Optimal. Leaf size=98 \[ \frac{b \left (a+b x^2\right )^{p+1} \left (c+d x^2\right )^q \left (\frac{b \left (c+d x^2\right )}{b c-a d}\right )^{-q} F_1\left (p+1;-q,2;p+2;-\frac{d \left (b x^2+a\right )}{b c-a d},\frac{b x^2+a}{a}\right )}{2 a^2 (p+1)} \]
[Out]
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Rubi [A] time = 0.260212, antiderivative size = 98, 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 \[ \frac{b \left (a+b x^2\right )^{p+1} \left (c+d x^2\right )^q \left (\frac{b \left (c+d x^2\right )}{b c-a d}\right )^{-q} F_1\left (p+1;-q,2;p+2;-\frac{d \left (b x^2+a\right )}{b c-a d},\frac{b x^2+a}{a}\right )}{2 a^2 (p+1)} \]
Antiderivative was successfully verified.
[In] Int[((a + b*x^2)^p*(c + d*x^2)^q)/x^3,x]
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Rubi in Sympy [A] time = 29.0642, size = 75, normalized size = 0.77 \[ \frac{b \left (\frac{b \left (- c - d x^{2}\right )}{a d - b c}\right )^{- q} \left (a + b x^{2}\right )^{p + 1} \left (c + d x^{2}\right )^{q} \operatorname{appellf_{1}}{\left (p + 1,2,- q,p + 2,\frac{a + b x^{2}}{a},\frac{d \left (a + b x^{2}\right )}{a d - b c} \right )}}{2 a^{2} \left (p + 1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((b*x**2+a)**p*(d*x**2+c)**q/x**3,x)
[Out]
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Mathematica [B] time = 0.470141, size = 225, normalized size = 2.3 \[ \frac{b d (p+q-2) \left (a+b x^2\right )^p \left (c+d x^2\right )^q F_1\left (-p-q+1;-p,-q;-p-q+2;-\frac{a}{b x^2},-\frac{c}{d x^2}\right )}{2 (p+q-1) \left (b d x^2 (p+q-2) F_1\left (-p-q+1;-p,-q;-p-q+2;-\frac{a}{b x^2},-\frac{c}{d x^2}\right )-a d p F_1\left (-p-q+2;1-p,-q;-p-q+3;-\frac{a}{b x^2},-\frac{c}{d x^2}\right )-b c q F_1\left (-p-q+2;-p,1-q;-p-q+3;-\frac{a}{b x^2},-\frac{c}{d x^2}\right )\right )} \]
Warning: Unable to verify antiderivative.
[In] Integrate[((a + b*x^2)^p*(c + d*x^2)^q)/x^3,x]
[Out]
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Maple [F] time = 0.086, size = 0, normalized size = 0. \[ \int{\frac{ \left ( b{x}^{2}+a \right ) ^{p} \left ( d{x}^{2}+c \right ) ^{q}}{{x}^{3}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((b*x^2+a)^p*(d*x^2+c)^q/x^3,x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (b x^{2} + a\right )}^{p}{\left (d x^{2} + c\right )}^{q}}{x^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^2 + a)^p*(d*x^2 + c)^q/x^3,x, algorithm="maxima")
[Out]
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (b x^{2} + a\right )}^{p}{\left (d x^{2} + c\right )}^{q}}{x^{3}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^2 + a)^p*(d*x^2 + c)^q/x^3,x, algorithm="fricas")
[Out]
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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**3,x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (b x^{2} + a\right )}^{p}{\left (d x^{2} + c\right )}^{q}}{x^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^2 + a)^p*(d*x^2 + c)^q/x^3,x, algorithm="giac")
[Out]