3.804 \(\int (d+e x)^m (d f-e f x)^m \left (a+c x^2\right )^p \, dx\)

Optimal. Leaf size=92 \[ x \left (a+c x^2\right )^p \left (\frac{c x^2}{a}+1\right )^{-p} (d+e x)^m \left (1-\frac{e^2 x^2}{d^2}\right )^{-m} (d f-e f x)^m F_1\left (\frac{1}{2};-p,-m;\frac{3}{2};-\frac{c x^2}{a},\frac{e^2 x^2}{d^2}\right ) \]

[Out]

(x*(d + e*x)^m*(d*f - e*f*x)^m*(a + c*x^2)^p*AppellF1[1/2, -p, -m, 3/2, -((c*x^2
)/a), (e^2*x^2)/d^2])/((1 + (c*x^2)/a)^p*(1 - (e^2*x^2)/d^2)^m)

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Rubi [A]  time = 0.193132, antiderivative size = 92, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.107 \[ x \left (a+c x^2\right )^p \left (\frac{c x^2}{a}+1\right )^{-p} (d+e x)^m \left (1-\frac{e^2 x^2}{d^2}\right )^{-m} (d f-e f x)^m F_1\left (\frac{1}{2};-p,-m;\frac{3}{2};-\frac{c x^2}{a},\frac{e^2 x^2}{d^2}\right ) \]

Antiderivative was successfully verified.

[In]  Int[(d + e*x)^m*(d*f - e*f*x)^m*(a + c*x^2)^p,x]

[Out]

(x*(d + e*x)^m*(d*f - e*f*x)^m*(a + c*x^2)^p*AppellF1[1/2, -p, -m, 3/2, -((c*x^2
)/a), (e^2*x^2)/d^2])/((1 + (c*x^2)/a)^p*(1 - (e^2*x^2)/d^2)^m)

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Rubi in Sympy [A]  time = 48.8009, size = 75, normalized size = 0.82 \[ x \left (1 + \frac{c x^{2}}{a}\right )^{- p} \left (1 - \frac{e^{2} x^{2}}{d^{2}}\right )^{- m} \left (a + c x^{2}\right )^{p} \left (d + e x\right )^{m} \left (d f - e f x\right )^{m} \operatorname{appellf_{1}}{\left (\frac{1}{2},- m,- p,\frac{3}{2},\frac{e^{2} x^{2}}{d^{2}},- \frac{c x^{2}}{a} \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((e*x+d)**m*(-e*f*x+d*f)**m*(c*x**2+a)**p,x)

[Out]

x*(1 + c*x**2/a)**(-p)*(1 - e**2*x**2/d**2)**(-m)*(a + c*x**2)**p*(d + e*x)**m*(
d*f - e*f*x)**m*appellf1(1/2, -m, -p, 3/2, e**2*x**2/d**2, -c*x**2/a)

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Mathematica [A]  time = 0.116997, size = 0, normalized size = 0. \[ \int (d+e x)^m (d f-e f x)^m \left (a+c x^2\right )^p \, dx \]

Verification is Not applicable to the result.

[In]  Integrate[(d + e*x)^m*(d*f - e*f*x)^m*(a + c*x^2)^p,x]

[Out]

Integrate[(d + e*x)^m*(d*f - e*f*x)^m*(a + c*x^2)^p, x]

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Maple [F]  time = 0.23, size = 0, normalized size = 0. \[ \int \left ( ex+d \right ) ^{m} \left ( -efx+df \right ) ^{m} \left ( c{x}^{2}+a \right ) ^{p}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x+d)^m*(-e*f*x+d*f)^m*(c*x^2+a)^p,x)

[Out]

int((e*x+d)^m*(-e*f*x+d*f)^m*(c*x^2+a)^p,x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-e*f*x + d*f)^m*(c*x^2 + a)^p*(e*x + d)^m,x, algorithm="maxima")

[Out]

integrate((-e*f*x + d*f)^m*(c*x^2 + a)^p*(e*x + d)^m, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-e*f*x + d*f)^m*(c*x^2 + a)^p*(e*x + d)^m,x, algorithm="fricas")

[Out]

integral((-e*f*x + d*f)^m*(c*x^2 + a)^p*(e*x + d)^m, x)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x+d)**m*(-e*f*x+d*f)**m*(c*x**2+a)**p,x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-e*f*x + d*f)^m*(c*x^2 + a)^p*(e*x + d)^m,x, algorithm="giac")

[Out]

integrate((-e*f*x + d*f)^m*(c*x^2 + a)^p*(e*x + d)^m, x)