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

Optimal. Leaf size=336 \[ -\frac{b p x \left (a+b x^2\right )^p \left (\frac{b x^2}{a}+1\right )^{-p} \left (3 a e^2+b d^2 (2 p+1)\right ) F_1\left (\frac{1}{2};-p,1;\frac{3}{2};-\frac{b x^2}{a},\frac{e^2 x^2}{d^2}\right )}{e \left (a e^2+b d^2\right )^2}+\frac{b (2 p+1) x \left (a+b x^2\right )^p \left (\frac{b x^2}{a}+1\right )^{-p} \left (a e^2+b d^2 p\right ) \, _2F_1\left (\frac{1}{2},-p;\frac{3}{2};-\frac{b x^2}{a}\right )}{e \left (a e^2+b d^2\right )^2}+\frac{b d p \left (a+b x^2\right )^{p+1} \left (3 a e^2+b d^2 (2 p+1)\right ) \, _2F_1\left (1,p+1;p+2;\frac{e^2 \left (b x^2+a\right )}{b d^2+a e^2}\right )}{2 (p+1) \left (a e^2+b d^2\right )^3}-\frac{\left (a+b x^2\right )^{p+1} \left (a e^2+b d^2 p\right )}{(d+e x) \left (a e^2+b d^2\right )^2}+\frac{d \left (a+b x^2\right )^{p+1}}{2 (d+e x)^2 \left (a e^2+b d^2\right )} \]

[Out]

(d*(a + b*x^2)^(1 + p))/(2*(b*d^2 + a*e^2)*(d + e*x)^2) - ((a*e^2 + b*d^2*p)*(a
+ b*x^2)^(1 + p))/((b*d^2 + a*e^2)^2*(d + e*x)) - (b*p*(3*a*e^2 + b*d^2*(1 + 2*p
))*x*(a + b*x^2)^p*AppellF1[1/2, -p, 1, 3/2, -((b*x^2)/a), (e^2*x^2)/d^2])/(e*(b
*d^2 + a*e^2)^2*(1 + (b*x^2)/a)^p) + (b*(1 + 2*p)*(a*e^2 + b*d^2*p)*x*(a + b*x^2
)^p*Hypergeometric2F1[1/2, -p, 3/2, -((b*x^2)/a)])/(e*(b*d^2 + a*e^2)^2*(1 + (b*
x^2)/a)^p) + (b*d*p*(3*a*e^2 + b*d^2*(1 + 2*p))*(a + b*x^2)^(1 + p)*Hypergeometr
ic2F1[1, 1 + p, 2 + p, (e^2*(a + b*x^2))/(b*d^2 + a*e^2)])/(2*(b*d^2 + a*e^2)^3*
(1 + p))

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Rubi [A]  time = 0.832627, antiderivative size = 336, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 9, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5 \[ -\frac{b p x \left (a+b x^2\right )^p \left (\frac{b x^2}{a}+1\right )^{-p} \left (3 a e^2+b d^2 (2 p+1)\right ) F_1\left (\frac{1}{2};-p,1;\frac{3}{2};-\frac{b x^2}{a},\frac{e^2 x^2}{d^2}\right )}{e \left (a e^2+b d^2\right )^2}+\frac{b (2 p+1) x \left (a+b x^2\right )^p \left (\frac{b x^2}{a}+1\right )^{-p} \left (a e^2+b d^2 p\right ) \, _2F_1\left (\frac{1}{2},-p;\frac{3}{2};-\frac{b x^2}{a}\right )}{e \left (a e^2+b d^2\right )^2}+\frac{b d p \left (a+b x^2\right )^{p+1} \left (3 a e^2+b d^2 (2 p+1)\right ) \, _2F_1\left (1,p+1;p+2;\frac{e^2 \left (b x^2+a\right )}{b d^2+a e^2}\right )}{2 (p+1) \left (a e^2+b d^2\right )^3}-\frac{\left (a+b x^2\right )^{p+1} \left (a e^2+b d^2 p\right )}{(d+e x) \left (a e^2+b d^2\right )^2}+\frac{d \left (a+b x^2\right )^{p+1}}{2 (d+e x)^2 \left (a e^2+b d^2\right )} \]

Antiderivative was successfully verified.

