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

Optimal. Leaf size=754 \[ \frac{2 e^4 x^3 \left (a+b x^2\right )^p \left (\frac{b x^2}{a}+1\right )^{-p} F_1\left (\frac{3}{2};-p,2;\frac{5}{2};-\frac{b x^2}{a},\frac{e^2 x^2}{d^2}\right )}{3 d^7}+\frac{e^4 x^3 \left (a+b x^2\right )^p \left (\frac{b x^2}{a}+1\right )^{-p} F_1\left (\frac{3}{2};-p,3;\frac{5}{2};-\frac{b x^2}{a},\frac{e^2 x^2}{d^2}\right )}{d^7}+\frac{3 e^2 x \left (a+b x^2\right )^p \left (\frac{b x^2}{a}+1\right )^{-p} F_1\left (\frac{1}{2};-p,1;\frac{3}{2};-\frac{b x^2}{a},\frac{e^2 x^2}{d^2}\right )}{d^5}+\frac{2 e^2 x \left (a+b x^2\right )^p \left (\frac{b x^2}{a}+1\right )^{-p} F_1\left (\frac{1}{2};-p,2;\frac{3}{2};-\frac{b x^2}{a},\frac{e^2 x^2}{d^2}\right )}{d^5}+\frac{e^2 x \left (a+b x^2\right )^p \left (\frac{b x^2}{a}+1\right )^{-p} F_1\left (\frac{1}{2};-p,3;\frac{3}{2};-\frac{b x^2}{a},\frac{e^2 x^2}{d^2}\right )}{d^5}-\frac{3 b^2 e^3 \left (a+b x^2\right )^{p+1} \, _2F_1\left (3,p+1;p+2;\frac{e^2 \left (b x^2+a\right )}{b d^2+a e^2}\right )}{2 (p+1) \left (a e^2+b d^2\right )^3}+\frac{3 e \left (a+b x^2\right )^{p+1} \, _2F_1\left (1,p+1;p+2;\frac{b x^2}{a}+1\right )}{2 a d^4 (p+1)}-\frac{\left (a+b x^2\right )^p \left (\frac{b x^2}{a}+1\right )^{-p} \, _2F_1\left (-\frac{1}{2},-p;\frac{1}{2};-\frac{b x^2}{a}\right )}{d^3 x}+\frac{b e^3 \left (a+b x^2\right )^{p+1} \left (2 a e^2+b d^2 (p+1)\right ) \, _2F_1\left (2,p+1;p+2;\frac{e^2 \left (b x^2+a\right )}{b d^2+a e^2}\right )}{4 d^2 (p+1) \left (a e^2+b d^2\right )^3}-\frac{2 b e^3 \left (a+b x^2\right )^{p+1} \, _2F_1\left (2,p+1;p+2;\frac{e^2 \left (b x^2+a\right )}{b d^2+a e^2}\right )}{d^2 (p+1) \left (a e^2+b d^2\right )^2}-\frac{e^3 \left (a+b x^2\right )^{p+1}}{4 \left (d^2-e^2 x^2\right )^2 \left (a e^2+b d^2\right )}-\frac{3 e^3 \left (a+b x^2\right )^{p+1} \, _2F_1\left (1,p+1;p+2;\frac{e^2 \left (b x^2+a\right )}{b d^2+a e^2}\right )}{2 d^4 (p+1) \left (a e^2+b d^2\right )} \]

[Out]

