3.503 \(\int x^3 (d+e x) \left (a+c x^2\right )^p \, dx\)

Optimal. Leaf size=100 \[ -\frac{a d \left (a+c x^2\right )^{p+1}}{2 c^2 (p+1)}+\frac{d \left (a+c x^2\right )^{p+2}}{2 c^2 (p+2)}+\frac{1}{5} e x^5 \left (a+c x^2\right )^p \left (\frac{c x^2}{a}+1\right )^{-p} \, _2F_1\left (\frac{5}{2},-p;\frac{7}{2};-\frac{c x^2}{a}\right ) \]

[Out]

-(a*d*(a + c*x^2)^(1 + p))/(2*c^2*(1 + p)) + (d*(a + c*x^2)^(2 + p))/(2*c^2*(2 +
p)) + (e*x^5*(a + c*x^2)^p*Hypergeometric2F1[5/2, -p, 7/2, -((c*x^2)/a)])/(5*(1
+ (c*x^2)/a)^p)

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Rubi [A]  time = 0.158682, antiderivative size = 100, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.278 \[ -\frac{a d \left (a+c x^2\right )^{p+1}}{2 c^2 (p+1)}+\frac{d \left (a+c x^2\right )^{p+2}}{2 c^2 (p+2)}+\frac{1}{5} e x^5 \left (a+c x^2\right )^p \left (\frac{c x^2}{a}+1\right )^{-p} \, _2F_1\left (\frac{5}{2},-p;\frac{7}{2};-\frac{c x^2}{a}\right ) \]

Antiderivative was successfully verified.

[In]  Int[x^3*(d + e*x)*(a + c*x^2)^p,x]

[Out]

-(a*d*(a + c*x^2)^(1 + p))/(2*c^2*(1 + p)) + (d*(a + c*x^2)^(2 + p))/(2*c^2*(2 +
p)) + (e*x^5*(a + c*x^2)^p*Hypergeometric2F1[5/2, -p, 7/2, -((c*x^2)/a)])/(5*(1
+ (c*x^2)/a)^p)

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Rubi in Sympy [A]  time = 20.9585, size = 82, normalized size = 0.82 \[ - \frac{a d \left (a + c x^{2}\right )^{p + 1}}{2 c^{2} \left (p + 1\right )} + \frac{e x^{5} \left (1 + \frac{c x^{2}}{a}\right )^{- p} \left (a + c x^{2}\right )^{p}{{}_{2}F_{1}\left (\begin{matrix} - p, \frac{5}{2} \\ \frac{7}{2} \end{matrix}\middle |{- \frac{c x^{2}}{a}} \right )}}{5} + \frac{d \left (a + c x^{2}\right )^{p + 2}}{2 c^{2} \left (p + 2\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**3*(e*x+d)*(c*x**2+a)**p,x)

[Out]

-a*d*(a + c*x**2)**(p + 1)/(2*c**2*(p + 1)) + e*x**5*(1 + c*x**2/a)**(-p)*(a + c
*x**2)**p*hyper((-p, 5/2), (7/2,), -c*x**2/a)/5 + d*(a + c*x**2)**(p + 2)/(2*c**
2*(p + 2))

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Mathematica [A]  time = 0.157128, size = 141, normalized size = 1.41 \[ \frac{\left (a+c x^2\right )^p \left (\frac{c x^2}{a}+1\right )^{-p} \left (5 d \left (-a^2 \left (\left (\frac{c x^2}{a}+1\right )^p-1\right )+c^2 (p+1) x^4 \left (\frac{c x^2}{a}+1\right )^p+a c p x^2 \left (\frac{c x^2}{a}+1\right )^p\right )+2 c^2 e \left (p^2+3 p+2\right ) x^5 \, _2F_1\left (\frac{5}{2},-p;\frac{7}{2};-\frac{c x^2}{a}\right )\right )}{10 c^2 (p+1) (p+2)} \]

Antiderivative was successfully verified.

[In]  Integrate[x^3*(d + e*x)*(a + c*x^2)^p,x]

[Out]

((a + c*x^2)^p*(5*d*(a*c*p*x^2*(1 + (c*x^2)/a)^p + c^2*(1 + p)*x^4*(1 + (c*x^2)/
a)^p - a^2*(-1 + (1 + (c*x^2)/a)^p)) + 2*c^2*e*(2 + 3*p + p^2)*x^5*Hypergeometri
c2F1[5/2, -p, 7/2, -((c*x^2)/a)]))/(10*c^2*(1 + p)*(2 + p)*(1 + (c*x^2)/a)^p)

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Maple [F]  time = 0.067, size = 0, normalized size = 0. \[ \int{x}^{3} \left ( ex+d \right ) \left ( c{x}^{2}+a \right ) ^{p}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^3*(e*x+d)*(c*x^2+a)^p,x)