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

[Out]

(d*(a + b*x^2)^(1 + p))/(2*(b*d^2 + a*e^2)*(d + e*x)^2) - ((a*e^2 + b*d^2*p)*(a
+ b*x^2)^(1 + p))/((b*d^2 + a*e^2)^2*(d + e*x)) - (b*p*(3*a*e^2 + b*d^2*(1 + 2*p
))*x*(a + b*x^2)^p*AppellF1[1/2, -p, 1, 3/2, -((b*x^2)/a), (e^2*x^2)/d^2])/(e*(b
*d^2 + a*e^2)^2*(1 + (b*x^2)/a)^p) + (b*(1 + 2*p)*(a*e^2 + b*d^2*p)*x*(a + b*x^2
)^p*Hypergeometric2F1[1/2, -p, 3/2, -((b*x^2)/a)])/(e*(b*d^2 + a*e^2)^2*(1 + (b*
x^2)/a)^p) + (b*d*p*(3*a*e^2 + b*d^2*(1 + 2*p))*(a + b*x^2)^(1 + p)*Hypergeometr
ic2F1[1, 1 + p, 2 + p, (e^2*(a + b*x^2))/(b*d^2 + a*e^2)])/(2*(b*d^2 + a*e^2)^3*
(1 + p))

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Rubi in Sympy [A]  time = 87.0114, size = 292, normalized size = 0.87 \[ - \frac{b d \left (\frac{e \left (\sqrt{b} x + \sqrt{- a}\right )}{\sqrt{b} \left (d + e x\right )}\right )^{- p} \left (- \frac{e \left (- \sqrt{b} x + \sqrt{- a}\right )}{\sqrt{b} \left (d + e x\right )}\right )^{- p} \left (a + b x^{2}\right )^{p} \left (3 a e^{2} + 2 b d^{2} p + b d^{2}\right ) \operatorname{appellf_{1}}{\left (- 2 p,- p,- p,- 2 p + 1,\frac{d - \frac{e \sqrt{- a}}{\sqrt{b}}}{d + e x},\frac{d + \frac{e \sqrt{- a}}{\sqrt{b}}}{d + e x} \right )}}{2 e^{2} \left (a e^{2} + b d^{2}\right )^{2}} + \frac{b x \left (1 + \frac{b x^{2}}{a}\right )^{- p} \left (a + b x^{2}\right )^{p} \left (2 p + 1\right ) \left (a e^{2} + b d^{2} p\right ){{}_{2}F_{1}\left (\begin{matrix} - p, \frac{1}{2} \\ \frac{3}{2} \end{matrix}\middle |{- \frac{b x^{2}}{a}} \right )}}{e \left (a e^{2} + b d^{2}\right )^{2}} + \frac{d \left (a + b x^{2}\right )^{p + 1}}{2 \left (d + e x\right )^{2} \left (a e^{2} + b d^{2}\right )} - \frac{\left (a + b x^{2}\right )^{p + 1} \left (a e^{2} + b d^{2} p\right )}{\left (d + e x\right ) \left (a e^{2} + b d^{2}\right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.29078, size = 229, normalized size = 0.68 \[ \frac{\left (a+b x^2\right )^p \left (\frac{e \left (x-\sqrt{-\frac{a}{b}}\right )}{d+e x}\right )^{-p} \left (\frac{e \left (\sqrt{-\frac{a}{b}}+x\right )}{d+e x}\right )^{-p} \left (2 (p-1) (d+e x) F_1\left (1-2 p;-p,-p;2-2 p;\frac{d-\sqrt{-\frac{a}{b}} e}{d+e x},\frac{d+\sqrt{-\frac{a}{b}} e}{d+e x}\right )+d (1-2 p) F_1\left (2-2 p;-p,-p;3-2 p;\frac{d-\sqrt{-\frac{a}{b}} e}{d+e x},\frac{d+\sqrt{-\frac{a}{b}} e}{d+e x}\right )\right )}{2 e^2 (p-1) (2 p-1) (d+e x)^2} \]

Warning: Unable to verify antiderivative.

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

[Out]

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

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Maple [F]  time = 0.1, size = 0, normalized size = 0. \[ \int{\frac{x \left ( b{x}^{2}+a \right ) ^{p}}{ \left ( ex+d \right ) ^{3}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x*(b*x^2+a)^p/(e*x+d)^3,x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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