Optimal. Leaf size=177 \[ -\frac {b c^3 d x^2 \sqrt {d+c^2 d x^2}}{4 \sqrt {1+c^2 x^2}}+\frac {3}{2} c^2 d x \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )-\frac {\left (d+c^2 d x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )}{x}+\frac {3 c d \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{4 b \sqrt {1+c^2 x^2}}+\frac {b c d \sqrt {d+c^2 d x^2} \log (x)}{\sqrt {1+c^2 x^2}} \]
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Rubi [A]
time = 0.13, antiderivative size = 177, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 5, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.192, Rules used = {5807, 5785,
5783, 30, 14} \begin {gather*} \frac {3}{2} c^2 d x \sqrt {c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )+\frac {3 c d \sqrt {c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )^2}{4 b \sqrt {c^2 x^2+1}}-\frac {\left (c^2 d x^2+d\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )}{x}+\frac {b c d \log (x) \sqrt {c^2 d x^2+d}}{\sqrt {c^2 x^2+1}}-\frac {b c^3 d x^2 \sqrt {c^2 d x^2+d}}{4 \sqrt {c^2 x^2+1}} \end {gather*}
Antiderivative was successfully verified.
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Rule 14
Rule 30
Rule 5783
Rule 5785
Rule 5807
Rubi steps
\begin {align*} \int \frac {\left (d+c^2 d x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )}{x^2} \, dx &=-\frac {\left (d+c^2 d x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )}{x}+\left (3 c^2 d\right ) \int \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right ) \, dx+\frac {\left (b c d \sqrt {d+c^2 d x^2}\right ) \int \frac {1+c^2 x^2}{x} \, dx}{\sqrt {1+c^2 x^2}}\\ &=\frac {3}{2} c^2 d x \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )-\frac {\left (d+c^2 d x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )}{x}+\frac {\left (b c d \sqrt {d+c^2 d x^2}\right ) \int \left (\frac {1}{x}+c^2 x\right ) \, dx}{\sqrt {1+c^2 x^2}}+\frac {\left (3 c^2 d \sqrt {d+c^2 d x^2}\right ) \int \frac {a+b \sinh ^{-1}(c x)}{\sqrt {1+c^2 x^2}} \, dx}{2 \sqrt {1+c^2 x^2}}-\frac {\left (3 b c^3 d \sqrt {d+c^2 d x^2}\right ) \int x \, dx}{2 \sqrt {1+c^2 x^2}}\\ &=-\frac {b c^3 d x^2 \sqrt {d+c^2 d x^2}}{4 \sqrt {1+c^2 x^2}}+\frac {3}{2} c^2 d x \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )-\frac {\left (d+c^2 d x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )}{x}+\frac {3 c d \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{4 b \sqrt {1+c^2 x^2}}+\frac {b c d \sqrt {d+c^2 d x^2} \log (x)}{\sqrt {1+c^2 x^2}}\\ \end {align*}
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Mathematica [A]
time = 0.55, size = 200, normalized size = 1.13 \begin {gather*} \frac {1}{8} \left (\frac {4 a d \left (-2+c^2 x^2\right ) \sqrt {d+c^2 d x^2}}{x}+\frac {4 b d \sqrt {d+c^2 d x^2} \left (-2 \sqrt {1+c^2 x^2} \sinh ^{-1}(c x)+c x \sinh ^{-1}(c x)^2+2 c x \log (c x)\right )}{x \sqrt {1+c^2 x^2}}+12 a c d^{3/2} \log \left (c d x+\sqrt {d} \sqrt {d+c^2 d x^2}\right )+\frac {b c d \sqrt {d+c^2 d x^2} \left (-\cosh \left (2 \sinh ^{-1}(c x)\right )+2 \sinh ^{-1}(c x) \left (\sinh ^{-1}(c x)+\sinh \left (2 \sinh ^{-1}(c x)\right )\right )\right )}{\sqrt {1+c^2 x^2}}\right ) \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. \(391\) vs.
\(2(155)=310\).
time = 2.16, size = 392, normalized size = 2.21
method | result | size |
default | \(-\frac {a \left (c^{2} d \,x^{2}+d \right )^{\frac {5}{2}}}{d x}+a \,c^{2} x \left (c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}+\frac {3 a \,c^{2} d x \sqrt {c^{2} d \,x^{2}+d}}{2}+\frac {3 a \,c^{2} d^{2} \ln \left (\frac {x \,c^{2} d}{\sqrt {c^{2} d}}+\sqrt {c^{2} d \,x^{2}+d}\right )}{2 \sqrt {c^{2} d}}+\frac {3 b \sqrt {d \left (c^{2} x^{2}+1\right )}\, \arcsinh \left (c x \right )^{2} c d}{4 \sqrt {c^{2} x^{2}+1}}+\frac {b \sqrt {d \left (c^{2} x^{2}+1\right )}\, c^{4} d \arcsinh \left (c x \right ) x^{3}}{2 c^{2} x^{2}+2}-\frac {b \sqrt {d \left (c^{2} x^{2}+1\right )}\, c^{3} d \,x^{2}}{4 \sqrt {c^{2} x^{2}+1}}-\frac {b \sqrt {d \left (c^{2} x^{2}+1\right )}\, c^{2} d \arcsinh \left (c x \right ) x}{2 \left (c^{2} x^{2}+1\right )}-\frac {b \sqrt {d \left (c^{2} x^{2}+1\right )}\, c d \arcsinh \left (c x \right )}{\sqrt {c^{2} x^{2}+1}}-\frac {b \sqrt {d \left (c^{2} x^{2}+1\right )}\, c d}{8 \sqrt {c^{2} x^{2}+1}}-\frac {b \sqrt {d \left (c^{2} x^{2}+1\right )}\, \arcsinh \left (c x \right ) d}{x \left (c^{2} x^{2}+1\right )}+\frac {b \sqrt {d \left (c^{2} x^{2}+1\right )}\, \ln \left (\left (c x +\sqrt {c^{2} x^{2}+1}\right )^{2}-1\right ) c d}{\sqrt {c^{2} x^{2}+1}}\) | \(392\) |
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*} \int \frac {\left (d \left (c^{2} x^{2} + 1\right )\right )^{\frac {3}{2}} \left (a + b \operatorname {asinh}{\left (c x \right )}\right )}{x^{2}}\, dx \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 (d\,c^2\,x^2+d\right )}^{3/2}}{x^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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