3.723 \(\int (d+e x) \left (a+c x^2\right )^p \, dx\)

Optimal. Leaf size=61 \[ \frac{d x \left (a+c x^2\right )^{p+1} \, _2F_1\left (1,p+\frac{3}{2};\frac{3}{2};-\frac{c x^2}{a}\right )}{a}+\frac{e \left (a+c x^2\right )^{p+1}}{2 c (p+1)} \]

[Out]

(e*(a + c*x^2)^(1 + p))/(2*c*(1 + p)) + (d*x*(a + c*x^2)^(1 + p)*Hypergeometric2
F1[1, 3/2 + p, 3/2, -((c*x^2)/a)])/a

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Rubi [A]  time = 0.0638907, antiderivative size = 70, normalized size of antiderivative = 1.15, number of steps used = 3, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ d x \left (a+c x^2\right )^p \left (\frac{c x^2}{a}+1\right )^{-p} \, _2F_1\left (\frac{1}{2},-p;\frac{3}{2};-\frac{c x^2}{a}\right )+\frac{e \left (a+c x^2\right )^{p+1}}{2 c (p+1)} \]

Antiderivative was successfully verified.

[In]  Int[(d + e*x)*(a + c*x^2)^p,x]

[Out]

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

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Rubi in Sympy [A]  time = 8.82142, size = 54, normalized size = 0.89 \[ d x \left (1 + \frac{c x^{2}}{a}\right )^{- p} \left (a + c x^{2}\right )^{p}{{}_{2}F_{1}\left (\begin{matrix} - p, \frac{1}{2} \\ \frac{3}{2} \end{matrix}\middle |{- \frac{c x^{2}}{a}} \right )} + \frac{e \left (a + c x^{2}\right )^{p + 1}}{2 c \left (p + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0835418, size = 98, normalized size = 1.61 \[ \frac{\left (a+c x^2\right )^p \left (\frac{c x^2}{a}+1\right )^{-p} \left (2 c d (p+1) x \, _2F_1\left (\frac{1}{2},-p;\frac{3}{2};-\frac{c x^2}{a}\right )+c e x^2 \left (\frac{c x^2}{a}+1\right )^p+a e \left (\left (\frac{c x^2}{a}+1\right )^p-1\right )\right )}{2 c (p+1)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [F]  time = 0.046, size = 0, normalized size = 0. \[ \int \left ( ex+d \right ) \left ( c{x}^{2}+a \right ) ^{p}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x+d)*(c*x^2+a)^p,x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((e*x + d)*(c*x^2 + a)^p, x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral((e*x + d)*(c*x^2 + a)^p, x)

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Sympy [A]  time = 21.3951, size = 61, normalized size = 1. \[ a^{p} d x{{}_{2}F_{1}\left (\begin{matrix} \frac{1}{2}, - p \\ \frac{3}{2} \end{matrix}\middle |{\frac{c x^{2} e^{i \pi }}{a}} \right )} + e \left (\begin{cases} \frac{a^{p} x^{2}}{2} & \text{for}\: c = 0 \\\frac{\begin{cases} \frac{\left (a + c x^{2}\right )^{p + 1}}{p + 1} & \text{for}\: p \neq -1 \\\log{\left (a + c x^{2} \right )} & \text{otherwise} \end{cases}}{2 c} & \text{otherwise} \end{cases}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a**p*d*x*hyper((1/2, -p), (3/2,), c*x**2*exp_polar(I*pi)/a) + e*Piecewise((a**p*
x**2/2, Eq(c, 0)), (Piecewise(((a + c*x**2)**(p + 1)/(p + 1), Ne(p, -1)), (log(a
+ c*x**2), True))/(2*c), True))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((e*x + d)*(c*x^2 + a)^p, x)