3.410 \(\int \frac{x^2 \left (a+b x^2\right )^p}{d+e x} \, dx\)

Optimal. Leaf size=161 \[ \frac{x^3 \left (a+b x^2\right )^p \left (\frac{b x^2}{a}+1\right )^{-p} F_1\left (\frac{3}{2};-p,1;\frac{5}{2};-\frac{b x^2}{a},\frac{e^2 x^2}{d^2}\right )}{3 d}-\frac{d^2 \left (a+b x^2\right )^{p+1} \, _2F_1\left (1,p+1;p+2;\frac{e^2 \left (b x^2+a\right )}{b d^2+a e^2}\right )}{2 e (p+1) \left (a e^2+b d^2\right )}+\frac{\left (a+b x^2\right )^{p+1}}{2 b e (p+1)} \]

[Out]

(a + b*x^2)^(1 + p)/(2*b*e*(1 + p)) + (x^3*(a + b*x^2)^p*AppellF1[3/2, -p, 1, 5/
2, -((b*x^2)/a), (e^2*x^2)/d^2])/(3*d*(1 + (b*x^2)/a)^p) - (d^2*(a + b*x^2)^(1 +
p)*Hypergeometric2F1[1, 1 + p, 2 + p, (e^2*(a + b*x^2))/(b*d^2 + a*e^2)])/(2*e*
(b*d^2 + a*e^2)*(1 + p))

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Rubi [A]  time = 0.366441, antiderivative size = 161, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3 \[ \frac{x^3 \left (a+b x^2\right )^p \left (\frac{b x^2}{a}+1\right )^{-p} F_1\left (\frac{3}{2};-p,1;\frac{5}{2};-\frac{b x^2}{a},\frac{e^2 x^2}{d^2}\right )}{3 d}-\frac{d^2 \left (a+b x^2\right )^{p+1} \, _2F_1\left (1,p+1;p+2;\frac{e^2 \left (b x^2+a\right )}{b d^2+a e^2}\right )}{2 e (p+1) \left (a e^2+b d^2\right )}+\frac{\left (a+b x^2\right )^{p+1}}{2 b e (p+1)} \]

Antiderivative was successfully verified.

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

[Out]

(a + b*x^2)^(1 + p)/(2*b*e*(1 + p)) + (x^3*(a + b*x^2)^p*AppellF1[3/2, -p, 1, 5/
2, -((b*x^2)/a), (e^2*x^2)/d^2])/(3*d*(1 + (b*x^2)/a)^p) - (d^2*(a + b*x^2)^(1 +
p)*Hypergeometric2F1[1, 1 + p, 2 + p, (e^2*(a + b*x^2))/(b*d^2 + a*e^2)])/(2*e*
(b*d^2 + a*e^2)*(1 + p))

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Rubi in Sympy [A]  time = 37.0512, size = 184, normalized size = 1.14 \[ \frac{d^{2} \left (\frac{e \left (\sqrt{b} x + \sqrt{- a}\right )}{\sqrt{b} \left (d + e x\right )}\right )^{- p} \left (- \frac{e \left (- \sqrt{b} x + \sqrt{- a}\right )}{\sqrt{b} \left (d + e x\right )}\right )^{- p} \left (a + b x^{2}\right )^{p} \operatorname{appellf_{1}}{\left (- 2 p,- p,- p,- 2 p + 1,\frac{d - \frac{e \sqrt{- a}}{\sqrt{b}}}{d + e x},\frac{d + \frac{e \sqrt{- a}}{\sqrt{b}}}{d + e x} \right )}}{2 e^{3} p} - \frac{d x \left (1 + \frac{b x^{2}}{a}\right )^{- p} \left (a + b x^{2}\right )^{p}{{}_{2}F_{1}\left (\begin{matrix} - p, \frac{1}{2} \\ \frac{3}{2} \end{matrix}\middle |{- \frac{b x^{2}}{a}} \right )}}{e^{2}} + \frac{\left (a + b x^{2}\right )^{p + 1}}{2 b e \left (p + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

d**2*(e*(sqrt(b)*x + sqrt(-a))/(sqrt(b)*(d + e*x)))**(-p)*(-e*(-sqrt(b)*x + sqrt
(-a))/(sqrt(b)*(d + e*x)))**(-p)*(a + b*x**2)**p*appellf1(-2*p, -p, -p, -2*p + 1
, (d - e*sqrt(-a)/sqrt(b))/(d + e*x), (d + e*sqrt(-a)/sqrt(b))/(d + e*x))/(2*e**
3*p) - d*x*(1 + b*x**2/a)**(-p)*(a + b*x**2)**p*hyper((-p, 1/2), (3/2,), -b*x**2
/a)/e**2 + (a + b*x**2)**(p + 1)/(2*b*e*(p + 1))

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Mathematica [A]  time = 0.396893, size = 227, normalized size = 1.41 \[ \frac{\left (a+b x^2\right )^p \left (b d^2 (p+1) \left (\frac{e \left (x-\sqrt{-\frac{a}{b}}\right )}{d+e x}\right )^{-p} \left (\frac{e \left (\sqrt{-\frac{a}{b}}+x\right )}{d+e x}\right )^{-p} F_1\left (-2 p;-p,-p;1-2 p;\frac{d-\sqrt{-\frac{a}{b}} e}{d+e x},\frac{d+\sqrt{-\frac{a}{b}} e}{d+e x}\right )-2 b d e p (p+1) x \left (\frac{b x^2}{a}+1\right )^{-p} \, _2F_1\left (\frac{1}{2},-p;\frac{3}{2};-\frac{b x^2}{a}\right )-a e^2 p \left (\frac{b x^2}{a}+1\right )^{-p}+a e^2 p+b e^2 p x^2\right )}{2 b e^3 p (p+1)} \]

Warning: Unable to verify antiderivative.

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

[Out]

((a + b*x^2)^p*(a*e^2*p + b*e^2*p*x^2 - (a*e^2*p)/(1 + (b*x^2)/a)^p + (b*d^2*(1
+ p)*AppellF1[-2*p, -p, -p, 1 - 2*p, (d - Sqrt[-(a/b)]*e)/(d + e*x), (d + Sqrt[-
(a/b)]*e)/(d + e*x)])/(((e*(-Sqrt[-(a/b)] + x))/(d + e*x))^p*((e*(Sqrt[-(a/b)] +
x))/(d + e*x))^p) - (2*b*d*e*p*(1 + p)*x*Hypergeometric2F1[1/2, -p, 3/2, -((b*x
^2)/a)])/(1 + (b*x^2)/a)^p))/(2*b*e^3*p*(1 + p))

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Maple [F]  time = 0.076, size = 0, normalized size = 0. \[ \int{\frac{{x}^{2} \left ( b{x}^{2}+a \right ) ^{p}}{ex+d}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^2*(b*x^2+a)^p/(e*x+d),x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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