3.2972 \(\int (d x)^m \sqrt{a+b \sqrt{c x^3}} \, dx\)

Optimal. Leaf size=84 \[ \frac{x (d x)^m \sqrt{a+b \sqrt{c x^3}} \, _2F_1\left (-\frac{1}{2},\frac{2 (m+1)}{3};\frac{1}{3} (2 m+5);-\frac{b \sqrt{c x^3}}{a}\right )}{(m+1) \sqrt{\frac{b \sqrt{c x^3}}{a}+1}} \]

[Out]

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

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Rubi [A]  time = 0.1712, antiderivative size = 84, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.217 \[ \frac{x (d x)^m \sqrt{a+b \sqrt{c x^3}} \, _2F_1\left (-\frac{1}{2},\frac{2 (m+1)}{3};\frac{1}{3} (2 m+5);-\frac{b \sqrt{c x^3}}{a}\right )}{(m+1) \sqrt{\frac{b \sqrt{c x^3}}{a}+1}} \]

Antiderivative was successfully verified.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0665177, size = 84, normalized size = 1. \[ \frac{x (d x)^m \sqrt{a+b \sqrt{c x^3}} \, _2F_1\left (-\frac{1}{2},\frac{2 (m+1)}{3};\frac{1}{3} (2 m+5);-\frac{b \sqrt{c x^3}}{a}\right )}{(m+1) \sqrt{\frac{b \sqrt{c x^3}}{a}+1}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [F]  time = 0.053, size = 0, normalized size = 0. \[ \int \left ( dx \right ) ^{m}\sqrt{a+b\sqrt{c{x}^{3}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(sqrt(sqrt(c*x^3)*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(sqrt(c*x^3)*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. \[ \int \left (d x\right )^{m} \sqrt{a + b \sqrt{c x^{3}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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