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

Optimal. Leaf size=75 \[ \frac{d \left (a+c x^2\right )^{p+1}}{2 c (p+1)}+\frac{1}{3} e x^3 \left (a+c x^2\right )^p \left (\frac{c x^2}{a}+1\right )^{-p} \, _2F_1\left (\frac{3}{2},-p;\frac{5}{2};-\frac{c x^2}{a}\right ) \]

[Out]

(d*(a + c*x^2)^(1 + p))/(2*c*(1 + p)) + (e*x^3*(a + c*x^2)^p*Hypergeometric2F1[3
/2, -p, 5/2, -((c*x^2)/a)])/(3*(1 + (c*x^2)/a)^p)

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Rubi [A]  time = 0.0872229, antiderivative size = 75, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25 \[ \frac{d \left (a+c x^2\right )^{p+1}}{2 c (p+1)}+\frac{1}{3} e x^3 \left (a+c x^2\right )^p \left (\frac{c x^2}{a}+1\right )^{-p} \, _2F_1\left (\frac{3}{2},-p;\frac{5}{2};-\frac{c x^2}{a}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(d*(a + c*x^2)^(1 + p))/(2*c*(1 + p)) + (e*x^3*(a + c*x^2)^p*Hypergeometric2F1[3
/2, -p, 5/2, -((c*x^2)/a)])/(3*(1 + (c*x^2)/a)^p)

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Rubi in Sympy [A]  time = 12.184, size = 58, normalized size = 0.77 \[ \frac{e x^{3} \left (1 + \frac{c x^{2}}{a}\right )^{- p} \left (a + c x^{2}\right )^{p}{{}_{2}F_{1}\left (\begin{matrix} - p, \frac{3}{2} \\ \frac{5}{2} \end{matrix}\middle |{- \frac{c x^{2}}{a}} \right )}}{3} + \frac{d \left (a + c x^{2}\right )^{p + 1}}{2 c \left (p + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

e*x**3*(1 + c*x**2/a)**(-p)*(a + c*x**2)**p*hyper((-p, 3/2), (5/2,), -c*x**2/a)/
3 + d*(a + c*x**2)**(p + 1)/(2*c*(p + 1))

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Mathematica [A]  time = 0.0913145, size = 102, normalized size = 1.36 \[ \frac{\left (a+c x^2\right )^p \left (\frac{c x^2}{a}+1\right )^{-p} \left (3 d \left (c x^2 \left (\frac{c x^2}{a}+1\right )^p+a \left (\left (\frac{c x^2}{a}+1\right )^p-1\right )\right )+2 c e (p+1) x^3 \, _2F_1\left (\frac{3}{2},-p;\frac{5}{2};-\frac{c x^2}{a}\right )\right )}{6 c (p+1)} \]

Antiderivative was successfully verified.

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

[Out]

((a + c*x^2)^p*(3*d*(c*x^2*(1 + (c*x^2)/a)^p + a*(-1 + (1 + (c*x^2)/a)^p)) + 2*c
*e*(1 + p)*x^3*Hypergeometric2F1[3/2, -p, 5/2, -((c*x^2)/a)]))/(6*c*(1 + p)*(1 +
(c*x^2)/a)^p)

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Maple [F]  time = 0.052, size = 0, normalized size = 0. \[ \int x \left ( ex+d \right ) \left ( c{x}^{2}+a \right ) ^{p}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x*(e*x+d)*(c*x^2+a)^p,x)

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: ValueError

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 32.5469, size = 65, normalized size = 0.87 \[ \frac{a^{p} e x^{3}{{}_{2}F_{1}\left (\begin{matrix} \frac{3}{2}, - p \\ \frac{5}{2} \end{matrix}\middle |{\frac{c x^{2} e^{i \pi }}{a}} \right )}}{3} + d \left (\begin{cases} \frac{a^{p} x^{2}}{2} & \text{for}\: c = 0 \\\frac{\begin{cases} \frac{\left (a + c x^{2}\right )^{p + 1}}{p + 1} & \text{for}\: p \neq -1 \\\log{\left (a + c x^{2} \right )} & \text{otherwise} \end{cases}}{2 c} & \text{otherwise} \end{cases}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a**p*e*x**3*hyper((3/2, -p), (5/2,), c*x**2*exp_polar(I*pi)/a)/3 + d*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. \[ \int{\left (e x + d\right )}{\left (c x^{2} + a\right )}^{p} x\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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