Optimal. Leaf size=70 \[ \frac{\left (a^2 d+b^2 c\right ) (a+b x)^{n+1}}{b^3 (n+1)}-\frac{2 a d (a+b x)^{n+2}}{b^3 (n+2)}+\frac{d (a+b x)^{n+3}}{b^3 (n+3)} \]
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Rubi [A] time = 0.031032, antiderivative size = 70, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067, Rules used = {697} \[ \frac{\left (a^2 d+b^2 c\right ) (a+b x)^{n+1}}{b^3 (n+1)}-\frac{2 a d (a+b x)^{n+2}}{b^3 (n+2)}+\frac{d (a+b x)^{n+3}}{b^3 (n+3)} \]
Antiderivative was successfully verified.
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Rule 697
Rubi steps
\begin{align*} \int (a+b x)^n \left (c+d x^2\right ) \, dx &=\int \left (\frac{\left (b^2 c+a^2 d\right ) (a+b x)^n}{b^2}-\frac{2 a d (a+b x)^{1+n}}{b^2}+\frac{d (a+b x)^{2+n}}{b^2}\right ) \, dx\\ &=\frac{\left (b^2 c+a^2 d\right ) (a+b x)^{1+n}}{b^3 (1+n)}-\frac{2 a d (a+b x)^{2+n}}{b^3 (2+n)}+\frac{d (a+b x)^{3+n}}{b^3 (3+n)}\\ \end{align*}
Mathematica [A] time = 0.0470745, size = 65, normalized size = 0.93 \[ \frac{(a+b x)^{n+1} \left (2 a^2 d-2 a b d (n+1) x+b^2 (n+2) \left (c (n+3)+d (n+1) x^2\right )\right )}{b^3 (n+1) (n+2) (n+3)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.049, size = 100, normalized size = 1.4 \begin{align*}{\frac{ \left ( bx+a \right ) ^{1+n} \left ({b}^{2}d{n}^{2}{x}^{2}+3\,{b}^{2}dn{x}^{2}-2\,abdnx+{b}^{2}c{n}^{2}+2\,d{x}^{2}{b}^{2}-2\,adxb+5\,{b}^{2}cn+2\,{a}^{2}d+6\,{b}^{2}c \right ) }{{b}^{3} \left ({n}^{3}+6\,{n}^{2}+11\,n+6 \right ) }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.90208, size = 308, normalized size = 4.4 \begin{align*} \frac{{\left (a b^{2} c n^{2} + 5 \, a b^{2} c n + 6 \, a b^{2} c + 2 \, a^{3} d +{\left (b^{3} d n^{2} + 3 \, b^{3} d n + 2 \, b^{3} d\right )} x^{3} +{\left (a b^{2} d n^{2} + a b^{2} d n\right )} x^{2} +{\left (b^{3} c n^{2} + 6 \, b^{3} c +{\left (5 \, b^{3} c - 2 \, a^{2} b d\right )} n\right )} x\right )}{\left (b x + a\right )}^{n}}{b^{3} n^{3} + 6 \, b^{3} n^{2} + 11 \, b^{3} n + 6 \, b^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 2.31337, size = 978, normalized size = 13.97 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.20527, size = 320, normalized size = 4.57 \begin{align*} \frac{{\left (b x + a\right )}^{n} b^{3} d n^{2} x^{3} +{\left (b x + a\right )}^{n} a b^{2} d n^{2} x^{2} + 3 \,{\left (b x + a\right )}^{n} b^{3} d n x^{3} +{\left (b x + a\right )}^{n} b^{3} c n^{2} x +{\left (b x + a\right )}^{n} a b^{2} d n x^{2} + 2 \,{\left (b x + a\right )}^{n} b^{3} d x^{3} +{\left (b x + a\right )}^{n} a b^{2} c n^{2} + 5 \,{\left (b x + a\right )}^{n} b^{3} c n x - 2 \,{\left (b x + a\right )}^{n} a^{2} b d n x + 5 \,{\left (b x + a\right )}^{n} a b^{2} c n + 6 \,{\left (b x + a\right )}^{n} b^{3} c x + 6 \,{\left (b x + a\right )}^{n} a b^{2} c + 2 \,{\left (b x + a\right )}^{n} a^{3} d}{b^{3} n^{3} + 6 \, b^{3} n^{2} + 11 \, b^{3} n + 6 \, b^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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