Optimal. Leaf size=73 \[ \frac{6}{5} a^2 b^2 c x^5+\frac{4}{3} a^3 b c x^3+a^4 c x+\frac{4}{7} a b^3 c x^7+\frac{d \left (a+b x^2\right )^5}{10 b}+\frac{1}{9} b^4 c x^9 \]
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Rubi [A] time = 0.0306908, antiderivative size = 73, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {641, 194} \[ \frac{6}{5} a^2 b^2 c x^5+\frac{4}{3} a^3 b c x^3+a^4 c x+\frac{4}{7} a b^3 c x^7+\frac{d \left (a+b x^2\right )^5}{10 b}+\frac{1}{9} b^4 c x^9 \]
Antiderivative was successfully verified.
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Rule 641
Rule 194
Rubi steps
\begin{align*} \int (c+d x) \left (a+b x^2\right )^4 \, dx &=\frac{d \left (a+b x^2\right )^5}{10 b}+c \int \left (a+b x^2\right )^4 \, dx\\ &=\frac{d \left (a+b x^2\right )^5}{10 b}+c \int \left (a^4+4 a^3 b x^2+6 a^2 b^2 x^4+4 a b^3 x^6+b^4 x^8\right ) \, dx\\ &=a^4 c x+\frac{4}{3} a^3 b c x^3+\frac{6}{5} a^2 b^2 c x^5+\frac{4}{7} a b^3 c x^7+\frac{1}{9} b^4 c x^9+\frac{d \left (a+b x^2\right )^5}{10 b}\\ \end{align*}
Mathematica [A] time = 0.0036308, size = 110, normalized size = 1.51 \[ \frac{6}{5} a^2 b^2 c x^5+a^2 b^2 d x^6+\frac{4}{3} a^3 b c x^3+a^3 b d x^4+a^4 c x+\frac{1}{2} a^4 d x^2+\frac{4}{7} a b^3 c x^7+\frac{1}{2} a b^3 d x^8+\frac{1}{9} b^4 c x^9+\frac{1}{10} b^4 d x^{10} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.042, size = 97, normalized size = 1.3 \begin{align*}{\frac{d{b}^{4}{x}^{10}}{10}}+{\frac{{b}^{4}c{x}^{9}}{9}}+{\frac{a{b}^{3}d{x}^{8}}{2}}+{\frac{4\,a{b}^{3}c{x}^{7}}{7}}+{a}^{2}{b}^{2}d{x}^{6}+{\frac{6\,{a}^{2}{b}^{2}c{x}^{5}}{5}}+d{a}^{3}b{x}^{4}+{\frac{4\,{a}^{3}bc{x}^{3}}{3}}+{\frac{{a}^{4}d{x}^{2}}{2}}+{a}^{4}cx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.06903, size = 130, normalized size = 1.78 \begin{align*} \frac{1}{10} \, b^{4} d x^{10} + \frac{1}{9} \, b^{4} c x^{9} + \frac{1}{2} \, a b^{3} d x^{8} + \frac{4}{7} \, a b^{3} c x^{7} + a^{2} b^{2} d x^{6} + \frac{6}{5} \, a^{2} b^{2} c x^{5} + a^{3} b d x^{4} + \frac{4}{3} \, a^{3} b c x^{3} + \frac{1}{2} \, a^{4} d x^{2} + a^{4} c x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.62656, size = 220, normalized size = 3.01 \begin{align*} \frac{1}{10} x^{10} d b^{4} + \frac{1}{9} x^{9} c b^{4} + \frac{1}{2} x^{8} d b^{3} a + \frac{4}{7} x^{7} c b^{3} a + x^{6} d b^{2} a^{2} + \frac{6}{5} x^{5} c b^{2} a^{2} + x^{4} d b a^{3} + \frac{4}{3} x^{3} c b a^{3} + \frac{1}{2} x^{2} d a^{4} + x c a^{4} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.082166, size = 112, normalized size = 1.53 \begin{align*} a^{4} c x + \frac{a^{4} d x^{2}}{2} + \frac{4 a^{3} b c x^{3}}{3} + a^{3} b d x^{4} + \frac{6 a^{2} b^{2} c x^{5}}{5} + a^{2} b^{2} d x^{6} + \frac{4 a b^{3} c x^{7}}{7} + \frac{a b^{3} d x^{8}}{2} + \frac{b^{4} c x^{9}}{9} + \frac{b^{4} d x^{10}}{10} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.27079, size = 130, normalized size = 1.78 \begin{align*} \frac{1}{10} \, b^{4} d x^{10} + \frac{1}{9} \, b^{4} c x^{9} + \frac{1}{2} \, a b^{3} d x^{8} + \frac{4}{7} \, a b^{3} c x^{7} + a^{2} b^{2} d x^{6} + \frac{6}{5} \, a^{2} b^{2} c x^{5} + a^{3} b d x^{4} + \frac{4}{3} \, a^{3} b c x^{3} + \frac{1}{2} \, a^{4} d x^{2} + a^{4} c x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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