3.707 \(\int x^m (a+b x)^{5/2} \, dx\)

Optimal. Leaf size=48 \[ \frac{2 x^m (a+b x)^{7/2} \left (-\frac{b x}{a}\right )^{-m} \, _2F_1\left (\frac{7}{2},-m;\frac{9}{2};\frac{b x}{a}+1\right )}{7 b} \]

[Out]

(2*x^m*(a + b*x)^(7/2)*Hypergeometric2F1[7/2, -m, 9/2, 1 + (b*x)/a])/(7*b*(-((b*
x)/a))^m)

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Rubi [A]  time = 0.0399649, 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 \[ \frac{2 x^m (a+b x)^{7/2} \left (-\frac{b x}{a}\right )^{-m} \, _2F_1\left (\frac{7}{2},-m;\frac{9}{2};\frac{b x}{a}+1\right )}{7 b} \]

Antiderivative was successfully verified.

[In]  Int[x^m*(a + b*x)^(5/2),x]

[Out]

(2*x^m*(a + b*x)^(7/2)*Hypergeometric2F1[7/2, -m, 9/2, 1 + (b*x)/a])/(7*b*(-((b*
x)/a))^m)

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Rubi in Sympy [A]  time = 5.94886, size = 37, normalized size = 0.77 \[ \frac{2 x^{m} \left (- \frac{b x}{a}\right )^{- m} \left (a + b x\right )^{\frac{7}{2}}{{}_{2}F_{1}\left (\begin{matrix} - m, \frac{7}{2} \\ \frac{9}{2} \end{matrix}\middle |{1 + \frac{b x}{a}} \right )}}{7 b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**m*(b*x+a)**(5/2),x)

[Out]

2*x**m*(-b*x/a)**(-m)*(a + b*x)**(7/2)*hyper((-m, 7/2), (9/2,), 1 + b*x/a)/(7*b)

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Mathematica [B]  time = 0.148806, size = 125, normalized size = 2.6 \[ \frac{x^{m+1} \sqrt{a+b x} \left (a^2 \left (m^2+5 m+6\right ) \, _2F_1\left (-\frac{1}{2},m+1;m+2;-\frac{b x}{a}\right )+b (m+1) x \left (2 a (m+3) \, _2F_1\left (-\frac{1}{2},m+2;m+3;-\frac{b x}{a}\right )+b (m+2) x \, _2F_1\left (-\frac{1}{2},m+3;m+4;-\frac{b x}{a}\right )\right )\right )}{(m+1) (m+2) (m+3) \sqrt{\frac{b x}{a}+1}} \]

Antiderivative was successfully verified.

[In]  Integrate[x^m*(a + b*x)^(5/2),x]

[Out]

(x^(1 + m)*Sqrt[a + b*x]*(a^2*(6 + 5*m + m^2)*Hypergeometric2F1[-1/2, 1 + m, 2 +
m, -((b*x)/a)] + b*(1 + m)*x*(2*a*(3 + m)*Hypergeometric2F1[-1/2, 2 + m, 3 + m,
-((b*x)/a)] + b*(2 + m)*x*Hypergeometric2F1[-1/2, 3 + m, 4 + m, -((b*x)/a)])))/
((1 + m)*(2 + m)*(3 + m)*Sqrt[1 + (b*x)/a])

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Maple [F]  time = 0.027, size = 0, normalized size = 0. \[ \int{x}^{m} \left ( bx+a \right ) ^{{\frac{5}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^m*(b*x+a)^(5/2),x)

[Out]

int(x^m*(b*x+a)^(5/2),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int{\left (b x + a\right )}^{\frac{5}{2}} x^{m}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^(5/2)*x^m,x, algorithm="maxima")

[Out]

integrate((b*x + a)^(5/2)*x^m, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^(5/2)*x^m,x, algorithm="fricas")

[Out]

integral((b^2*x^2 + 2*a*b*x + a^2)*sqrt(b*x + a)*x^m, x)

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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**m*(b*x+a)**(5/2),x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int{\left (b x + a\right )}^{\frac{5}{2}} x^{m}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^(5/2)*x^m,x, algorithm="giac")

[Out]

integrate((b*x + a)^(5/2)*x^m, x)