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

Optimal. Leaf size=213 \[ -\frac{\left (a+b x^2\right )^{p+1} \left (a e^2+b d^2 p\right ) \, _2F_1\left (1,p+1;p+2;\frac{b x^2}{a}+1\right )}{2 a^2 d^3 (p+1)}+\frac{e \left (a+b x^2\right )^p \left (\frac{b x^2}{a}+1\right )^{-p} F_1\left (-\frac{1}{2};-p,1;\frac{1}{2};-\frac{b x^2}{a},\frac{e^2 x^2}{d^2}\right )}{d^2 x}+\frac{e^4 \left (a+b x^2\right )^{p+1} \, _2F_1\left (1,p+1;p+2;\frac{e^2 \left (b x^2+a\right )}{b d^2+a e^2}\right )}{2 d^3 (p+1) \left (a e^2+b d^2\right )}-\frac{\left (a+b x^2\right )^{p+1}}{2 a d x^2} \]

[Out]

-(a + b*x^2)^(1 + p)/(2*a*d*x^2) + (e*(a + b*x^2)^p*AppellF1[-1/2, -p, 1, 1/2, -
((b*x^2)/a), (e^2*x^2)/d^2])/(d^2*x*(1 + (b*x^2)/a)^p) + (e^4*(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*d^3*
(b*d^2 + a*e^2)*(1 + p)) - ((a*e^2 + b*d^2*p)*(a + b*x^2)^(1 + p)*Hypergeometric
2F1[1, 1 + p, 2 + p, 1 + (b*x^2)/a])/(2*a^2*d^3*(1 + p))

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Rubi [A]  time = 0.582735, antiderivative size = 213, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 8, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4 \[ -\frac{\left (a+b x^2\right )^{p+1} \left (a e^2+b d^2 p\right ) \, _2F_1\left (1,p+1;p+2;\frac{b x^2}{a}+1\right )}{2 a^2 d^3 (p+1)}+\frac{e \left (a+b x^2\right )^p \left (\frac{b x^2}{a}+1\right )^{-p} F_1\left (-\frac{1}{2};-p,1;\frac{1}{2};-\frac{b x^2}{a},\frac{e^2 x^2}{d^2}\right )}{d^2 x}+\frac{e^4 \left (a+b x^2\right )^{p+1} \, _2F_1\left (1,p+1;p+2;\frac{e^2 \left (b x^2+a\right )}{b d^2+a e^2}\right )}{2 d^3 (p+1) \left (a e^2+b d^2\right )}-\frac{\left (a+b x^2\right )^{p+1}}{2 a d x^2} \]

Antiderivative was successfully verified.

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

[Out]

-(a + b*x^2)^(1 + p)/(2*a*d*x^2) + (e*(a + b*x^2)^p*AppellF1[-1/2, -p, 1, 1/2, -
((b*x^2)/a), (e^2*x^2)/d^2])/(d^2*x*(1 + (b*x^2)/a)^p) + (e^4*(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*d^3*
(b*d^2 + a*e^2)*(1 + p)) - ((a*e^2 + b*d^2*p)*(a + b*x^2)^(1 + p)*Hypergeometric
2F1[1, 1 + p, 2 + p, 1 + (b*x^2)/a])/(2*a^2*d^3*(1 + p))

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Rubi in Sympy [A]  time = 51.6582, size = 240, normalized size = 1.13 \[ \frac{e \left (1 + \frac{b x^{2}}{a}\right )^{- p} \left (a + b x^{2}\right )^{p}{{}_{2}F_{1}\left (\begin{matrix} - p, - \frac{1}{2} \\ \frac{1}{2} \end{matrix}\middle |{- \frac{b x^{2}}{a}} \right )}}{d^{2} x} - \frac{e^{2} \left (\frac{e \left (\sqrt{b} x + \sqrt{- a}\right )}{\sqrt{b} \left (d + e x\right )}\right )^{- p} \left (- \frac{e \left (- \sqrt{b} x + \sqrt{- a}\right )}{\sqrt{b} \left (d + e x\right )}\right )^{- p} \left (a + b x^{2}\right )^{p} \operatorname{appellf_{1}}{\left (- 2 p,- p,- p,- 2 p + 1,\frac{d - \frac{e \sqrt{- a}}{\sqrt{b}}}{d + e x},\frac{d + \frac{e \sqrt{- a}}{\sqrt{b}}}{d + e x} \right )}}{2 d^{3} p} - \frac{e^{2} \left (a + b x^{2}\right )^{p + 1}{{}_{2}F_{1}\left (\begin{matrix} 1, p + 1 \\ p + 2 \end{matrix}\middle |{1 + \frac{b x^{2}}{a}} \right )}}{2 a d^{3} \left (p + 1\right )} + \frac{b \left (a + b x^{2}\right )^{p + 1}{{}_{2}F_{1}\left (\begin{matrix} 2, p + 1 \\ p + 2 \end{matrix}\middle |{1 + \frac{b x^{2}}{a}} \right )}}{2 a^{2} d \left (p + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.519981, size = 256, normalized size = 1.2 \[ \frac{\left (a+b x^2\right )^p \left (-\frac{e^2 \left (\frac{e \left (x-\sqrt{-\frac{a}{b}}\right )}{d+e x}\right )^{-p} \left (\frac{e \left (\sqrt{-\frac{a}{b}}+x\right )}{d+e x}\right )^{-p} F_1\left (-2 p;-p,-p;1-2 p;\frac{d-\sqrt{-\frac{a}{b}} e}{d+e x},\frac{d+\sqrt{-\frac{a}{b}} e}{d+e x}\right )}{p}+\left (\frac{a}{b x^2}+1\right )^{-p} \left (\frac{d^2 \, _2F_1\left (1-p,-p;2-p;-\frac{a}{b x^2}\right )}{(p-1) x^2}+\frac{e^2 \, _2F_1\left (-p,-p;1-p;-\frac{a}{b x^2}\right )}{p}\right )+\frac{2 d e \left (\frac{b x^2}{a}+1\right )^{-p} \, _2F_1\left (-\frac{1}{2},-p;\frac{1}{2};-\frac{b x^2}{a}\right )}{x}\right )}{2 d^3} \]

Warning: Unable to verify antiderivative.

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

[Out]

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

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Maple [F]  time = 0.092, size = 0, normalized size = 0. \[ \int{\frac{ \left ( b{x}^{2}+a \right ) ^{p}}{{x}^{3} \left ( ex+d \right ) }}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^2+a)^p/x^3/(e*x+d),x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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