3.409 \(\int \frac{x^3 \left (a+b x^2\right )^p}{d+e x} \, dx\)

Optimal. Leaf size=163 \[ -\frac{e x^5 \left (a+b x^2\right )^p \left (\frac{b x^2}{a}+1\right )^{-p} F_1\left (\frac{5}{2};-p,1;\frac{7}{2};-\frac{b x^2}{a},\frac{e^2 x^2}{d^2}\right )}{5 d^2}+\frac{d^3 \left (a+b x^2\right )^{p+1} \, _2F_1\left (1,p+1;p+2;\frac{e^2 \left (b x^2+a\right )}{b d^2+a e^2}\right )}{2 e^2 (p+1) \left (a e^2+b d^2\right )}-\frac{d \left (a+b x^2\right )^{p+1}}{2 b e^2 (p+1)} \]

[Out]

-(d*(a + b*x^2)^(1 + p))/(2*b*e^2*(1 + p)) - (e*x^5*(a + b*x^2)^p*AppellF1[5/2,
-p, 1, 7/2, -((b*x^2)/a), (e^2*x^2)/d^2])/(5*d^2*(1 + (b*x^2)/a)^p) + (d^3*(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^2*(b*d^2 + a*e^2)*(1 + p))

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Rubi [A]  time = 0.370748, antiderivative size = 163, 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 \[ -\frac{e x^5 \left (a+b x^2\right )^p \left (\frac{b x^2}{a}+1\right )^{-p} F_1\left (\frac{5}{2};-p,1;\frac{7}{2};-\frac{b x^2}{a},\frac{e^2 x^2}{d^2}\right )}{5 d^2}+\frac{d^3 \left (a+b x^2\right )^{p+1} \, _2F_1\left (1,p+1;p+2;\frac{e^2 \left (b x^2+a\right )}{b d^2+a e^2}\right )}{2 e^2 (p+1) \left (a e^2+b d^2\right )}-\frac{d \left (a+b x^2\right )^{p+1}}{2 b e^2 (p+1)} \]

Antiderivative was successfully verified.

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

[Out]

-(d*(a + b*x^2)^(1 + p))/(2*b*e^2*(1 + p)) - (e*x^5*(a + b*x^2)^p*AppellF1[5/2,
-p, 1, 7/2, -((b*x^2)/a), (e^2*x^2)/d^2])/(5*d^2*(1 + (b*x^2)/a)^p) + (d^3*(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^2*(b*d^2 + a*e^2)*(1 + p))

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Rubi in Sympy [A]  time = 46.6381, size = 230, normalized size = 1.41 \[ - \frac{d^{3} \left (\frac{e \left (\sqrt{b} x + \sqrt{- a}\right )}{\sqrt{b} \left (d + e x\right )}\right )^{- p} \left (- \frac{e \left (- \sqrt{b} x + \sqrt{- a}\right )}{\sqrt{b} \left (d + e x\right )}\right )^{- p} \left (a + b x^{2}\right )^{p} \operatorname{appellf_{1}}{\left (- 2 p,- p,- p,- 2 p + 1,\frac{d - \frac{e \sqrt{- a}}{\sqrt{b}}}{d + e x},\frac{d + \frac{e \sqrt{- a}}{\sqrt{b}}}{d + e x} \right )}}{2 e^{4} p} + \frac{d^{2} x \left (1 + \frac{b x^{2}}{a}\right )^{- p} \left (a + b x^{2}\right )^{p}{{}_{2}F_{1}\left (\begin{matrix} - p, \frac{1}{2} \\ \frac{3}{2} \end{matrix}\middle |{- \frac{b x^{2}}{a}} \right )}}{e^{3}} + \frac{x^{3} \left (1 + \frac{b x^{2}}{a}\right )^{- p} \left (a + b x^{2}\right )^{p}{{}_{2}F_{1}\left (\begin{matrix} - p, \frac{3}{2} \\ \frac{5}{2} \end{matrix}\middle |{- \frac{b x^{2}}{a}} \right )}}{3 e} - \frac{d \left (a + b x^{2}\right )^{p + 1}}{2 b e^{2} \left (p + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.531567, size = 260, normalized size = 1.6 \[ \frac{\left (a+b x^2\right )^p \left (\frac{e \left (\frac{b x^2}{a}+1\right )^{-p} \left (6 b d^2 (p+1) x \, _2F_1\left (\frac{1}{2},-p;\frac{3}{2};-\frac{b x^2}{a}\right )+e \left (2 b e (p+1) x^3 \, _2F_1\left (\frac{3}{2},-p;\frac{5}{2};-\frac{b x^2}{a}\right )-3 d \left (b x^2 \left (\frac{b x^2}{a}+1\right )^p+a \left (\left (\frac{b x^2}{a}+1\right )^p-1\right )\right )\right )\right )}{b (p+1)}-\frac{3 d^3 \left (\frac{e \left (x-\sqrt{-\frac{a}{b}}\right )}{d+e x}\right )^{-p} \left (\frac{e \left (\sqrt{-\frac{a}{b}}+x\right )}{d+e x}\right )^{-p} F_1\left (-2 p;-p,-p;1-2 p;\frac{d-\sqrt{-\frac{a}{b}} e}{d+e x},\frac{d+\sqrt{-\frac{a}{b}} e}{d+e x}\right )}{p}\right )}{6 e^4} \]

Warning: Unable to verify antiderivative.

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

[Out]

((a + b*x^2)^p*((-3*d^3*AppellF1[-2*p, -p, -p, 1 - 2*p, (d - Sqrt[-(a/b)]*e)/(d
+ e*x), (d + Sqrt[-(a/b)]*e)/(d + e*x)])/(p*((e*(-Sqrt[-(a/b)] + x))/(d + e*x))^
p*((e*(Sqrt[-(a/b)] + x))/(d + e*x))^p) + (e*(6*b*d^2*(1 + p)*x*Hypergeometric2F
1[1/2, -p, 3/2, -((b*x^2)/a)] + e*(-3*d*(b*x^2*(1 + (b*x^2)/a)^p + a*(-1 + (1 +
(b*x^2)/a)^p)) + 2*b*e*(1 + p)*x^3*Hypergeometric2F1[3/2, -p, 5/2, -((b*x^2)/a)]
)))/(b*(1 + p)*(1 + (b*x^2)/a)^p)))/(6*e^4)

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Maple [F]  time = 0.095, size = 0, normalized size = 0. \[ \int{\frac{{x}^{3} \left ( b{x}^{2}+a \right ) ^{p}}{ex+d}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^3*(b*x^2+a)^p/(e*x+d),x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral((b*x^2 + a)^p*x^3/(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**3*(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. \[ \int \frac{{\left (b x^{2} + a\right )}^{p} x^{3}}{e x + d}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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