Optimal. Leaf size=146 \[ \frac{\left (a+b x^2\right )^{p+1} \left (c+d x^2\right )^{q+1}}{2 b d (p+q+2)}-\frac{\left (a+b x^2\right )^{p+1} \left (c+d x^2\right )^q (a d (q+1)+b c (p+1)) \left (\frac{b \left (c+d x^2\right )}{b c-a d}\right )^{-q} \, _2F_1\left (p+1,-q;p+2;-\frac{d \left (b x^2+a\right )}{b c-a d}\right )}{2 b^2 d (p+1) (p+q+2)} \]
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Rubi [A] time = 0.318021, antiderivative size = 146, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182 \[ \frac{\left (a+b x^2\right )^{p+1} \left (c+d x^2\right )^{q+1}}{2 b d (p+q+2)}-\frac{\left (a+b x^2\right )^{p+1} \left (c+d x^2\right )^q (a d (q+1)+b c (p+1)) \left (\frac{b \left (c+d x^2\right )}{b c-a d}\right )^{-q} \, _2F_1\left (p+1,-q;p+2;-\frac{d \left (b x^2+a\right )}{b c-a d}\right )}{2 b^2 d (p+1) (p+q+2)} \]
Antiderivative was successfully verified.
[In] Int[x^3*(a + b*x^2)^p*(c + d*x^2)^q,x]
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Rubi in Sympy [A] time = 46.5865, size = 117, normalized size = 0.8 \[ \frac{\left (a + b x^{2}\right )^{p + 1} \left (c + d x^{2}\right )^{q + 1}}{2 b d \left (p + q + 2\right )} - \frac{\left (\frac{b \left (- c - d x^{2}\right )}{a d - b c}\right )^{- q} \left (a + b x^{2}\right )^{p + 1} \left (c + d x^{2}\right )^{q} \left (a d \left (q + 1\right ) + b c \left (p + 1\right )\right ){{}_{2}F_{1}\left (\begin{matrix} - q, p + 1 \\ p + 2 \end{matrix}\middle |{\frac{d \left (a + b x^{2}\right )}{a d - b c}} \right )}}{2 b^{2} d \left (p + 1\right ) \left (p + q + 2\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**3*(b*x**2+a)**p*(d*x**2+c)**q,x)
[Out]
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Mathematica [C] time = 0.410162, size = 159, normalized size = 1.09 \[ \frac{3 a c x^4 \left (a+b x^2\right )^p \left (c+d x^2\right )^q F_1\left (2;-p,-q;3;-\frac{b x^2}{a},-\frac{d x^2}{c}\right )}{4 \left (x^2 \left (b c p F_1\left (3;1-p,-q;4;-\frac{b x^2}{a},-\frac{d x^2}{c}\right )+a d q F_1\left (3;-p,1-q;4;-\frac{b x^2}{a},-\frac{d x^2}{c}\right )\right )+3 a c F_1\left (2;-p,-q;3;-\frac{b x^2}{a},-\frac{d x^2}{c}\right )\right )} \]
Warning: Unable to verify antiderivative.
[In] Integrate[x^3*(a + b*x^2)^p*(c + d*x^2)^q,x]
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Maple [F] time = 0.088, size = 0, normalized size = 0. \[ \int{x}^{3} \left ( b{x}^{2}+a \right ) ^{p} \left ( d{x}^{2}+c \right ) ^{q}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^3*(b*x^2+a)^p*(d*x^2+c)^q,x)
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int{\left (b x^{2} + a\right )}^{p}{\left (d x^{2} + c\right )}^{q} x^{3}\,{d x} \]
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. \[{\rm integral}\left ({\left (b x^{2} + a\right )}^{p}{\left (d x^{2} + c\right )}^{q} x^{3}, x\right ) \]
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(x**3*(b*x**2+a)**p*(d*x**2+c)**q,x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int{\left (b x^{2} + a\right )}^{p}{\left (d x^{2} + c\right )}^{q} x^{3}\,{d x} \]
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]