Optimal. Leaf size=165 \[ -\frac {b \pi ^{3/2} x^2}{32 c}-\frac {7}{96} b c \pi ^{3/2} x^4-\frac {1}{36} b c^3 \pi ^{3/2} x^6+\frac {\pi ^{3/2} x \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{16 c^2}+\frac {1}{8} \pi x^3 \sqrt {\pi +c^2 \pi x^2} \left (a+b \sinh ^{-1}(c x)\right )+\frac {1}{6} x^3 \left (\pi +c^2 \pi x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )-\frac {\pi ^{3/2} \left (a+b \sinh ^{-1}(c x)\right )^2}{32 b c^3} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.22, antiderivative size = 165, normalized size of antiderivative = 1.00, number of steps
used = 8, number of rules used = 6, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {5808, 5806,
5812, 5783, 30, 14} \begin {gather*} -\frac {\pi ^{3/2} \left (a+b \sinh ^{-1}(c x)\right )^2}{32 b c^3}+\frac {\pi ^{3/2} x \sqrt {c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )}{16 c^2}+\frac {1}{6} x^3 \left (\pi c^2 x^2+\pi \right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )+\frac {1}{8} \pi x^3 \sqrt {\pi c^2 x^2+\pi } \left (a+b \sinh ^{-1}(c x)\right )-\frac {1}{36} \pi ^{3/2} b c^3 x^6-\frac {7}{96} \pi ^{3/2} b c x^4-\frac {\pi ^{3/2} b x^2}{32 c} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 14
Rule 30
Rule 5783
Rule 5806
Rule 5808
Rule 5812
Rubi steps
\begin {align*} \int x^2 \left (\pi +c^2 \pi x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right ) \, dx &=\frac {1}{6} x^3 \left (\pi +c^2 \pi x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )+\frac {1}{2} \pi \int x^2 \sqrt {\pi +c^2 \pi x^2} \left (a+b \sinh ^{-1}(c x)\right ) \, dx-\frac {\left (b c \pi \sqrt {\pi +c^2 \pi x^2}\right ) \int x^3 \left (1+c^2 x^2\right ) \, dx}{6 \sqrt {1+c^2 x^2}}\\ &=\frac {1}{8} \pi x^3 \sqrt {\pi +c^2 \pi x^2} \left (a+b \sinh ^{-1}(c x)\right )+\frac {1}{6} x^3 \left (\pi +c^2 \pi x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )+\frac {\left (\pi \sqrt {\pi +c^2 \pi x^2}\right ) \int \frac {x^2 \left (a+b \sinh ^{-1}(c x)\right )}{\sqrt {1+c^2 x^2}} \, dx}{8 \sqrt {1+c^2 x^2}}-\frac {\left (b c \pi \sqrt {\pi +c^2 \pi x^2}\right ) \int x^3 \, dx}{8 \sqrt {1+c^2 x^2}}-\frac {\left (b c \pi \sqrt {\pi +c^2 \pi x^2}\right ) \int \left (x^3+c^2 x^5\right ) \, dx}{6 \sqrt {1+c^2 x^2}}\\ &=-\frac {7 b c \pi x^4 \sqrt {\pi +c^2 \pi x^2}}{96 \sqrt {1+c^2 x^2}}-\frac {b c^3 \pi x^6 \sqrt {\pi +c^2 \pi x^2}}{36 \sqrt {1+c^2 x^2}}+\frac {\pi x \sqrt {\pi +c^2 \pi x^2} \left (a+b \sinh ^{-1}(c x)\right )}{16 c^2}+\frac {1}{8} \pi x^3 \sqrt {\pi +c^2 \pi x^2} \left (a+b \sinh ^{-1}(c x)\right )+\frac {1}{6} x^3 \left (\pi +c^2 \pi x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )-\frac {\left (\pi \sqrt {\pi +c^2 \pi x^2}\right ) \int \frac {a+b \sinh ^{-1}(c x)}{\sqrt {1+c^2 x^2}} \, dx}{16 c^2 \sqrt {1+c^2 x^2}}-\frac {\left (b \pi \sqrt {\pi +c^2 \pi x^2}\right ) \int x \, dx}{16 c \sqrt {1+c^2 x^2}}\\ &=-\frac {b \pi x^2 \sqrt {\pi +c^2 \pi x^2}}{32 c \sqrt {1+c^2 x^2}}-\frac {7 b c \pi x^4 \sqrt {\pi +c^2 \pi x^2}}{96 \sqrt {1+c^2 x^2}}-\frac {b c^3 \pi x^6 \sqrt {\pi +c^2 \pi x^2}}{36 \sqrt {1+c^2 x^2}}+\frac {\pi x \sqrt {\pi +c^2 \pi x^2} \left (a+b \sinh ^{-1}(c x)\right )}{16 c^2}+\frac {1}{8} \pi x^3 \sqrt {\pi +c^2 \pi x^2} \left (a+b \sinh ^{-1}(c x)\right )+\frac {1}{6} x^3 \left (\pi +c^2 \pi x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )-\frac {\pi \sqrt {\pi +c^2 \pi x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{32 b c^3 \sqrt {1+c^2 x^2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.24, size = 154, normalized size = 0.93 \begin {gather*} \frac {\pi ^{3/2} \left (144 a c x \sqrt {1+c^2 x^2}+672 a c^3 x^3 \sqrt {1+c^2 x^2}+384 a c^5 x^5 \sqrt {1+c^2 x^2}-72 b \sinh ^{-1}(c x)^2+18 b \cosh \left (2 \sinh ^{-1}(c x)\right )-9 b \cosh \left (4 \sinh ^{-1}(c x)\right )-2 b \cosh \left (6 \sinh ^{-1}(c x)\right )-12 \sinh ^{-1}(c x) \left (12 a+3 b \sinh \left (2 \sinh ^{-1}(c x)\right )-3 b \sinh \left (4 \sinh ^{-1}(c x)\right )-b \sinh \left (6 \sinh ^{-1}(c x)\right )\right )\right )}{2304 c^3} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F]
time = 180.00, size = 0, normalized size = 0.00 \[\int x^{2} \left (\pi \,c^{2} x^{2}+\pi \right )^{\frac {3}{2}} \left (a +b \arcsinh \left (c x \right )\right )\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: RuntimeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A]
time = 7.01, size = 262, normalized size = 1.59 \begin {gather*} \begin {cases} \frac {\pi ^{\frac {3}{2}} a c^{2} x^{5} \sqrt {c^{2} x^{2} + 1}}{6} + \frac {7 \pi ^{\frac {3}{2}} a x^{3} \sqrt {c^{2} x^{2} + 1}}{24} + \frac {\pi ^{\frac {3}{2}} a x \sqrt {c^{2} x^{2} + 1}}{16 c^{2}} - \frac {\pi ^{\frac {3}{2}} a \operatorname {asinh}{\left (c x \right )}}{16 c^{3}} - \frac {\pi ^{\frac {3}{2}} b c^{3} x^{6}}{36} + \frac {\pi ^{\frac {3}{2}} b c^{2} x^{5} \sqrt {c^{2} x^{2} + 1} \operatorname {asinh}{\left (c x \right )}}{6} - \frac {7 \pi ^{\frac {3}{2}} b c x^{4}}{96} + \frac {7 \pi ^{\frac {3}{2}} b x^{3} \sqrt {c^{2} x^{2} + 1} \operatorname {asinh}{\left (c x \right )}}{24} - \frac {\pi ^{\frac {3}{2}} b x^{2}}{32 c} + \frac {\pi ^{\frac {3}{2}} b x \sqrt {c^{2} x^{2} + 1} \operatorname {asinh}{\left (c x \right )}}{16 c^{2}} - \frac {\pi ^{\frac {3}{2}} b \operatorname {asinh}^{2}{\left (c x \right )}}{32 c^{3}} & \text {for}\: c \neq 0 \\\frac {\pi ^{\frac {3}{2}} a x^{3}}{3} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int x^2\,\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )\,{\left (\Pi \,c^2\,x^2+\Pi \right )}^{3/2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________