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

Optimal. Leaf size=152 \[ -\frac{a d e \left (a+b x^2\right )^{p+1}}{b^2 (p+1)}+\frac{d e \left (a+b x^2\right )^{p+2}}{b^2 (p+2)}-\frac{x^3 \left (a+b x^2\right )^p \left (\frac{b x^2}{a}+1\right )^{-p} \left (3 a e^2-b d^2 (2 p+5)\right ) \, _2F_1\left (\frac{3}{2},-p;\frac{5}{2};-\frac{b x^2}{a}\right )}{3 b (2 p+5)}+\frac{e^2 x^3 \left (a+b x^2\right )^{p+1}}{b (2 p+5)} \]

[Out]

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

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Rubi [A]  time = 0.278796, antiderivative size = 144, normalized size of antiderivative = 0.95, number of steps used = 8, number of rules used = 7, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.35 \[ -\frac{a d e \left (a+b x^2\right )^{p+1}}{b^2 (p+1)}+\frac{d e \left (a+b x^2\right )^{p+2}}{b^2 (p+2)}+\frac{1}{3} x^3 \left (a+b x^2\right )^p \left (\frac{b x^2}{a}+1\right )^{-p} \left (d^2-\frac{3 a e^2}{2 b p+5 b}\right ) \, _2F_1\left (\frac{3}{2},-p;\frac{5}{2};-\frac{b x^2}{a}\right )+\frac{e^2 x^3 \left (a+b x^2\right )^{p+1}}{b (2 p+5)} \]

Antiderivative was successfully verified.

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

[Out]

-((a*d*e*(a + b*x^2)^(1 + p))/(b^2*(1 + p))) + (e^2*x^3*(a + b*x^2)^(1 + p))/(b*
(5 + 2*p)) + (d*e*(a + b*x^2)^(2 + p))/(b^2*(2 + p)) + ((d^2 - (3*a*e^2)/(5*b +
2*b*p))*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 = 34.3618, size = 126, normalized size = 0.83 \[ - \frac{a d e \left (a + b x^{2}\right )^{p + 1}}{b^{2} \left (p + 1\right )} + \frac{d^{2} x^{3} \left (1 + \frac{b x^{2}}{a}\right )^{- p} \left (a + b x^{2}\right )^{p}{{}_{2}F_{1}\left (\begin{matrix} - p, \frac{3}{2} \\ \frac{5}{2} \end{matrix}\middle |{- \frac{b x^{2}}{a}} \right )}}{3} + \frac{e^{2} x^{5} \left (1 + \frac{b x^{2}}{a}\right )^{- p} \left (a + b x^{2}\right )^{p}{{}_{2}F_{1}\left (\begin{matrix} - p, \frac{5}{2} \\ \frac{7}{2} \end{matrix}\middle |{- \frac{b x^{2}}{a}} \right )}}{5} + \frac{d e \left (a + b x^{2}\right )^{p + 2}}{b^{2} \left (p + 2\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.26068, size = 182, normalized size = 1.2 \[ \frac{\left (a+b x^2\right )^p \left (\frac{b x^2}{a}+1\right )^{-p} \left (3 e \left (5 d \left (-a^2 \left (\left (\frac{b x^2}{a}+1\right )^p-1\right )+b^2 (p+1) x^4 \left (\frac{b x^2}{a}+1\right )^p+a b p x^2 \left (\frac{b x^2}{a}+1\right )^p\right )+b^2 e \left (p^2+3 p+2\right ) x^5 \, _2F_1\left (\frac{5}{2},-p;\frac{7}{2};-\frac{b x^2}{a}\right )\right )+5 b^2 d^2 \left (p^2+3 p+2\right ) x^3 \, _2F_1\left (\frac{3}{2},-p;\frac{5}{2};-\frac{b x^2}{a}\right )\right )}{15 b^2 (p+1) (p+2)} \]

Antiderivative was successfully verified.

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

[Out]

((a + b*x^2)^p*(5*b^2*d^2*(2 + 3*p + p^2)*x^3*Hypergeometric2F1[3/2, -p, 5/2, -(
(b*x^2)/a)] + 3*e*(5*d*(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)) + b^2*e*(2 + 3*p + p^2)*x^5*Hypergeometr
ic2F1[5/2, -p, 7/2, -((b*x^2)/a)])))/(15*b^2*(1 + p)*(2 + p)*(1 + (b*x^2)/a)^p)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 81.6281, size = 430, normalized size = 2.83 \[ \frac{a^{p} d^{2} x^{3}{{}_{2}F_{1}\left (\begin{matrix} \frac{3}{2}, - p \\ \frac{5}{2} \end{matrix}\middle |{\frac{b x^{2} e^{i \pi }}{a}} \right )}}{3} + \frac{a^{p} e^{2} x^{5}{{}_{2}F_{1}\left (\begin{matrix} \frac{5}{2}, - p \\ \frac{7}{2} \end{matrix}\middle |{\frac{b x^{2} e^{i \pi }}{a}} \right )}}{5} + 2 d e \left (\begin{cases} \frac{a^{p} x^{4}}{4} & \text{for}\: b = 0 \\\frac{a \log{\left (- i \sqrt{a} \sqrt{\frac{1}{b}} + x \right )}}{2 a b^{2} + 2 b^{3} x^{2}} + \frac{a \log{\left (i \sqrt{a} \sqrt{\frac{1}{b}} + x \right )}}{2 a b^{2} + 2 b^{3} x^{2}} + \frac{a}{2 a b^{2} + 2 b^{3} x^{2}} + \frac{b x^{2} \log{\left (- i \sqrt{a} \sqrt{\frac{1}{b}} + x \right )}}{2 a b^{2} + 2 b^{3} x^{2}} + \frac{b x^{2} \log{\left (i \sqrt{a} \sqrt{\frac{1}{b}} + x \right )}}{2 a b^{2} + 2 b^{3} x^{2}} & \text{for}\: p = -2 \\- \frac{a \log{\left (- i \sqrt{a} \sqrt{\frac{1}{b}} + x \right )}}{2 b^{2}} - \frac{a \log{\left (i \sqrt{a} \sqrt{\frac{1}{b}} + x \right )}}{2 b^{2}} + \frac{x^{2}}{2 b} & \text{for}\: p = -1 \\- \frac{a^{2} \left (a + b x^{2}\right )^{p}}{2 b^{2} p^{2} + 6 b^{2} p + 4 b^{2}} + \frac{a b p x^{2} \left (a + b x^{2}\right )^{p}}{2 b^{2} p^{2} + 6 b^{2} p + 4 b^{2}} + \frac{b^{2} p x^{4} \left (a + b x^{2}\right )^{p}}{2 b^{2} p^{2} + 6 b^{2} p + 4 b^{2}} + \frac{b^{2} x^{4} \left (a + b x^{2}\right )^{p}}{2 b^{2} p^{2} + 6 b^{2} p + 4 b^{2}} & \text{otherwise} \end{cases}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a**p*d**2*x**3*hyper((3/2, -p), (5/2,), b*x**2*exp_polar(I*pi)/a)/3 + a**p*e**2*
x**5*hyper((5/2, -p), (7/2,), b*x**2*exp_polar(I*pi)/a)/5 + 2*d*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*l
og(I*sqrt(a)*sqrt(1/b) + x)/(2*a*b**2 + 2*b**3*x**2), Eq(p, -2)), (-a*log(-I*sqr
t(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. \[ \int{\left (e x + d\right )}^{2}{\left (b x^{2} + a\right )}^{p} x^{2}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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