3.2949 \(\int (d x)^m \sqrt{a+b \sqrt{c x^2}} \, dx\)

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]

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

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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]

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

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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]

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

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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]

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

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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]

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

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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]

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

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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]

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

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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]

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

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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]

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