Optimal. Leaf size=157 \[ -\frac {9}{16} b c^3 \pi ^{5/2} x^2-\frac {1}{16} b c^5 \pi ^{5/2} x^4+\frac {15}{8} c^2 \pi ^2 x \sqrt {\pi +c^2 \pi x^2} \left (a+b \sinh ^{-1}(c x)\right )+\frac {5}{4} c^2 \pi x \left (\pi +c^2 \pi x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )-\frac {\left (\pi +c^2 \pi x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )}{x}+\frac {15 c \pi ^{5/2} \left (a+b \sinh ^{-1}(c x)\right )^2}{16 b}+b c \pi ^{5/2} \log (x) \]
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Rubi [A]
time = 0.16, antiderivative size = 157, normalized size of antiderivative = 1.00, number of steps
used = 10, number of rules used = 8, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {5807, 5786,
5785, 5783, 30, 14, 272, 45} \begin {gather*} \frac {5}{4} \pi c^2 x \left (\pi c^2 x^2+\pi \right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )+\frac {15}{8} \pi ^2 c^2 x \sqrt {\pi c^2 x^2+\pi } \left (a+b \sinh ^{-1}(c x)\right )-\frac {\left (\pi c^2 x^2+\pi \right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )}{x}+\frac {15 \pi ^{5/2} c \left (a+b \sinh ^{-1}(c x)\right )^2}{16 b}-\frac {1}{16} \pi ^{5/2} b c^5 x^4-\frac {9}{16} \pi ^{5/2} b c^3 x^2+\pi ^{5/2} b c \log (x) \end {gather*}
Antiderivative was successfully verified.
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Rule 14
Rule 30
Rule 45
Rule 272
Rule 5783
Rule 5785
Rule 5786
Rule 5807
Rubi steps
\begin {align*} \int \frac {\left (\pi +c^2 \pi x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )}{x^2} \, dx &=-\frac {\left (\pi +c^2 \pi x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )}{x}+\left (5 c^2 \pi \right ) \int \left (\pi +c^2 \pi x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right ) \, dx+\frac {\left (b c \pi ^2 \sqrt {\pi +c^2 \pi x^2}\right ) \int \frac {\left (1+c^2 x^2\right )^2}{x} \, dx}{\sqrt {1+c^2 x^2}}\\ &=\frac {5}{4} c^2 \pi x \left (\pi +c^2 \pi x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )-\frac {\left (\pi +c^2 \pi x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )}{x}+\frac {1}{4} \left (15 c^2 \pi ^2\right ) \int \sqrt {\pi +c^2 \pi x^2} \left (a+b \sinh ^{-1}(c x)\right ) \, dx+\frac {\left (b c \pi ^2 \sqrt {\pi +c^2 \pi x^2}\right ) \text {Subst}\left (\int \frac {\left (1+c^2 x\right )^2}{x} \, dx,x,x^2\right )}{2 \sqrt {1+c^2 x^2}}-\frac {\left (5 b c^3 \pi ^2 \sqrt {\pi +c^2 \pi x^2}\right ) \int x \left (1+c^2 x^2\right ) \, dx}{4 \sqrt {1+c^2 x^2}}\\ &=\frac {15}{8} c^2 \pi ^2 x \sqrt {\pi +c^2 \pi x^2} \left (a+b \sinh ^{-1}(c x)\right )+\frac {5}{4} c^2 \pi x \left (\pi +c^2 \pi x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )-\frac {\left (\pi +c^2 \pi x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )}{x}+\frac {\left (b c \pi ^2 \sqrt {\pi +c^2 \pi x^2}\right ) \text {Subst}\left (\int \left (2 c^2+\frac {1}{x}+c^4 x\right ) \, dx,x,x^2\right )}{2 \sqrt {1+c^2 x^2}}+\frac {\left (15 c^2 \pi ^2 \sqrt {\pi +c^2 \pi x^2}\right ) \int \frac {a+b \sinh ^{-1}(c x)}{\sqrt {1+c^2 x^2}} \, dx}{8 \sqrt {1+c^2 x^2}}-\frac {\left (5 b c^3 \pi ^2 \sqrt {\pi +c^2 \pi x^2}\right ) \int \left (x+c^2 x^3\right ) \, dx}{4 \sqrt {1+c^2 x^2}}-\frac {\left (15 b c^3 \pi ^2 \sqrt {\pi +c^2 \pi x^2}\right ) \int x \, dx}{8 \sqrt {1+c^2 x^2}}\\ &=-\frac {9 b c^3 \pi ^2 x^2 \sqrt {\pi +c^2 \pi x^2}}{16 \sqrt {1+c^2 x^2}}-\frac {b c^5 \pi ^2 x^4 \sqrt {\pi +c^2 \pi x^2}}{16 \sqrt {1+c^2 x^2}}+\frac {15}{8} c^2 \pi ^2 x \sqrt {\pi +c^2 \pi x^2} \left (a+b \sinh ^{-1}(c x)\right )+\frac {5}{4} c^2 \pi x \left (\pi +c^2 \pi x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )-\frac {\left (\pi +c^2 \pi x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )}{x}+\frac {15 c \pi ^2 \sqrt {\pi +c^2 \pi x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{16 b \sqrt {1+c^2 x^2}}+\frac {b c \pi ^2 \sqrt {\pi +c^2 \pi x^2} \log (x)}{\sqrt {1+c^2 x^2}}\\ \end {align*}
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Mathematica [A]
time = 0.27, size = 168, normalized size = 1.07 \begin {gather*} \frac {\pi ^{5/2} \left (-128 a \sqrt {1+c^2 x^2}+144 a c^2 x^2 \sqrt {1+c^2 x^2}+32 a c^4 x^4 \sqrt {1+c^2 x^2}+120 b c x \sinh ^{-1}(c x)^2-32 b c x \cosh \left (2 \sinh ^{-1}(c x)\right )-b c x \cosh \left (4 \sinh ^{-1}(c x)\right )+128 b c x \log (c x)+4 \sinh ^{-1}(c x) \left (60 a c x-32 b \sqrt {1+c^2 x^2}+16 b c x \sinh \left (2 \sinh ^{-1}(c x)\right )+b c x \sinh \left (4 \sinh ^{-1}(c x)\right )\right )\right )}{128 x} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(282\) vs.
\(2(133)=266\).
time = 6.40, size = 283, normalized size = 1.80
method | result | size |
default | \(-\frac {a \left (\pi \,c^{2} x^{2}+\pi \right )^{\frac {7}{2}}}{\pi x}+a \,c^{2} x \left (\pi \,c^{2} x^{2}+\pi \right )^{\frac {5}{2}}+\frac {5 a \,c^{2} \pi x \left (\pi \,c^{2} x^{2}+\pi \right )^{\frac {3}{2}}}{4}+\frac {15 a \,c^{2} \pi ^{2} x \sqrt {\pi \,c^{2} x^{2}+\pi }}{8}+\frac {15 a \,c^{2} \pi ^{3} \ln \left (\frac {\pi \,c^{2} x}{\sqrt {\pi \,c^{2}}}+\sqrt {\pi \,c^{2} x^{2}+\pi }\right )}{8 \sqrt {\pi \,c^{2}}}-\frac {33 b \,\pi ^{\frac {5}{2}} c}{128}+\frac {15 b c \,\pi ^{\frac {5}{2}} \arcsinh \left (c x \right )^{2}}{16}-b c \,\pi ^{\frac {5}{2}} \arcsinh \left (c x \right )+b c \,\pi ^{\frac {5}{2}} \ln \left (\left (c x +\sqrt {c^{2} x^{2}+1}\right )^{2}-1\right )-\frac {b \,\pi ^{\frac {5}{2}} \arcsinh \left (c x \right ) \sqrt {c^{2} x^{2}+1}}{x}+\frac {b \arcsinh \left (c x \right ) \pi ^{\frac {5}{2}} \sqrt {c^{2} x^{2}+1}\, x^{3} c^{4}}{4}+\frac {9 b \arcsinh \left (c x \right ) \pi ^{\frac {5}{2}} \sqrt {c^{2} x^{2}+1}\, x \,c^{2}}{8}-\frac {b \,c^{5} \pi ^{\frac {5}{2}} x^{4}}{16}-\frac {9 b \,c^{3} \pi ^{\frac {5}{2}} x^{2}}{16}\) | \(283\) |
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 [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \pi ^{\frac {5}{2}} \left (\int 2 a c^{2} \sqrt {c^{2} x^{2} + 1}\, dx + \int \frac {a \sqrt {c^{2} x^{2} + 1}}{x^{2}}\, dx + \int a c^{4} x^{2} \sqrt {c^{2} x^{2} + 1}\, dx + \int 2 b c^{2} \sqrt {c^{2} x^{2} + 1} \operatorname {asinh}{\left (c x \right )}\, dx + \int \frac {b \sqrt {c^{2} x^{2} + 1} \operatorname {asinh}{\left (c x \right )}}{x^{2}}\, dx + \int b c^{4} x^{2} \sqrt {c^{2} x^{2} + 1} \operatorname {asinh}{\left (c x \right )}\, dx\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \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 {\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )\,{\left (\Pi \,c^2\,x^2+\Pi \right )}^{5/2}}{x^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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