Optimal. Leaf size=126 \[ \frac {3 b x^2}{16 c^3 \sqrt {\pi }}-\frac {b x^4}{16 c \sqrt {\pi }}-\frac {3 x \sqrt {\pi +c^2 \pi x^2} \left (a+b \sinh ^{-1}(c x)\right )}{8 c^4 \pi }+\frac {x^3 \sqrt {\pi +c^2 \pi x^2} \left (a+b \sinh ^{-1}(c x)\right )}{4 c^2 \pi }+\frac {3 \left (a+b \sinh ^{-1}(c x)\right )^2}{16 b c^5 \sqrt {\pi }} \]
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Rubi [A]
time = 0.15, antiderivative size = 126, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 3, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.115, Rules used = {5812, 5783, 30}
\begin {gather*} \frac {3 \left (a+b \sinh ^{-1}(c x)\right )^2}{16 \sqrt {\pi } b c^5}+\frac {x^3 \sqrt {\pi c^2 x^2+\pi } \left (a+b \sinh ^{-1}(c x)\right )}{4 \pi c^2}-\frac {3 x \sqrt {\pi c^2 x^2+\pi } \left (a+b \sinh ^{-1}(c x)\right )}{8 \pi c^4}+\frac {3 b x^2}{16 \sqrt {\pi } c^3}-\frac {b x^4}{16 \sqrt {\pi } c} \end {gather*}
Antiderivative was successfully verified.
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Rule 30
Rule 5783
Rule 5812
Rubi steps
\begin {align*} \int \frac {x^4 \left (a+b \sinh ^{-1}(c x)\right )}{\sqrt {\pi +c^2 \pi x^2}} \, dx &=\frac {x^3 \sqrt {\pi +c^2 \pi x^2} \left (a+b \sinh ^{-1}(c x)\right )}{4 c^2 \pi }-\frac {3 \int \frac {x^2 \left (a+b \sinh ^{-1}(c x)\right )}{\sqrt {\pi +c^2 \pi x^2}} \, dx}{4 c^2}-\frac {\left (b \sqrt {1+c^2 x^2}\right ) \int x^3 \, dx}{4 c \sqrt {\pi +c^2 \pi x^2}}\\ &=-\frac {b x^4 \sqrt {1+c^2 x^2}}{16 c \sqrt {\pi +c^2 \pi x^2}}-\frac {3 x \sqrt {\pi +c^2 \pi x^2} \left (a+b \sinh ^{-1}(c x)\right )}{8 c^4 \pi }+\frac {x^3 \sqrt {\pi +c^2 \pi x^2} \left (a+b \sinh ^{-1}(c x)\right )}{4 c^2 \pi }+\frac {3 \int \frac {a+b \sinh ^{-1}(c x)}{\sqrt {\pi +c^2 \pi x^2}} \, dx}{8 c^4}+\frac {\left (3 b \sqrt {1+c^2 x^2}\right ) \int x \, dx}{8 c^3 \sqrt {\pi +c^2 \pi x^2}}\\ &=\frac {3 b x^2 \sqrt {1+c^2 x^2}}{16 c^3 \sqrt {\pi +c^2 \pi x^2}}-\frac {b x^4 \sqrt {1+c^2 x^2}}{16 c \sqrt {\pi +c^2 \pi x^2}}-\frac {3 x \sqrt {\pi +c^2 \pi x^2} \left (a+b \sinh ^{-1}(c x)\right )}{8 c^4 \pi }+\frac {x^3 \sqrt {\pi +c^2 \pi x^2} \left (a+b \sinh ^{-1}(c x)\right )}{4 c^2 \pi }+\frac {3 \left (a+b \sinh ^{-1}(c x)\right )^2}{16 b c^5 \sqrt {\pi }}\\ \end {align*}
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Mathematica [A]
time = 0.15, size = 111, normalized size = 0.88 \begin {gather*} \frac {-48 a c x \sqrt {1+c^2 x^2}+32 a c^3 x^3 \sqrt {1+c^2 x^2}+24 b \sinh ^{-1}(c x)^2+16 b \cosh \left (2 \sinh ^{-1}(c x)\right )-b \cosh \left (4 \sinh ^{-1}(c x)\right )+4 \sinh ^{-1}(c x) \left (12 a-8 b \sinh \left (2 \sinh ^{-1}(c x)\right )+b \sinh \left (4 \sinh ^{-1}(c x)\right )\right )}{128 c^5 \sqrt {\pi }} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 180.00, size = 0, normalized size = 0.00 \[\int \frac {x^{4} \left (a +b \arcsinh \left (c x \right )\right )}{\sqrt {\pi \,c^{2} x^{2}+\pi }}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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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.
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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.
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Sympy [A]
time = 5.31, size = 185, normalized size = 1.47 \begin {gather*} \frac {a x^{5}}{4 \sqrt {\pi } \sqrt {c^{2} x^{2} + 1}} - \frac {a x^{3}}{8 \sqrt {\pi } c^{2} \sqrt {c^{2} x^{2} + 1}} - \frac {3 a x}{8 \sqrt {\pi } c^{4} \sqrt {c^{2} x^{2} + 1}} + \frac {3 a \operatorname {asinh}{\left (c x \right )}}{8 \sqrt {\pi } c^{5}} + \frac {b \left (\begin {cases} - \frac {x^{4}}{16 c} + \frac {x^{3} \sqrt {c^{2} x^{2} + 1} \operatorname {asinh}{\left (c x \right )}}{4 c^{2}} + \frac {3 x^{2}}{16 c^{3}} - \frac {3 x \sqrt {c^{2} x^{2} + 1} \operatorname {asinh}{\left (c x \right )}}{8 c^{4}} + \frac {3 \operatorname {asinh}^{2}{\left (c x \right )}}{16 c^{5}} & \text {for}\: c \neq 0 \\0 & \text {otherwise} \end {cases}\right )}{\sqrt {\pi }} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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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.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {x^4\,\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}{\sqrt {\Pi \,c^2\,x^2+\Pi }} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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