Optimal. Leaf size=169 \[ \frac{a^4 c \sqrt{c x^2} (a+b x)^{n+1}}{b^5 (n+1) x}-\frac{4 a^3 c \sqrt{c x^2} (a+b x)^{n+2}}{b^5 (n+2) x}+\frac{6 a^2 c \sqrt{c x^2} (a+b x)^{n+3}}{b^5 (n+3) x}-\frac{4 a c \sqrt{c x^2} (a+b x)^{n+4}}{b^5 (n+4) x}+\frac{c \sqrt{c x^2} (a+b x)^{n+5}}{b^5 (n+5) x} \]
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Rubi [A] time = 0.0553475, antiderivative size = 169, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111, Rules used = {15, 43} \[ \frac{a^4 c \sqrt{c x^2} (a+b x)^{n+1}}{b^5 (n+1) x}-\frac{4 a^3 c \sqrt{c x^2} (a+b x)^{n+2}}{b^5 (n+2) x}+\frac{6 a^2 c \sqrt{c x^2} (a+b x)^{n+3}}{b^5 (n+3) x}-\frac{4 a c \sqrt{c x^2} (a+b x)^{n+4}}{b^5 (n+4) x}+\frac{c \sqrt{c x^2} (a+b x)^{n+5}}{b^5 (n+5) x} \]
Antiderivative was successfully verified.
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Rule 15
Rule 43
Rubi steps
\begin{align*} \int x \left (c x^2\right )^{3/2} (a+b x)^n \, dx &=\frac{\left (c \sqrt{c x^2}\right ) \int x^4 (a+b x)^n \, dx}{x}\\ &=\frac{\left (c \sqrt{c x^2}\right ) \int \left (\frac{a^4 (a+b x)^n}{b^4}-\frac{4 a^3 (a+b x)^{1+n}}{b^4}+\frac{6 a^2 (a+b x)^{2+n}}{b^4}-\frac{4 a (a+b x)^{3+n}}{b^4}+\frac{(a+b x)^{4+n}}{b^4}\right ) \, dx}{x}\\ &=\frac{a^4 c \sqrt{c x^2} (a+b x)^{1+n}}{b^5 (1+n) x}-\frac{4 a^3 c \sqrt{c x^2} (a+b x)^{2+n}}{b^5 (2+n) x}+\frac{6 a^2 c \sqrt{c x^2} (a+b x)^{3+n}}{b^5 (3+n) x}-\frac{4 a c \sqrt{c x^2} (a+b x)^{4+n}}{b^5 (4+n) x}+\frac{c \sqrt{c x^2} (a+b x)^{5+n}}{b^5 (5+n) x}\\ \end{align*}
Mathematica [A] time = 0.0744413, size = 132, normalized size = 0.78 \[ \frac{\left (c x^2\right )^{3/2} (a+b x)^{n+1} \left (12 a^2 b^2 \left (n^2+3 n+2\right ) x^2-24 a^3 b (n+1) x+24 a^4-4 a b^3 \left (n^3+6 n^2+11 n+6\right ) x^3+b^4 \left (n^4+10 n^3+35 n^2+50 n+24\right ) x^4\right )}{b^5 (n+1) (n+2) (n+3) (n+4) (n+5) x^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.005, size = 199, normalized size = 1.2 \begin{align*}{\frac{ \left ( bx+a \right ) ^{1+n} \left ({b}^{4}{n}^{4}{x}^{4}+10\,{b}^{4}{n}^{3}{x}^{4}-4\,a{b}^{3}{n}^{3}{x}^{3}+35\,{b}^{4}{n}^{2}{x}^{4}-24\,a{b}^{3}{n}^{2}{x}^{3}+50\,{b}^{4}n{x}^{4}+12\,{a}^{2}{b}^{2}{n}^{2}{x}^{2}-44\,a{b}^{3}n{x}^{3}+24\,{b}^{4}{x}^{4}+36\,{a}^{2}{b}^{2}n{x}^{2}-24\,{x}^{3}a{b}^{3}-24\,{a}^{3}bnx+24\,{x}^{2}{a}^{2}{b}^{2}-24\,bx{a}^{3}+24\,{a}^{4} \right ) }{{x}^{3}{b}^{5} \left ({n}^{5}+15\,{n}^{4}+85\,{n}^{3}+225\,{n}^{2}+274\,n+120 \right ) } \left ( c{x}^{2} \right ) ^{{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.07084, size = 212, normalized size = 1.25 \begin{align*} \frac{{\left ({\left (n^{4} + 10 \, n^{3} + 35 \, n^{2} + 50 \, n + 24\right )} b^{5} c^{\frac{3}{2}} x^{5} +{\left (n^{4} + 6 \, n^{3} + 11 \, n^{2} + 6 \, n\right )} a b^{4} c^{\frac{3}{2}} x^{4} - 4 \,{\left (n^{3} + 3 \, n^{2} + 2 \, n\right )} a^{2} b^{3} c^{\frac{3}{2}} x^{3} + 12 \,{\left (n^{2} + n\right )} a^{3} b^{2} c^{\frac{3}{2}} x^{2} - 24 \, a^{4} b c^{\frac{3}{2}} n x + 24 \, a^{5} c^{\frac{3}{2}}\right )}{\left (b x + a\right )}^{n}}{{\left (n^{5} + 15 \, n^{4} + 85 \, n^{3} + 225 \, n^{2} + 274 \, n + 120\right )} b^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.57399, size = 494, normalized size = 2.92 \begin{align*} -\frac{{\left (24 \, a^{4} b c n x - 24 \, a^{5} c -{\left (b^{5} c n^{4} + 10 \, b^{5} c n^{3} + 35 \, b^{5} c n^{2} + 50 \, b^{5} c n + 24 \, b^{5} c\right )} x^{5} -{\left (a b^{4} c n^{4} + 6 \, a b^{4} c n^{3} + 11 \, a b^{4} c n^{2} + 6 \, a b^{4} c n\right )} x^{4} + 4 \,{\left (a^{2} b^{3} c n^{3} + 3 \, a^{2} b^{3} c n^{2} + 2 \, a^{2} b^{3} c n\right )} x^{3} - 12 \,{\left (a^{3} b^{2} c n^{2} + a^{3} b^{2} c n\right )} x^{2}\right )} \sqrt{c x^{2}}{\left (b x + a\right )}^{n}}{{\left (b^{5} n^{5} + 15 \, b^{5} n^{4} + 85 \, b^{5} n^{3} + 225 \, b^{5} n^{2} + 274 \, b^{5} n + 120 \, b^{5}\right )} x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int x \left (c x^{2}\right )^{\frac{3}{2}} \left (a + b x\right )^{n}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.07302, size = 575, normalized size = 3.4 \begin{align*} -{\left (\frac{24 \, a^{5} a^{n} \mathrm{sgn}\left (x\right )}{b^{5} n^{5} + 15 \, b^{5} n^{4} + 85 \, b^{5} n^{3} + 225 \, b^{5} n^{2} + 274 \, b^{5} n + 120 \, b^{5}} - \frac{{\left (b x + a\right )}^{n} b^{5} n^{4} x^{5} \mathrm{sgn}\left (x\right ) +{\left (b x + a\right )}^{n} a b^{4} n^{4} x^{4} \mathrm{sgn}\left (x\right ) + 10 \,{\left (b x + a\right )}^{n} b^{5} n^{3} x^{5} \mathrm{sgn}\left (x\right ) + 6 \,{\left (b x + a\right )}^{n} a b^{4} n^{3} x^{4} \mathrm{sgn}\left (x\right ) + 35 \,{\left (b x + a\right )}^{n} b^{5} n^{2} x^{5} \mathrm{sgn}\left (x\right ) - 4 \,{\left (b x + a\right )}^{n} a^{2} b^{3} n^{3} x^{3} \mathrm{sgn}\left (x\right ) + 11 \,{\left (b x + a\right )}^{n} a b^{4} n^{2} x^{4} \mathrm{sgn}\left (x\right ) + 50 \,{\left (b x + a\right )}^{n} b^{5} n x^{5} \mathrm{sgn}\left (x\right ) - 12 \,{\left (b x + a\right )}^{n} a^{2} b^{3} n^{2} x^{3} \mathrm{sgn}\left (x\right ) + 6 \,{\left (b x + a\right )}^{n} a b^{4} n x^{4} \mathrm{sgn}\left (x\right ) + 24 \,{\left (b x + a\right )}^{n} b^{5} x^{5} \mathrm{sgn}\left (x\right ) + 12 \,{\left (b x + a\right )}^{n} a^{3} b^{2} n^{2} x^{2} \mathrm{sgn}\left (x\right ) - 8 \,{\left (b x + a\right )}^{n} a^{2} b^{3} n x^{3} \mathrm{sgn}\left (x\right ) + 12 \,{\left (b x + a\right )}^{n} a^{3} b^{2} n x^{2} \mathrm{sgn}\left (x\right ) - 24 \,{\left (b x + a\right )}^{n} a^{4} b n x \mathrm{sgn}\left (x\right ) + 24 \,{\left (b x + a\right )}^{n} a^{5} \mathrm{sgn}\left (x\right )}{b^{5} n^{5} + 15 \, b^{5} n^{4} + 85 \, b^{5} n^{3} + 225 \, b^{5} n^{2} + 274 \, b^{5} n + 120 \, b^{5}}\right )} c^{\frac{3}{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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