Optimal. Leaf size=86 \[ \frac{(d x)^{m+1} \sqrt{a+b \left (c x^2\right )^{3/2}} \, _2F_1\left (-\frac{1}{2},\frac{m+1}{3};\frac{m+4}{3};-\frac{b \left (c x^2\right )^{3/2}}{a}\right )}{d (m+1) \sqrt{\frac{b \left (c x^2\right )^{3/2}}{a}+1}} \]
[Out]
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Rubi [A] time = 0.143826, antiderivative size = 86, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.13 \[ \frac{(d x)^{m+1} \sqrt{a+b \left (c x^2\right )^{3/2}} \, _2F_1\left (-\frac{1}{2},\frac{m+1}{3};\frac{m+4}{3};-\frac{b \left (c x^2\right )^{3/2}}{a}\right )}{d (m+1) \sqrt{\frac{b \left (c x^2\right )^{3/2}}{a}+1}} \]
Antiderivative was successfully verified.
[In] Int[(d*x)^m*Sqrt[a + b*(c*x^2)^(3/2)],x]
[Out]
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Rubi in Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \left (d x\right )^{m} \sqrt{a + b \left (c x^{2}\right )^{\frac{3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((d*x)**m*(a+b*(c*x**2)**(3/2))**(1/2),x)
[Out]
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Mathematica [A] time = 0.0485507, size = 0, normalized size = 0. \[ \int (d x)^m \sqrt{a+b \left (c x^2\right )^{3/2}} \, dx \]
Verification is Not applicable to the result.
[In] Integrate[(d*x)^m*Sqrt[a + b*(c*x^2)^(3/2)],x]
[Out]
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Maple [F] time = 0.044, size = 0, normalized size = 0. \[ \int \left ( dx \right ) ^{m}\sqrt{a+b \left ( c{x}^{2} \right ) ^{{\frac{3}{2}}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((d*x)^m*(a+b*(c*x^2)^(3/2))^(1/2),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \sqrt{\left (c x^{2}\right )^{\frac{3}{2}} b + a} \left (d x\right )^{m}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt((c*x^2)^(3/2)*b + a)*(d*x)^m,x, algorithm="maxima")
[Out]
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\sqrt{\sqrt{c x^{2}} b c x^{2} + a} \left (d x\right )^{m}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt((c*x^2)^(3/2)*b + a)*(d*x)^m,x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \left (d x\right )^{m} \sqrt{a + b \left (c x^{2}\right )^{\frac{3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x)**m*(a+b*(c*x**2)**(3/2))**(1/2),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \sqrt{\left (c x^{2}\right )^{\frac{3}{2}} b + a} \left (d x\right )^{m}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt((c*x^2)^(3/2)*b + a)*(d*x)^m,x, algorithm="giac")
[Out]