Optimal. Leaf size=253 \[ -\frac{(e x)^{m+1} \left (a+b x^2\right )^p \left (\frac{b x^2}{a}+1\right )^{-p} \, _2F_1\left (\frac{m+1}{2},-p;\frac{m+3}{2};-\frac{b x^2}{a}\right ) (A b (m+2 p+3) (a d (m+1)-b c (m+2 p+5))-a (m+1) (a B d (m+3)-b (2 A d+B c (m+2 p+5))))}{b^2 e (m+1) (m+2 p+3) (m+2 p+5)}-\frac{(e x)^{m+1} \left (a+b x^2\right )^{p+1} (a B d (m+3)-b (2 A d+B c (m+2 p+5)))}{b^2 e (m+2 p+3) (m+2 p+5)}+\frac{d \left (A+B x^2\right ) (e x)^{m+1} \left (a+b x^2\right )^{p+1}}{b e (m+2 p+5)} \]
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Rubi [A] time = 0.615577, antiderivative size = 238, normalized size of antiderivative = 0.94, number of steps used = 4, number of rules used = 4, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.138 \[ \frac{(e x)^{m+1} \left (a+b x^2\right )^{p+1} (-a B d (m+3)+2 A b d+b B c (m+2 p+5))}{b^2 e (m+2 p+3) (m+2 p+5)}-\frac{(e x)^{m+1} \left (a+b x^2\right )^p \left (\frac{b x^2}{a}+1\right )^{-p} \, _2F_1\left (\frac{m+1}{2},-p;\frac{m+3}{2};-\frac{b x^2}{a}\right ) \left (\frac{a (-a B d (m+3)+2 A b d+b B c (m+2 p+5))}{b (m+2 p+3)}+a A d-\frac{A b c (m+2 p+5)}{m+1}\right )}{b e (m+2 p+5)}+\frac{d \left (A+B x^2\right ) (e x)^{m+1} \left (a+b x^2\right )^{p+1}}{b e (m+2 p+5)} \]
Antiderivative was successfully verified.
[In] Int[(e*x)^m*(a + b*x^2)^p*(A + B*x^2)*(c + d*x^2),x]
[Out]
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Rubi in Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((e*x)**m*(b*x**2+a)**p*(B*x**2+A)*(d*x**2+c),x)
[Out]
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Mathematica [A] time = 0.226034, size = 147, normalized size = 0.58 \[ x (e x)^m \left (a+b x^2\right )^p \left (\frac{b x^2}{a}+1\right )^{-p} \left (\frac{x^2 (A d+B c) \, _2F_1\left (\frac{m+3}{2},-p;\frac{m+5}{2};-\frac{b x^2}{a}\right )}{m+3}+\frac{A c \, _2F_1\left (\frac{m+1}{2},-p;\frac{m+3}{2};-\frac{b x^2}{a}\right )}{m+1}+\frac{B d x^4 \, _2F_1\left (\frac{m+5}{2},-p;\frac{m+7}{2};-\frac{b x^2}{a}\right )}{m+5}\right ) \]
Antiderivative was successfully verified.
[In] Integrate[(e*x)^m*(a + b*x^2)^p*(A + B*x^2)*(c + d*x^2),x]
[Out]
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Maple [F] time = 0.08, size = 0, normalized size = 0. \[ \int \left ( ex \right ) ^{m} \left ( b{x}^{2}+a \right ) ^{p} \left ( B{x}^{2}+A \right ) \left ( d{x}^{2}+c \right ) \, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((e*x)^m*(b*x^2+a)^p*(B*x^2+A)*(d*x^2+c),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int{\left (B x^{2} + A\right )}{\left (d x^{2} + c\right )}{\left (b x^{2} + a\right )}^{p} \left (e x\right )^{m}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x^2 + A)*(d*x^2 + c)*(b*x^2 + a)^p*(e*x)^m,x, algorithm="maxima")
[Out]
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left ({\left (B d x^{4} +{\left (B c + A d\right )} x^{2} + A c\right )}{\left (b x^{2} + a\right )}^{p} \left (e x\right )^{m}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x^2 + A)*(d*x^2 + c)*(b*x^2 + a)^p*(e*x)^m,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((e*x)**m*(b*x**2+a)**p*(B*x**2+A)*(d*x**2+c),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int{\left (B x^{2} + A\right )}{\left (d x^{2} + c\right )}{\left (b x^{2} + a\right )}^{p} \left (e x\right )^{m}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x^2 + A)*(d*x^2 + c)*(b*x^2 + a)^p*(e*x)^m,x, algorithm="giac")
[Out]