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

Optimal. Leaf size=100 \[ -\frac{a e \left (a+b x^2\right )^{p+1}}{2 b^2 (p+1)}+\frac{e \left (a+b x^2\right )^{p+2}}{2 b^2 (p+2)}+\frac{1}{3} d x^3 \left (a+b x^2\right )^p \left (\frac{b x^2}{a}+1\right )^{-p} \, _2F_1\left (\frac{3}{2},-p;\frac{5}{2};-\frac{b x^2}{a}\right ) \]

[Out]

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

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Rubi [A]  time = 0.147994, 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 \[ -\frac{a e \left (a+b x^2\right )^{p+1}}{2 b^2 (p+1)}+\frac{e \left (a+b x^2\right )^{p+2}}{2 b^2 (p+2)}+\frac{1}{3} d x^3 \left (a+b x^2\right )^p \left (\frac{b x^2}{a}+1\right )^{-p} \, _2F_1\left (\frac{3}{2},-p;\frac{5}{2};-\frac{b x^2}{a}\right ) \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 21.4733, size = 82, normalized size = 0.82 \[ - \frac{a e \left (a + b x^{2}\right )^{p + 1}}{2 b^{2} \left (p + 1\right )} + \frac{d x^{3} \left (1 + \frac{b x^{2}}{a}\right )^{- p} \left (a + b x^{2}\right )^{p}{{}_{2}F_{1}\left (\begin{matrix} - p, \frac{3}{2} \\ \frac{5}{2} \end{matrix}\middle |{- \frac{b x^{2}}{a}} \right )}}{3} + \frac{e \left (a + b x^{2}\right )^{p + 2}}{2 b^{2} \left (p + 2\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.133301, size = 141, normalized size = 1.41 \[ \frac{\left (a+b x^2\right )^p \left (\frac{b x^2}{a}+1\right )^{-p} \left (3 e \left (-a^2 \left (\left (\frac{b x^2}{a}+1\right )^p-1\right )+b^2 (p+1) x^4 \left (\frac{b x^2}{a}+1\right )^p+a b p x^2 \left (\frac{b x^2}{a}+1\right )^p\right )+2 b^2 d \left (p^2+3 p+2\right ) x^3 \, _2F_1\left (\frac{3}{2},-p;\frac{5}{2};-\frac{b x^2}{a}\right )\right )}{6 b^2 (p+1) (p+2)} \]

Antiderivative was successfully verified.

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

[Out]

((a + b*x^2)^p*(3*e*(a*b*p*x^2*(1 + (b*x^2)/a)^p + b^2*(1 + p)*x^4*(1 + (b*x^2)/
a)^p - a^2*(-1 + (1 + (b*x^2)/a)^p)) + 2*b^2*d*(2 + 3*p + p^2)*x^3*Hypergeometri
c2F1[3/2, -p, 5/2, -((b*x^2)/a)]))/(6*b^2*(1 + p)*(2 + p)*(1 + (b*x^2)/a)^p)

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Maple [F]  time = 0.053, size = 0, normalized size = 0. \[ \int{x}^{2} \left ( ex+d \right ) \left ( b{x}^{2}+a \right ) ^{p}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^2*(e*x+d)*(b*x^2+a)^p,x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 42.36, size = 394, normalized size = 3.94 \[ \frac{a^{p} d x^{3}{{}_{2}F_{1}\left (\begin{matrix} \frac{3}{2}, - p \\ \frac{5}{2} \end{matrix}\middle |{\frac{b x^{2} e^{i \pi }}{a}} \right )}}{3} + e \left (\begin{cases} \frac{a^{p} x^{4}}{4} & \text{for}\: b = 0 \\\frac{a \log{\left (- i \sqrt{a} \sqrt{\frac{1}{b}} + x \right )}}{2 a b^{2} + 2 b^{3} x^{2}} + \frac{a \log{\left (i \sqrt{a} \sqrt{\frac{1}{b}} + x \right )}}{2 a b^{2} + 2 b^{3} x^{2}} + \frac{a}{2 a b^{2} + 2 b^{3} x^{2}} + \frac{b x^{2} \log{\left (- i \sqrt{a} \sqrt{\frac{1}{b}} + x \right )}}{2 a b^{2} + 2 b^{3} x^{2}} + \frac{b x^{2} \log{\left (i \sqrt{a} \sqrt{\frac{1}{b}} + x \right )}}{2 a b^{2} + 2 b^{3} x^{2}} & \text{for}\: p = -2 \\- \frac{a \log{\left (- i \sqrt{a} \sqrt{\frac{1}{b}} + x \right )}}{2 b^{2}} - \frac{a \log{\left (i \sqrt{a} \sqrt{\frac{1}{b}} + x \right )}}{2 b^{2}} + \frac{x^{2}}{2 b} & \text{for}\: p = -1 \\- \frac{a^{2} \left (a + b x^{2}\right )^{p}}{2 b^{2} p^{2} + 6 b^{2} p + 4 b^{2}} + \frac{a b p x^{2} \left (a + b x^{2}\right )^{p}}{2 b^{2} p^{2} + 6 b^{2} p + 4 b^{2}} + \frac{b^{2} p x^{4} \left (a + b x^{2}\right )^{p}}{2 b^{2} p^{2} + 6 b^{2} p + 4 b^{2}} + \frac{b^{2} x^{4} \left (a + b x^{2}\right )^{p}}{2 b^{2} p^{2} + 6 b^{2} p + 4 b^{2}} & \text{otherwise} \end{cases}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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