3.1148 \(\int \frac{\left (a+b x^2\right )^p \left (c+d x^2\right )^q}{x} \, dx\)

Optimal. Leaf size=97 \[ -\frac{\left (a+b x^2\right )^{p+1} \left (c+d x^2\right )^q \left (\frac{b \left (c+d x^2\right )}{b c-a d}\right )^{-q} F_1\left (p+1;-q,1;p+2;-\frac{d \left (b x^2+a\right )}{b c-a d},\frac{b x^2+a}{a}\right )}{2 a (p+1)} \]

[Out]

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

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Rubi [A]  time = 0.245747, antiderivative size = 97, 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 \[ -\frac{\left (a+b x^2\right )^{p+1} \left (c+d x^2\right )^q \left (\frac{b \left (c+d x^2\right )}{b c-a d}\right )^{-q} F_1\left (p+1;-q,1;p+2;-\frac{d \left (b x^2+a\right )}{b c-a d},\frac{b x^2+a}{a}\right )}{2 a (p+1)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 29.5256, size = 73, normalized size = 0.75 \[ - \frac{\left (\frac{b \left (- c - d x^{2}\right )}{a d - b c}\right )^{- q} \left (a + b x^{2}\right )^{p + 1} \left (c + d x^{2}\right )^{q} \operatorname{appellf_{1}}{\left (p + 1,1,- q,p + 2,\frac{a + b x^{2}}{a},\frac{d \left (a + b x^{2}\right )}{a d - b c} \right )}}{2 a \left (p + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-(b*(-c - d*x**2)/(a*d - b*c))**(-q)*(a + b*x**2)**(p + 1)*(c + d*x**2)**q*appel
lf1(p + 1, 1, -q, p + 2, (a + b*x**2)/a, d*(a + b*x**2)/(a*d - b*c))/(2*a*(p + 1
))

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Mathematica [B]  time = 0.427312, size = 225, normalized size = 2.32 \[ \frac{b d x^2 (p+q-1) \left (a+b x^2\right )^p \left (c+d x^2\right )^q F_1\left (-p-q;-p,-q;-p-q+1;-\frac{a}{b x^2},-\frac{c}{d x^2}\right )}{2 (p+q) \left (b d x^2 (p+q-1) F_1\left (-p-q;-p,-q;-p-q+1;-\frac{a}{b x^2},-\frac{c}{d x^2}\right )-a d p F_1\left (-p-q+1;1-p,-q;-p-q+2;-\frac{a}{b x^2},-\frac{c}{d x^2}\right )-b c q F_1\left (-p-q+1;-p,1-q;-p-q+2;-\frac{a}{b x^2},-\frac{c}{d x^2}\right )\right )} \]

Warning: Unable to verify antiderivative.

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

[Out]

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

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Maple [F]  time = 0.081, size = 0, normalized size = 0. \[ \int{\frac{ \left ( b{x}^{2}+a \right ) ^{p} \left ( d{x}^{2}+c \right ) ^{q}}{x}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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