3.1150 \(\int \frac{\left (a+b x^2\right )^p \left (c+d x^2\right )^q}{x^5} \, dx\)

Optimal. Leaf size=100 \[ -\frac{b^2 \left (a+b x^2\right )^{p+1} \left (c+d x^2\right )^q \left (\frac{b \left (c+d x^2\right )}{b c-a d}\right )^{-q} F_1\left (p+1;-q,3;p+2;-\frac{d \left (b x^2+a\right )}{b c-a d},\frac{b x^2+a}{a}\right )}{2 a^3 (p+1)} \]

[Out]

-(b^2*(a + b*x^2)^(1 + p)*(c + d*x^2)^q*AppellF1[1 + p, -q, 3, 2 + p, -((d*(a +
b*x^2))/(b*c - a*d)), (a + b*x^2)/a])/(2*a^3*(1 + p)*((b*(c + d*x^2))/(b*c - a*d
))^q)

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Rubi [A]  time = 0.260977, antiderivative size = 100, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.136 \[ -\frac{b^2 \left (a+b x^2\right )^{p+1} \left (c+d x^2\right )^q \left (\frac{b \left (c+d x^2\right )}{b c-a d}\right )^{-q} F_1\left (p+1;-q,3;p+2;-\frac{d \left (b x^2+a\right )}{b c-a d},\frac{b x^2+a}{a}\right )}{2 a^3 (p+1)} \]

Antiderivative was successfully verified.

[In]  Int[((a + b*x^2)^p*(c + d*x^2)^q)/x^5,x]

[Out]

-(b^2*(a + b*x^2)^(1 + p)*(c + d*x^2)^q*AppellF1[1 + p, -q, 3, 2 + p, -((d*(a +
b*x^2))/(b*c - a*d)), (a + b*x^2)/a])/(2*a^3*(1 + p)*((b*(c + d*x^2))/(b*c - a*d
))^q)

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Rubi in Sympy [A]  time = 30.6111, size = 78, normalized size = 0.78 \[ - \frac{b^{2} \left (\frac{b \left (- c - d x^{2}\right )}{a d - b c}\right )^{- q} \left (a + b x^{2}\right )^{p + 1} \left (c + d x^{2}\right )^{q} \operatorname{appellf_{1}}{\left (p + 1,3,- q,p + 2,\frac{a + b x^{2}}{a},\frac{d \left (a + b x^{2}\right )}{a d - b c} \right )}}{2 a^{3} \left (p + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x**2+a)**p*(d*x**2+c)**q/x**5,x)

[Out]

-b**2*(b*(-c - d*x**2)/(a*d - b*c))**(-q)*(a + b*x**2)**(p + 1)*(c + d*x**2)**q*
appellf1(p + 1, 3, -q, p + 2, (a + b*x**2)/a, d*(a + b*x**2)/(a*d - b*c))/(2*a**
3*(p + 1))

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Mathematica [B]  time = 0.527402, size = 228, normalized size = 2.28 \[ \frac{b d (p+q-3) \left (a+b x^2\right )^p \left (c+d x^2\right )^q F_1\left (-p-q+2;-p,-q;-p-q+3;-\frac{a}{b x^2},-\frac{c}{d x^2}\right )}{2 x^2 (p+q-2) \left (b d x^2 (p+q-3) F_1\left (-p-q+2;-p,-q;-p-q+3;-\frac{a}{b x^2},-\frac{c}{d x^2}\right )-a d p F_1\left (-p-q+3;1-p,-q;-p-q+4;-\frac{a}{b x^2},-\frac{c}{d x^2}\right )-b c q F_1\left (-p-q+3;-p,1-q;-p-q+4;-\frac{a}{b x^2},-\frac{c}{d x^2}\right )\right )} \]

Warning: Unable to verify antiderivative.

[In]  Integrate[((a + b*x^2)^p*(c + d*x^2)^q)/x^5,x]

[Out]

(b*d*(-3 + p + q)*(a + b*x^2)^p*(c + d*x^2)^q*AppellF1[2 - p - q, -p, -q, 3 - p
- q, -(a/(b*x^2)), -(c/(d*x^2))])/(2*(-2 + p + q)*x^2*(b*d*(-3 + p + q)*x^2*Appe
llF1[2 - p - q, -p, -q, 3 - p - q, -(a/(b*x^2)), -(c/(d*x^2))] - a*d*p*AppellF1[
3 - p - q, 1 - p, -q, 4 - p - q, -(a/(b*x^2)), -(c/(d*x^2))] - b*c*q*AppellF1[3
- p - q, -p, 1 - q, 4 - p - q, -(a/(b*x^2)), -(c/(d*x^2))]))

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Maple [F]  time = 0.091, size = 0, normalized size = 0. \[ \int{\frac{ \left ( b{x}^{2}+a \right ) ^{p} \left ( d{x}^{2}+c \right ) ^{q}}{{x}^{5}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^2+a)^p*(d*x^2+c)^q/x^5,x)

[Out]

int((b*x^2+a)^p*(d*x^2+c)^q/x^5,x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (b x^{2} + a\right )}^{p}{\left (d x^{2} + c\right )}^{q}}{x^{5}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^p*(d*x^2 + c)^q/x^5,x, algorithm="maxima")

[Out]

integrate((b*x^2 + a)^p*(d*x^2 + c)^q/x^5, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^p*(d*x^2 + c)^q/x^5,x, algorithm="fricas")

[Out]

integral((b*x^2 + a)^p*(d*x^2 + c)^q/x^5, 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*(d*x**2+c)**q/x**5,x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (b x^{2} + a\right )}^{p}{\left (d x^{2} + c\right )}^{q}}{x^{5}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^p*(d*x^2 + c)^q/x^5,x, algorithm="giac")

[Out]

integrate((b*x^2 + a)^p*(d*x^2 + c)^q/x^5, x)