-(e^3*(a + b*x^2)^(1 + p))/(4*(b*d^2 + a*e^2)*(d^2 - e^2*x^2)^2) + (3*e^2*x*(a +
b*x^2)^p*AppellF1[1/2, -p, 1, 3/2, -((b*x^2)/a), (e^2*x^2)/d^2])/(d^5*(1 + (b*x
^2)/a)^p) + (2*e^2*x*(a + b*x^2)^p*AppellF1[1/2, -p, 2, 3/2, -((b*x^2)/a), (e^2*
x^2)/d^2])/(d^5*(1 + (b*x^2)/a)^p) + (e^2*x*(a + b*x^2)^p*AppellF1[1/2, -p, 3, 3
/2, -((b*x^2)/a), (e^2*x^2)/d^2])/(d^5*(1 + (b*x^2)/a)^p) + (2*e^4*x^3*(a + b*x^
2)^p*AppellF1[3/2, -p, 2, 5/2, -((b*x^2)/a), (e^2*x^2)/d^2])/(3*d^7*(1 + (b*x^2)
/a)^p) + (e^4*x^3*(a + b*x^2)^p*AppellF1[3/2, -p, 3, 5/2, -((b*x^2)/a), (e^2*x^2
)/d^2])/(d^7*(1 + (b*x^2)/a)^p) - ((a + b*x^2)^p*Hypergeometric2F1[-1/2, -p, 1/2
, -((b*x^2)/a)])/(d^3*x*(1 + (b*x^2)/a)^p) - (3*e^3*(a + b*x^2)^(1 + p)*Hypergeo
metric2F1[1, 1 + p, 2 + p, (e^2*(a + b*x^2))/(b*d^2 + a*e^2)])/(2*d^4*(b*d^2 + a
*e^2)*(1 + p)) + (3*e*(a + b*x^2)^(1 + p)*Hypergeometric2F1[1, 1 + p, 2 + p, 1 +
(b*x^2)/a])/(2*a*d^4*(1 + p)) - (2*b*e^3*(a + b*x^2)^(1 + p)*Hypergeometric2F1[
2, 1 + p, 2 + p, (e^2*(a + b*x^2))/(b*d^2 + a*e^2)])/(d^2*(b*d^2 + a*e^2)^2*(1 +
p)) + (b*e^3*(2*a*e^2 + b*d^2*(1 + p))*(a + b*x^2)^(1 + p)*Hypergeometric2F1[2,
1 + p, 2 + p, (e^2*(a + b*x^2))/(b*d^2 + a*e^2)])/(4*d^2*(b*d^2 + a*e^2)^3*(1 +
p)) - (3*b^2*e^3*(a + b*x^2)^(1 + p)*Hypergeometric2F1[3, 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 = 1.82706, antiderivative size = 754, normalized size of antiderivative = 1., number of steps used = 31, number of rules used = 15, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.75 \[ \frac{2 e^4 x^3 \left (a+b x^2\right )^p \left (\frac{b x^2}{a}+1\right )^{-p} F_1\left (\frac{3}{2};-p,2;\frac{5}{2};-\frac{b x^2}{a},\frac{e^2 x^2}{d^2}\right )}{3 d^7}+\frac{e^4 x^3 \left (a+b x^2\right )^p \left (\frac{b x^2}{a}+1\right )^{-p} F_1\left (\frac{3}{2};-p,3;\frac{5}{2};-\frac{b x^2}{a},\frac{e^2 x^2}{d^2}\right )}{d^7}+\frac{3 e^2 x \left (a+b x^2\right )^p \left (\frac{b x^2}{a}+1\right )^{-p} F_1\left (\frac{1}{2};-p,1;\frac{3}{2};-\frac{b x^2}{a},\frac{e^2 x^2}{d^2}\right )}{d^5}+\frac{2 e^2 x \left (a+b x^2\right )^p \left (\frac{b x^2}{a}+1\right )^{-p} F_1\left (\frac{1}{2};-p,2;\frac{3}{2};-\frac{b x^2}{a},\frac{e^2 x^2}{d^2}\right )}{d^5}+\frac{e^2 x \left (a+b x^2\right )^p \left (\frac{b x^2}{a}+1\right )^{-p} F_1\left (\frac{1}{2};-p,3;\frac{3}{2};-\frac{b x^2}{a},\frac{e^2 x^2}{d^2}\right )}{d^5}-\frac{3 b^2 e^3 \left (a+b x^2\right )^{p+1} \, _2F_1\left (3,p+1;p+2;\frac{e^2 \left (b x^2+a\right )}{b d^2+a e^2}\right )}{2 (p+1) \left (a e^2+b d^2\right )^3}+\frac{3 e \left (a+b x^2\right )^{p+1} \, _2F_1\left (1,p+1;p+2;\frac{b x^2}{a}+1\right )}{2 a d^4 (p+1)}-\frac{\left (a+b x^2\right )^p \left (\frac{b x^2}{a}+1\right )^{-p} \, _2F_1\left (-\frac{1}{2},-p;\frac{1}{2};-\frac{b x^2}{a}\right )}{d^3 x}+\frac{b e^3 \left (a+b x^2\right )^{p+1} \left (2 a e^2+b d^2 (p+1)\right ) \, _2F_1\left (2,p+1;p+2;\frac{e^2 \left (b x^2+a\right )}{b d^2+a e^2}\right )}{4 d^2 (p+1) \left (a e^2+b d^2\right )^3}-\frac{2 b e^3 \left (a+b x^2\right )^{p+1} \, _2F_1\left (2,p+1;p+2;\frac{e^2 \left (b x^2+a\right )}{b d^2+a e^2}\right )}{d^2 (p+1) \left (a e^2+b d^2\right )^2}-\frac{e^3 \left (a+b x^2\right )^{p+1}}{4 \left (d^2-e^2 x^2\right )^2 \left (a e^2+b d^2\right )}-\frac{3 e^3 \left (a+b x^2\right )^{p+1} \, _2F_1\left (1,p+1;p+2;\frac{e^2 \left (b x^2+a\right )}{b d^2+a e^2}\right )}{2 d^4 (p+1) \left (a e^2+b d^2\right )} \]

