3.2912 \(\int (d x)^m \sqrt{a+b (c x)^{3/2}} \, dx\)

Optimal. Leaf size=78 \[ \frac{x (d x)^m \sqrt{a+b (c x)^{3/2}} \, _2F_1\left (-\frac{1}{2},\frac{2 (m+1)}{3};\frac{1}{3} (2 m+5);-\frac{b (c x)^{3/2}}{a}\right )}{(m+1) \sqrt{\frac{b (c x)^{3/2}}{a}+1}} \]

[Out]

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

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Rubi [A]  time = 0.197142, antiderivative size = 78, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.238 \[ \frac{x (d x)^m \sqrt{a+b (c x)^{3/2}} \, _2F_1\left (-\frac{1}{2},\frac{2 (m+1)}{3};\frac{1}{3} (2 m+5);-\frac{b (c x)^{3/2}}{a}\right )}{(m+1) \sqrt{\frac{b (c x)^{3/2}}{a}+1}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 13.8224, size = 80, normalized size = 1.03 \[ \frac{\left (c x\right )^{- m} \left (c x\right )^{m + 1} \left (d x\right )^{m} \sqrt{a + b \left (c x\right )^{\frac{3}{2}}}{{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{2}, \frac{2 m}{3} + \frac{2}{3} \\ \frac{2 m}{3} + \frac{5}{3} \end{matrix}\middle |{- \frac{b \left (c x\right )^{\frac{3}{2}}}{a}} \right )}}{c \sqrt{1 + \frac{b \left (c x\right )^{\frac{3}{2}}}{a}} \left (m + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(c*x)**(-m)*(c*x)**(m + 1)*(d*x)**m*sqrt(a + b*(c*x)**(3/2))*hyper((-1/2, 2*m/3
+ 2/3), (2*m/3 + 5/3,), -b*(c*x)**(3/2)/a)/(c*sqrt(1 + b*(c*x)**(3/2)/a)*(m + 1)
)

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Mathematica [A]  time = 0.0873848, size = 79, normalized size = 1.01 \[ \frac{x (d x)^m \sqrt{a+b (c x)^{3/2}} \, _2F_1\left (-\frac{1}{2},\frac{2 (m+1)}{3};\frac{2 (m+1)}{3}+1;-\frac{b (c x)^{3/2}}{a}\right )}{(m+1) \sqrt{\frac{a+b (c x)^{3/2}}{a}}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [F]  time = 0.067, size = 0, normalized size = 0. \[ \int \left ( dx \right ) ^{m}\sqrt{a+b \left ( cx \right ) ^{{\frac{3}{2}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

int((d*x)^m*(a+b*(c*x)^(3/2))^(1/2),x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Exception raised: TypeError

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x)**m*(a+b*(c*x)**(3/2))**(1/2),x)

[Out]

Integral((d*x)**m*sqrt(a + b*(c*x)**(3/2)), x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(sqrt((c*x)^(3/2)*b + a)*(d*x)^m, x)