Optimal. Leaf size=217 \[ -\frac{a^5 c^2 \sqrt{c x^2} (a+b x)^{n+1}}{b^6 (n+1) x}+\frac{5 a^4 c^2 \sqrt{c x^2} (a+b x)^{n+2}}{b^6 (n+2) x}-\frac{10 a^3 c^2 \sqrt{c x^2} (a+b x)^{n+3}}{b^6 (n+3) x}+\frac{10 a^2 c^2 \sqrt{c x^2} (a+b x)^{n+4}}{b^6 (n+4) x}-\frac{5 a c^2 \sqrt{c x^2} (a+b x)^{n+5}}{b^6 (n+5) x}+\frac{c^2 \sqrt{c x^2} (a+b x)^{n+6}}{b^6 (n+6) x} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.073794, antiderivative size = 217, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118, Rules used = {15, 43} \[ -\frac{a^5 c^2 \sqrt{c x^2} (a+b x)^{n+1}}{b^6 (n+1) x}+\frac{5 a^4 c^2 \sqrt{c x^2} (a+b x)^{n+2}}{b^6 (n+2) x}-\frac{10 a^3 c^2 \sqrt{c x^2} (a+b x)^{n+3}}{b^6 (n+3) x}+\frac{10 a^2 c^2 \sqrt{c x^2} (a+b x)^{n+4}}{b^6 (n+4) x}-\frac{5 a c^2 \sqrt{c x^2} (a+b x)^{n+5}}{b^6 (n+5) x}+\frac{c^2 \sqrt{c x^2} (a+b x)^{n+6}}{b^6 (n+6) x} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 15
Rule 43
Rubi steps
\begin{align*} \int \left (c x^2\right )^{5/2} (a+b x)^n \, dx &=\frac{\left (c^2 \sqrt{c x^2}\right ) \int x^5 (a+b x)^n \, dx}{x}\\ &=\frac{\left (c^2 \sqrt{c x^2}\right ) \int \left (-\frac{a^5 (a+b x)^n}{b^5}+\frac{5 a^4 (a+b x)^{1+n}}{b^5}-\frac{10 a^3 (a+b x)^{2+n}}{b^5}+\frac{10 a^2 (a+b x)^{3+n}}{b^5}-\frac{5 a (a+b x)^{4+n}}{b^5}+\frac{(a+b x)^{5+n}}{b^5}\right ) \, dx}{x}\\ &=-\frac{a^5 c^2 \sqrt{c x^2} (a+b x)^{1+n}}{b^6 (1+n) x}+\frac{5 a^4 c^2 \sqrt{c x^2} (a+b x)^{2+n}}{b^6 (2+n) x}-\frac{10 a^3 c^2 \sqrt{c x^2} (a+b x)^{3+n}}{b^6 (3+n) x}+\frac{10 a^2 c^2 \sqrt{c x^2} (a+b x)^{4+n}}{b^6 (4+n) x}-\frac{5 a c^2 \sqrt{c x^2} (a+b x)^{5+n}}{b^6 (5+n) x}+\frac{c^2 \sqrt{c x^2} (a+b x)^{6+n}}{b^6 (6+n) x}\\ \end{align*}
Mathematica [A] time = 0.1091, size = 172, normalized size = 0.79 \[ \frac{c^3 x (a+b x)^{n+1} \left (-60 a^3 b^2 \left (n^2+3 n+2\right ) x^2+20 a^2 b^3 \left (n^3+6 n^2+11 n+6\right ) x^3+120 a^4 b (n+1) x-120 a^5-5 a b^4 \left (n^4+10 n^3+35 n^2+50 n+24\right ) x^4+b^5 \left (n^5+15 n^4+85 n^3+225 n^2+274 n+120\right ) x^5\right )}{b^6 (n+1) (n+2) (n+3) (n+4) (n+5) (n+6) \sqrt{c x^2}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.006, size = 280, normalized size = 1.3 \begin{align*} -{\frac{ \left ( bx+a \right ) ^{1+n} \left ( -{b}^{5}{n}^{5}{x}^{5}-15\,{b}^{5}{n}^{4}{x}^{5}+5\,a{b}^{4}{n}^{4}{x}^{4}-85\,{b}^{5}{n}^{3}{x}^{5}+50\,a{b}^{4}{n}^{3}{x}^{4}-225\,{b}^{5}{n}^{2}{x}^{5}-20\,{a}^{2}{b}^{3}{n}^{3}{x}^{3}+175\,a{b}^{4}{n}^{2}{x}^{4}-274\,{b}^{5}n{x}^{5}-120\,{a}^{2}{b}^{3}{n}^{2}{x}^{3}+250\,a{b}^{4}n{x}^{4}-120\,{b}^{5}{x}^{5}+60\,{a}^{3}{b}^{2}{n}^{2}{x}^{2}-220\,{a}^{2}{b}^{3}n{x}^{3}+120\,a{b}^{4}{x}^{4}+180\,{a}^{3}{b}^{2}n{x}^{2}-120\,{a}^{2}{b}^{3}{x}^{3}-120\,{a}^{4}bnx+120\,{a}^{3}{b}^{2}{x}^{2}-120\,{a}^{4}bx+120\,{a}^{5} \right ) }{{x}^{5}{b}^{6} \left ({n}^{6}+21\,{n}^{5}+175\,{n}^{4}+735\,{n}^{3}+1624\,{n}^{2}+1764\,n+720 \right ) } \left ( c{x}^{2} \right ) ^{{\frac{5}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.12252, size = 274, normalized size = 1.26 \begin{align*} \frac{{\left ({\left (n^{5} + 15 \, n^{4} + 85 \, n^{3} + 225 \, n^{2} + 274 \, n + 120\right )} b^{6} c^{\frac{5}{2}} x^{6} +{\left (n^{5} + 10 \, n^{4} + 35 \, n^{3} + 50 \, n^{2} + 24 \, n\right )} a b^{5} c^{\frac{5}{2}} x^{5} - 5 \,{\left (n^{4} + 6 \, n^{3} + 11 \, n^{2} + 6 \, n\right )} a^{2} b^{4} c^{\frac{5}{2}} x^{4} + 20 \,{\left (n^{3} + 3 \, n^{2} + 2 \, n\right )} a^{3} b^{3} c^{\frac{5}{2}} x^{3} - 60 \,{\left (n^{2} + n\right )} a^{4} b^{2} c^{\frac{5}{2}} x^{2} + 120 \, a^{5} b c^{\frac{5}{2}} n x - 120 \, a^{6} c^{\frac{5}{2}}\right )}{\left (b x + a\right )}^{n}}{{\left (n^{6} + 21 \, n^{5} + 175 \, n^{4} + 735 \, n^{3} + 1624 \, n^{2} + 1764 \, n + 720\right )} b^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.71717, size = 733, normalized size = 3.38 \begin{align*} \frac{{\left (120 \, a^{5} b c^{2} n x - 120 \, a^{6} c^{2} +{\left (b^{6} c^{2} n^{5} + 15 \, b^{6} c^{2} n^{4} + 85 \, b^{6} c^{2} n^{3} + 225 \, b^{6} c^{2} n^{2} + 274 \, b^{6} c^{2} n + 120 \, b^{6} c^{2}\right )} x^{6} +{\left (a b^{5} c^{2} n^{5} + 10 \, a b^{5} c^{2} n^{4} + 35 \, a b^{5} c^{2} n^{3} + 50 \, a b^{5} c^{2} n^{2} + 24 \, a b^{5} c^{2} n\right )} x^{5} - 5 \,{\left (a^{2} b^{4} c^{2} n^{4} + 6 \, a^{2} b^{4} c^{2} n^{3} + 11 \, a^{2} b^{4} c^{2} n^{2} + 6 \, a^{2} b^{4} c^{2} n\right )} x^{4} + 20 \,{\left (a^{3} b^{3} c^{2} n^{3} + 3 \, a^{3} b^{3} c^{2} n^{2} + 2 \, a^{3} b^{3} c^{2} n\right )} x^{3} - 60 \,{\left (a^{4} b^{2} c^{2} n^{2} + a^{4} b^{2} c^{2} n\right )} x^{2}\right )} \sqrt{c x^{2}}{\left (b x + a\right )}^{n}}{{\left (b^{6} n^{6} + 21 \, b^{6} n^{5} + 175 \, b^{6} n^{4} + 735 \, b^{6} n^{3} + 1624 \, b^{6} n^{2} + 1764 \, b^{6} n + 720 \, b^{6}\right )} x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (c x^{2}\right )^{\frac{5}{2}} \left (a + b x\right )^{n}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [B] time = 1.08614, size = 864, normalized size = 3.98 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]