Optimal. Leaf size=86 \[ \frac{\left (a+b x^2\right )^3 \left (a^2+2 a b x^2+b^2 x^4\right )^p}{2 b^2 (2 p+3)}-\frac{a \left (a+b x^2\right )^2 \left (a^2+2 a b x^2+b^2 x^4\right )^p}{4 b^2 (p+1)} \]
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Rubi [A] time = 0.0887778, antiderivative size = 86, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.129, Rules used = {1249, 770, 21, 43} \[ \frac{\left (a+b x^2\right )^3 \left (a^2+2 a b x^2+b^2 x^4\right )^p}{2 b^2 (2 p+3)}-\frac{a \left (a+b x^2\right )^2 \left (a^2+2 a b x^2+b^2 x^4\right )^p}{4 b^2 (p+1)} \]
Antiderivative was successfully verified.
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Rule 1249
Rule 770
Rule 21
Rule 43
Rubi steps
\begin{align*} \int x^3 \left (a+b x^2\right ) \left (a^2+2 a b x^2+b^2 x^4\right )^p \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int x (a+b x) \left (a^2+2 a b x+b^2 x^2\right )^p \, dx,x,x^2\right )\\ &=\frac{1}{2} \left (\left (b \left (a+b x^2\right )\right )^{-2 p} \left (a^2+2 a b x^2+b^2 x^4\right )^p\right ) \operatorname{Subst}\left (\int x (a+b x) \left (a b+b^2 x\right )^{2 p} \, dx,x,x^2\right )\\ &=\frac{\left (\left (b \left (a+b x^2\right )\right )^{-2 p} \left (a^2+2 a b x^2+b^2 x^4\right )^p\right ) \operatorname{Subst}\left (\int x \left (a b+b^2 x\right )^{1+2 p} \, dx,x,x^2\right )}{2 b}\\ &=\frac{\left (\left (b \left (a+b x^2\right )\right )^{-2 p} \left (a^2+2 a b x^2+b^2 x^4\right )^p\right ) \operatorname{Subst}\left (\int \left (-\frac{a \left (a b+b^2 x\right )^{1+2 p}}{b}+\frac{\left (a b+b^2 x\right )^{2+2 p}}{b^2}\right ) \, dx,x,x^2\right )}{2 b}\\ &=-\frac{a \left (a+b x^2\right )^2 \left (a^2+2 a b x^2+b^2 x^4\right )^p}{4 b^2 (1+p)}+\frac{\left (a+b x^2\right )^3 \left (a^2+2 a b x^2+b^2 x^4\right )^p}{2 b^2 (3+2 p)}\\ \end{align*}
Mathematica [A] time = 0.0248835, size = 45, normalized size = 0.52 \[ \frac{\left (\left (a+b x^2\right )^2\right )^{p+1} \left (2 b (p+1) x^2-a\right )}{4 b^2 (p+1) (2 p+3)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.005, size = 62, normalized size = 0.7 \begin{align*} -{\frac{ \left ({b}^{2}{x}^{4}+2\,ab{x}^{2}+{a}^{2} \right ) ^{p} \left ( -2\,{x}^{2}pb-2\,b{x}^{2}+a \right ) \left ( b{x}^{2}+a \right ) ^{2}}{4\,{b}^{2} \left ( 2\,{p}^{2}+5\,p+3 \right ) }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.01171, size = 182, normalized size = 2.12 \begin{align*} \frac{{\left (b^{2}{\left (2 \, p + 1\right )} x^{4} + 2 \, a b p x^{2} - a^{2}\right )}{\left (b x^{2} + a\right )}^{2 \, p} a}{4 \,{\left (2 \, p^{2} + 3 \, p + 1\right )} b^{2}} + \frac{{\left ({\left (2 \, p^{2} + 3 \, p + 1\right )} b^{3} x^{6} +{\left (2 \, p^{2} + p\right )} a b^{2} x^{4} - 2 \, a^{2} b p x^{2} + a^{3}\right )}{\left (b x^{2} + a\right )}^{2 \, p}}{2 \,{\left (4 \, p^{3} + 12 \, p^{2} + 11 \, p + 3\right )} b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.55942, size = 185, normalized size = 2.15 \begin{align*} \frac{{\left (2 \,{\left (b^{3} p + b^{3}\right )} x^{6} + 2 \, a^{2} b p x^{2} +{\left (4 \, a b^{2} p + 3 \, a b^{2}\right )} x^{4} - a^{3}\right )}{\left (b^{2} x^{4} + 2 \, a b x^{2} + a^{2}\right )}^{p}}{4 \,{\left (2 \, b^{2} p^{2} + 5 \, b^{2} p + 3 \, b^{2}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.14165, size = 265, normalized size = 3.08 \begin{align*} \frac{2 \,{\left (b^{2} x^{4} + 2 \, a b x^{2} + a^{2}\right )}^{p} b^{3} p x^{6} + 2 \,{\left (b^{2} x^{4} + 2 \, a b x^{2} + a^{2}\right )}^{p} b^{3} x^{6} + 4 \,{\left (b^{2} x^{4} + 2 \, a b x^{2} + a^{2}\right )}^{p} a b^{2} p x^{4} + 3 \,{\left (b^{2} x^{4} + 2 \, a b x^{2} + a^{2}\right )}^{p} a b^{2} x^{4} + 2 \,{\left (b^{2} x^{4} + 2 \, a b x^{2} + a^{2}\right )}^{p} a^{2} b p x^{2} -{\left (b^{2} x^{4} + 2 \, a b x^{2} + a^{2}\right )}^{p} a^{3}}{4 \,{\left (2 \, b^{2} p^{2} + 5 \, b^{2} p + 3 \, b^{2}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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