Optimal. Leaf size=184 \[ -\frac {c^2 d \sqrt {c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )}{x}-\frac {\left (c^2 d x^2+d\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )}{3 x^3}+\frac {c^3 d \sqrt {c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )^2}{2 b \sqrt {c^2 x^2+1}}-\frac {b c d \sqrt {c^2 d x^2+d}}{6 x^2 \sqrt {c^2 x^2+1}}+\frac {4 b c^3 d \log (x) \sqrt {c^2 d x^2+d}}{3 \sqrt {c^2 x^2+1}} \]
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Rubi [A] time = 0.23, antiderivative size = 184, 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 = {5739, 5737, 29, 5675, 14} \[ \frac {c^3 d \sqrt {c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )^2}{2 b \sqrt {c^2 x^2+1}}-\frac {c^2 d \sqrt {c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )}{x}-\frac {\left (c^2 d x^2+d\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )}{3 x^3}-\frac {b c d \sqrt {c^2 d x^2+d}}{6 x^2 \sqrt {c^2 x^2+1}}+\frac {4 b c^3 d \log (x) \sqrt {c^2 d x^2+d}}{3 \sqrt {c^2 x^2+1}} \]
Antiderivative was successfully verified.
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Rule 14
Rule 29
Rule 5675
Rule 5737
Rule 5739
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^4} \, dx &=-\frac {\left (d+c^2 d x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )}{3 x^3}+\left (c^2 d\right ) \int \frac {\sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{x^2} \, dx+\frac {\left (b c d \sqrt {d+c^2 d x^2}\right ) \int \frac {1+c^2 x^2}{x^3} \, dx}{3 \sqrt {1+c^2 x^2}}\\ &=-\frac {c^2 d \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{x}-\frac {\left (d+c^2 d x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )}{3 x^3}+\frac {\left (b c d \sqrt {d+c^2 d x^2}\right ) \int \left (\frac {1}{x^3}+\frac {c^2}{x}\right ) \, dx}{3 \sqrt {1+c^2 x^2}}+\frac {\left (b c^3 d \sqrt {d+c^2 d x^2}\right ) \int \frac {1}{x} \, dx}{\sqrt {1+c^2 x^2}}+\frac {\left (c^4 d \sqrt {d+c^2 d x^2}\right ) \int \frac {a+b \sinh ^{-1}(c x)}{\sqrt {1+c^2 x^2}} \, dx}{\sqrt {1+c^2 x^2}}\\ &=-\frac {b c d \sqrt {d+c^2 d x^2}}{6 x^2 \sqrt {1+c^2 x^2}}-\frac {c^2 d \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{x}-\frac {\left (d+c^2 d x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )}{3 x^3}+\frac {c^3 d \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{2 b \sqrt {1+c^2 x^2}}+\frac {4 b c^3 d \sqrt {d+c^2 d x^2} \log (x)}{3 \sqrt {1+c^2 x^2}}\\ \end {align*}
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Mathematica [A] time = 0.77, size = 214, normalized size = 1.16 \[ \frac {1}{6} d \left (-\frac {2 a \left (4 c^2 x^2+1\right ) \sqrt {c^2 d x^2+d}}{x^3}+6 a c^3 \sqrt {d} \log \left (\sqrt {d} \sqrt {c^2 d x^2+d}+c d x\right )+\frac {3 b c^2 \sqrt {c^2 d x^2+d} \left (-2 \sqrt {c^2 x^2+1} \sinh ^{-1}(c x)+2 c x \log (c x)+c x \sinh ^{-1}(c x)^2\right )}{x \sqrt {c^2 x^2+1}}-\frac {b \sqrt {c^2 d x^2+d} \left (-2 c^3 x^3 \log (c x)+2 \left (c^2 x^2+1\right )^{3/2} \sinh ^{-1}(c x)+c x\right )}{x^3 \sqrt {c^2 x^2+1}}\right ) \]
Antiderivative was successfully verified.
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fricas [F] time = 0.57, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (a c^{2} d x^{2} + a d + {\left (b c^{2} d x^{2} + b d\right )} \operatorname {arsinh}\left (c x\right )\right )} \sqrt {c^{2} d x^{2} + d}}{x^{4}}, x\right ) \]
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 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.26, size = 1107, normalized size = 6.02 \[ \text {result too large to display} \]
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 \[ \text {Exception raised: RuntimeError} \]
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 \[ \int \frac {\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )\,{\left (d\,c^2\,x^2+d\right )}^{3/2}}{x^4} \,d x \]
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 \[ \int \frac {\left (d \left (c^{2} x^{2} + 1\right )\right )^{\frac {3}{2}} \left (a + b \operatorname {asinh}{\left (c x \right )}\right )}{x^{4}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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