Optimal. Leaf size=88 \[ \frac{2 (d x)^{m+1} \left (a+b \sqrt{c x^2}\right )^{3/2} \left (-\frac{b \sqrt{c x^2}}{a}\right )^{-m} \, _2F_1\left (\frac{3}{2},-m;\frac{5}{2};\frac{\sqrt{c x^2} b}{a}+1\right )}{3 b d \sqrt{c x^2}} \]
[Out]
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Rubi [A] time = 0.108945, antiderivative size = 88, 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 \[ \frac{2 (d x)^{m+1} \left (a+b \sqrt{c x^2}\right )^{3/2} \left (-\frac{b \sqrt{c x^2}}{a}\right )^{-m} \, _2F_1\left (\frac{3}{2},-m;\frac{5}{2};\frac{\sqrt{c x^2} b}{a}+1\right )}{3 b d \sqrt{c x^2}} \]
Antiderivative was successfully verified.
[In] Int[(d*x)^m*Sqrt[a + b*Sqrt[c*x^2]],x]
[Out]
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Rubi in Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \left (d x\right )^{m} \sqrt{a + b \sqrt{c x^{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((d*x)**m*(a+b*(c*x**2)**(1/2))**(1/2),x)
[Out]
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Mathematica [A] time = 0.0623717, size = 74, normalized size = 0.84 \[ \frac{x (d x)^m \sqrt{a+b \sqrt{c x^2}} \, _2F_1\left (-\frac{1}{2},m+1;m+2;-\frac{b \sqrt{c x^2}}{a}\right )}{(m+1) \sqrt{\frac{b \sqrt{c x^2}}{a}+1}} \]
Antiderivative was successfully verified.
[In] Integrate[(d*x)^m*Sqrt[a + b*Sqrt[c*x^2]],x]
[Out]
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Maple [F] time = 0.046, size = 0, normalized size = 0. \[ \int \left ( dx \right ) ^{m}\sqrt{a+b\sqrt{c{x}^{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((d*x)^m*(a+b*(c*x^2)^(1/2))^(1/2),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \sqrt{\sqrt{c x^{2}} b + a} \left (d x\right )^{m}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(sqrt(c*x^2)*b + a)*(d*x)^m,x, algorithm="maxima")
[Out]
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\sqrt{\sqrt{c x^{2}} b + a} \left (d x\right )^{m}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(sqrt(c*x^2)*b + a)*(d*x)^m,x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \left (d x\right )^{m} \sqrt{a + b \sqrt{c x^{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x)**m*(a+b*(c*x**2)**(1/2))**(1/2),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \sqrt{\sqrt{c x^{2}} b + a} \left (d x\right )^{m}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(sqrt(c*x^2)*b + a)*(d*x)^m,x, algorithm="giac")
[Out]