3.720 \(\int \left (\frac{a+b+c x^2}{d}\right )^m \, dx\)

Optimal. Leaf size=49 \[ \frac{d x \left (\frac{a+b}{d}+\frac{c x^2}{d}\right )^{m+1} \, _2F_1\left (1,m+\frac{3}{2};\frac{3}{2};-\frac{c x^2}{a+b}\right )}{a+b} \]

[Out]

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

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Rubi [A]  time = 0.0401009, antiderivative size = 57, normalized size of antiderivative = 1.16, number of steps used = 3, number of rules used = 3, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.214 \[ x \left (\frac{c x^2}{a+b}+1\right )^{-m} \left (\frac{a+b}{d}+\frac{c x^2}{d}\right )^m \, _2F_1\left (\frac{1}{2},-m;\frac{3}{2};-\frac{c x^2}{a+b}\right ) \]

Antiderivative was successfully verified.

[In]  Int[((a + b + c*x^2)/d)^m,x]

[Out]

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

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Rubi in Sympy [A]  time = 3.9821, size = 42, normalized size = 0.86 \[ x \left (\frac{c x^{2}}{d} + \frac{a + b}{d}\right )^{m} \left (\frac{c x^{2}}{a + b} + 1\right )^{- m}{{}_{2}F_{1}\left (\begin{matrix} - m, \frac{1}{2} \\ \frac{3}{2} \end{matrix}\middle |{- \frac{c x^{2}}{a + b}} \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0283969, size = 53, normalized size = 1.08 \[ x \left (\frac{c x^2}{a+b}+1\right )^{-m} \left (\frac{a+b+c x^2}{d}\right )^m \, _2F_1\left (\frac{1}{2},-m;\frac{3}{2};-\frac{c x^2}{a+b}\right ) \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [F]  time = 0.055, size = 0, normalized size = 0. \[ \int \left ({\frac{c{x}^{2}+a+b}{d}} \right ) ^{m}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(((c*x^2+a+b)/d)^m,x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(((c*x^2 + a + b)/d)^m, x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\left (\frac{c x^{2} + a + b}{d}\right )^{m}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral(((c*x^2 + a + b)/d)^m, x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(((a + b + c*x**2)/d)**m, x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(((c*x^2 + a + b)/d)^m, x)