Antiderivative was successfully verified.

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

[Out]

-(e^3*(a + b*x^2)^(1 + p))/(4*(b*d^2 + a*e^2)*(d^2 - e^2*x^2)^2) + (3*e^2*x*(a +
b*x^2)^p*AppellF1[1/2, -p, 1, 3/2, -((b*x^2)/a), (e^2*x^2)/d^2])/(d^5*(1 + (b*x
^2)/a)^p) + (2*e^2*x*(a + b*x^2)^p*AppellF1[1/2, -p, 2, 3/2, -((b*x^2)/a), (e^2*
x^2)/d^2])/(d^5*(1 + (b*x^2)/a)^p) + (e^2*x*(a + b*x^2)^p*AppellF1[1/2, -p, 3, 3
/2, -((b*x^2)/a), (e^2*x^2)/d^2])/(d^5*(1 + (b*x^2)/a)^p) + (2*e^4*x^3*(a + b*x^
2)^p*AppellF1[3/2, -p, 2, 5/2, -((b*x^2)/a), (e^2*x^2)/d^2])/(3*d^7*(1 + (b*x^2)
/a)^p) + (e^4*x^3*(a + b*x^2)^p*AppellF1[3/2, -p, 3, 5/2, -((b*x^2)/a), (e^2*x^2
)/d^2])/(d^7*(1 + (b*x^2)/a)^p) - ((a + b*x^2)^p*Hypergeometric2F1[-1/2, -p, 1/2
, -((b*x^2)/a)])/(d^3*x*(1 + (b*x^2)/a)^p) - (3*e^3*(a + b*x^2)^(1 + p)*Hypergeo
metric2F1[1, 1 + p, 2 + p, (e^2*(a + b*x^2))/(b*d^2 + a*e^2)])/(2*d^4*(b*d^2 + a
*e^2)*(1 + p)) + (3*e*(a + b*x^2)^(1 + p)*Hypergeometric2F1[1, 1 + p, 2 + p, 1 +
(b*x^2)/a])/(2*a*d^4*(1 + p)) - (2*b*e^3*(a + b*x^2)^(1 + p)*Hypergeometric2F1[
2, 1 + p, 2 + p, (e^2*(a + b*x^2))/(b*d^2 + a*e^2)])/(d^2*(b*d^2 + a*e^2)^2*(1 +
p)) + (b*e^3*(2*a*e^2 + b*d^2*(1 + p))*(a + b*x^2)^(1 + p)*Hypergeometric2F1[2,
1 + p, 2 + p, (e^2*(a + b*x^2))/(b*d^2 + a*e^2)])/(4*d^2*(b*d^2 + a*e^2)^3*(1 +
p)) - (3*b^2*e^3*(a + b*x^2)^(1 + p)*Hypergeometric2F1[3, 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 = 96.0753, size = 500, normalized size = 0.66 \[ - \frac{e \left (\frac{e \left (\sqrt{b} x + \sqrt{- a}\right )}{\sqrt{b} \left (d + e x\right )}\right )^{- p} \left (- \frac{e \left (- \sqrt{b} x + \sqrt{- a}\right )}{\sqrt{b} \left (d + e x\right )}\right )^{- p} \left (a + b x^{2}\right )^{p} \left (\frac{1}{d + e x}\right )^{2 p} \left (\frac{1}{d + e x}\right )^{- 2 p + 2} \operatorname{appellf_{1}}{\left (- 2 p + 2,- p,- p,- 2 p + 3,\frac{d - \frac{e \sqrt{- a}}{\sqrt{b}}}{d + e x},\frac{d + \frac{e \sqrt{- a}}{\sqrt{b}}}{d + e x} \right )}}{2 d^{2} \left (- p + 1\right )} - \frac{2 e \left (\frac{e \left (\sqrt{b} x + \sqrt{- a}\right )}{\sqrt{b} \left (d + e x\right )}\right )^{- p} \left (- \frac{e \left (- \sqrt{b} x + \sqrt{- a}\right )}{\sqrt{b} \left (d + e x\right )}\right )^{- p} \left (a + b x^{2}\right )^{p} \left (\frac{1}{d + e x}\right )^{2 p} \left (\frac{1}{d + e x}\right )^{- 2 p + 1} \operatorname{appellf_{1}}{\left (- 2 p + 1,- p,- p,- 2 p + 2,\frac{d - \frac{e \sqrt{- a}}{\sqrt{b}}}{d + e x},\frac{d + \frac{e \sqrt{- a}}{\sqrt{b}}}{d + e x} \right )}}{d^{3} \left (- 2 p + 1\right )} - \frac{\left (1 + \frac{b x^{2}}{a}\right )^{- p} \left (a + b x^{2}\right )^{p}{{}_{2}F_{1}\left (\begin{matrix} - p, - \frac{1}{2} \\ \frac{1}{2} \end{matrix}\middle |{- \frac{b x^{2}}{a}} \right )}}{d^{3} x} + \frac{3 e \left (\frac{e \left (\sqrt{b} x + \sqrt{- a}\right )}{\sqrt{b} \left (d + e x\right )}\right )^{- p} \left (- \frac{e \left (- \sqrt{b} x + \sqrt{- a}\right )}{\sqrt{b} \left (d + e x\right )}\right )^{- p} \left (a + b x^{2}\right )^{p} \operatorname{appellf_{1}}{\left (- 2 p,- p,- p,- 2 p + 1,\frac{d - \frac{e \sqrt{- a}}{\sqrt{b}}}{d + e x},\frac{d + \frac{e \sqrt{- a}}{\sqrt{b}}}{d + e x} \right )}}{2 d^{4} p} + \frac{3 e \left (a + b x^{2}\right )^{p + 1}{{}_{2}F_{1}\left (\begin{matrix} 1, p + 1 \\ p + 2 \end{matrix}\middle |{1 + \frac{b x^{2}}{a}} \right )}}{2 a d^{4} \left (p + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-e*(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*(1/(d + e*x))**(2*p)*(1/(d + e*x)
)**(-2*p + 2)*appellf1(-2*p + 2, -p, -p, -2*p + 3, (d - e*sqrt(-a)/sqrt(b))/(d +
e*x), (d + e*sqrt(-a)/sqrt(b))/(d + e*x))/(2*d**2*(-p + 1)) - 2*e*(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*(1/(d + e*x))**(2*p)*(1/(d + e*x))**(-2*p + 1)*ap
pellf1(-2*p + 1, -p, -p, -2*p + 2, (d - e*sqrt(-a)/sqrt(b))/(d + e*x), (d + e*sq
rt(-a)/sqrt(b))/(d + e*x))/(d**3*(-2*p + 1)) - (1 + b*x**2/a)**(-p)*(a + b*x**2)
**p*hyper((-p, -1/2), (1/2,), -b*x**2/a)/(d**3*x) + 3*e*(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*d**4*p) + 3*e*(a + b*x**2)**(p +
1)*hyper((1, p + 1), (p + 2,), 1 + b*x**2/a)/(2*a*d**4*(p + 1))

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Mathematica [A]  time = 0.156269, size = 0, normalized size = 0. \[ \int \frac{\left (a+b x^2\right )^p}{x^2 (d+e x)^3} \, dx \]

Verification is Not applicable to the result.

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

[Out]

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

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Maple [F]  time = 0.14, size = 0, normalized size = 0. \[ \int{\frac{ \left ( b{x}^{2}+a \right ) ^{p}}{{x}^{2} \left ( ex+d \right ) ^{3}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (b x^{2} + a\right )}^{p}}{e^{3} x^{5} + 3 \, d e^{2} x^{4} + 3 \, d^{2} e x^{3} + d^{3} x^{2}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral((b*x^2 + a)^p/(e^3*x^5 + 3*d*e^2*x^4 + 3*d^2*e*x^3 + d^3*x^2), 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((b*x**2+a)**p/x**2/(e*x+d)**3,x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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