2},m+2;m+3;-\frac{b x}{a}\right )\right )}{(m+1) (m+2) \sqrt{\frac{b x}{a}+1}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [F]  time = 0.026, size = 0, normalized size = 0. \[ \int{x}^{m} \left ( bx+a \right ) ^{{\frac{3}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^m*(b*x+a)^(3/2),x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((b*x + a)^(3/2)*x^m, x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left ({\left (b x + a\right )}^{\frac{3}{2}} x^{m}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral((b*x + a)^(3/2)*x^m, x)

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Sympy [A]  time = 23.4338, size = 37, normalized size = 0.77 \[ \frac{a^{\frac{3}{2}} x x^{m} \Gamma \left (m + 1\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{3}{2}, m + 1 \\ m + 2 \end{matrix}\middle |{\frac{b x e^{i \pi }}{a}} \right )}}{\Gamma \left (m + 2\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a**(3/2)*x*x**m*gamma(m + 1)*hyper((-3/2, m + 1), (m + 2,), b*x*exp_polar(I*pi)/
a)/gamma(m + 2)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((b*x + a)^(3/2)*x^m, x)