[Out]

int(x^3*(e*x+d)*(c*x^2+a)^p,x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ e \int{\left (c x^{2} + a\right )}^{p} x^{4}\,{d x} + \frac{{\left (c^{2}{\left (p + 1\right )} x^{4} + a c p x^{2} - a^{2}\right )}{\left (c x^{2} + a\right )}^{p} d}{2 \,{\left (p^{2} + 3 \, p + 2\right )} c^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)*(c*x^2 + a)^p*x^3,x, algorithm="maxima")

[Out]

e*integrate((c*x^2 + a)^p*x^4, x) + 1/2*(c^2*(p + 1)*x^4 + a*c*p*x^2 - a^2)*(c*x
^2 + a)^p*d/((p^2 + 3*p + 2)*c^2)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left ({\left (e x^{4} + d x^{3}\right )}{\left (c x^{2} + a\right )}^{p}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)*(c*x^2 + a)^p*x^3,x, algorithm="fricas")

[Out]

integral((e*x^4 + d*x^3)*(c*x^2 + a)^p, x)

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Sympy [A]  time = 65.8732, size = 394, normalized size = 3.94 \[ \frac{a^{p} e x^{5}{{}_{2}F_{1}\left (\begin{matrix} \frac{5}{2}, - p \\ \frac{7}{2} \end{matrix}\middle |{\frac{c x^{2} e^{i \pi }}{a}} \right )}}{5} + d \left (\begin{cases} \frac{a^{p} x^{4}}{4} & \text{for}\: c = 0 \\\frac{a \log{\left (- i \sqrt{a} \sqrt{\frac{1}{c}} + x \right )}}{2 a c^{2} + 2 c^{3} x^{2}} + \frac{a \log{\left (i \sqrt{a} \sqrt{\frac{1}{c}} + x \right )}}{2 a c^{2} + 2 c^{3} x^{2}} + \frac{a}{2 a c^{2} + 2 c^{3} x^{2}} + \frac{c x^{2} \log{\left (- i \sqrt{a} \sqrt{\frac{1}{c}} + x \right )}}{2 a c^{2} + 2 c^{3} x^{2}} + \frac{c x^{2} \log{\left (i \sqrt{a} \sqrt{\frac{1}{c}} + x \right )}}{2 a c^{2} + 2 c^{3} x^{2}} & \text{for}\: p = -2 \\- \frac{a \log{\left (- i \sqrt{a} \sqrt{\frac{1}{c}} + x \right )}}{2 c^{2}} - \frac{a \log{\left (i \sqrt{a} \sqrt{\frac{1}{c}} + x \right )}}{2 c^{2}} + \frac{x^{2}}{2 c} & \text{for}\: p = -1 \\- \frac{a^{2} \left (a + c x^{2}\right )^{p}}{2 c^{2} p^{2} + 6 c^{2} p + 4 c^{2}} + \frac{a c p x^{2} \left (a + c x^{2}\right )^{p}}{2 c^{2} p^{2} + 6 c^{2} p + 4 c^{2}} + \frac{c^{2} p x^{4} \left (a + c x^{2}\right )^{p}}{2 c^{2} p^{2} + 6 c^{2} p + 4 c^{2}} + \frac{c^{2} x^{4} \left (a + c x^{2}\right )^{p}}{2 c^{2} p^{2} + 6 c^{2} p + 4 c^{2}} & \text{otherwise} \end{cases}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**3*(e*x+d)*(c*x**2+a)**p,x)

[Out]

a**p*e*x**5*hyper((5/2, -p), (7/2,), c*x**2*exp_polar(I*pi)/a)/5 + d*Piecewise((
a**p*x**4/4, Eq(c, 0)), (a*log(-I*sqrt(a)*sqrt(1/c) + x)/(2*a*c**2 + 2*c**3*x**2
) + a*log(I*sqrt(a)*sqrt(1/c) + x)/(2*a*c**2 + 2*c**3*x**2) + a/(2*a*c**2 + 2*c*
*3*x**2) + c*x**2*log(-I*sqrt(a)*sqrt(1/c) + x)/(2*a*c**2 + 2*c**3*x**2) + c*x**
2*log(I*sqrt(a)*sqrt(1/c) + x)/(2*a*c**2 + 2*c**3*x**2), Eq(p, -2)), (-a*log(-I*
sqrt(a)*sqrt(1/c) + x)/(2*c**2) - a*log(I*sqrt(a)*sqrt(1/c) + x)/(2*c**2) + x**2
/(2*c), Eq(p, -1)), (-a**2*(a + c*x**2)**p/(2*c**2*p**2 + 6*c**2*p + 4*c**2) + a
*c*p*x**2*(a + c*x**2)**p/(2*c**2*p**2 + 6*c**2*p + 4*c**2) + c**2*p*x**4*(a + c
*x**2)**p/(2*c**2*p**2 + 6*c**2*p + 4*c**2) + c**2*x**4*(a + c*x**2)**p/(2*c**2*
p**2 + 6*c**2*p + 4*c**2), True))

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int{\left (e x + d\right )}{\left (c x^{2} + a\right )}^{p} x^{3}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)*(c*x^2 + a)^p*x^3,x, algorithm="giac")

[Out]

integrate((e*x + d)*(c*x^2 + a)^p*x^3, x)