Optimal. Leaf size=228 \[ \frac{8 c^4 x \left (a+b \sinh ^{-1}(c x)\right )}{3 d \sqrt{c^2 d x^2+d}}+\frac{4 c^2 \left (a+b \sinh ^{-1}(c x)\right )}{3 d x \sqrt{c^2 d x^2+d}}-\frac{a+b \sinh ^{-1}(c x)}{3 d x^3 \sqrt{c^2 d x^2+d}}-\frac{b c \sqrt{c^2 d x^2+d}}{6 d^2 x^2 \sqrt{c^2 x^2+1}}-\frac{5 b c^3 \log (x) \sqrt{c^2 d x^2+d}}{3 d^2 \sqrt{c^2 x^2+1}}-\frac{b c^3 \sqrt{c^2 d x^2+d} \log \left (c^2 x^2+1\right )}{2 d^2 \sqrt{c^2 x^2+1}} \]
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Rubi [A] time = 0.288848, antiderivative size = 228, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 8, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.308, Rules used = {5747, 5687, 260, 266, 36, 29, 31, 44} \[ \frac{8 c^4 x \left (a+b \sinh ^{-1}(c x)\right )}{3 d \sqrt{c^2 d x^2+d}}+\frac{4 c^2 \left (a+b \sinh ^{-1}(c x)\right )}{3 d x \sqrt{c^2 d x^2+d}}-\frac{a+b \sinh ^{-1}(c x)}{3 d x^3 \sqrt{c^2 d x^2+d}}-\frac{b c \sqrt{c^2 x^2+1}}{6 d x^2 \sqrt{c^2 d x^2+d}}-\frac{5 b c^3 \sqrt{c^2 x^2+1} \log (x)}{3 d \sqrt{c^2 d x^2+d}}-\frac{b c^3 \sqrt{c^2 x^2+1} \log \left (c^2 x^2+1\right )}{2 d \sqrt{c^2 d x^2+d}} \]
Antiderivative was successfully verified.
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Rule 5747
Rule 5687
Rule 260
Rule 266
Rule 36
Rule 29
Rule 31
Rule 44
Rubi steps
\begin{align*} \int \frac{a+b \sinh ^{-1}(c x)}{x^4 \left (d+c^2 d x^2\right )^{3/2}} \, dx &=-\frac{a+b \sinh ^{-1}(c x)}{3 d x^3 \sqrt{d+c^2 d x^2}}-\frac{1}{3} \left (4 c^2\right ) \int \frac{a+b \sinh ^{-1}(c x)}{x^2 \left (d+c^2 d x^2\right )^{3/2}} \, dx+\frac{\left (b c \sqrt{1+c^2 x^2}\right ) \int \frac{1}{x^3 \left (1+c^2 x^2\right )} \, dx}{3 d \sqrt{d+c^2 d x^2}}\\ &=-\frac{a+b \sinh ^{-1}(c x)}{3 d x^3 \sqrt{d+c^2 d x^2}}+\frac{4 c^2 \left (a+b \sinh ^{-1}(c x)\right )}{3 d x \sqrt{d+c^2 d x^2}}+\frac{1}{3} \left (8 c^4\right ) \int \frac{a+b \sinh ^{-1}(c x)}{\left (d+c^2 d x^2\right )^{3/2}} \, dx+\frac{\left (b c \sqrt{1+c^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{1}{x^2 \left (1+c^2 x\right )} \, dx,x,x^2\right )}{6 d \sqrt{d+c^2 d x^2}}-\frac{\left (4 b c^3 \sqrt{1+c^2 x^2}\right ) \int \frac{1}{x \left (1+c^2 x^2\right )} \, dx}{3 d \sqrt{d+c^2 d x^2}}\\ &=-\frac{a+b \sinh ^{-1}(c x)}{3 d x^3 \sqrt{d+c^2 d x^2}}+\frac{4 c^2 \left (a+b \sinh ^{-1}(c x)\right )}{3 d x \sqrt{d+c^2 d x^2}}+\frac{8 c^4 x \left (a+b \sinh ^{-1}(c x)\right )}{3 d \sqrt{d+c^2 d x^2}}+\frac{\left (b c \sqrt{1+c^2 x^2}\right ) \operatorname{Subst}\left (\int \left (\frac{1}{x^2}-\frac{c^2}{x}+\frac{c^4}{1+c^2 x}\right ) \, dx,x,x^2\right )}{6 d \sqrt{d+c^2 d x^2}}-\frac{\left (2 b c^3 \sqrt{1+c^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{1}{x \left (1+c^2 x\right )} \, dx,x,x^2\right )}{3 d \sqrt{d+c^2 d x^2}}-\frac{\left (8 b c^5 \sqrt{1+c^2 x^2}\right ) \int \frac{x}{1+c^2 x^2} \, dx}{3 d \sqrt{d+c^2 d x^2}}\\ &=-\frac{b c \sqrt{1+c^2 x^2}}{6 d x^2 \sqrt{d+c^2 d x^2}}-\frac{a+b \sinh ^{-1}(c x)}{3 d x^3 \sqrt{d+c^2 d x^2}}+\frac{4 c^2 \left (a+b \sinh ^{-1}(c x)\right )}{3 d x \sqrt{d+c^2 d x^2}}+\frac{8 c^4 x \left (a+b \sinh ^{-1}(c x)\right )}{3 d \sqrt{d+c^2 d x^2}}-\frac{b c^3 \sqrt{1+c^2 x^2} \log (x)}{3 d \sqrt{d+c^2 d x^2}}-\frac{7 b c^3 \sqrt{1+c^2 x^2} \log \left (1+c^2 x^2\right )}{6 d \sqrt{d+c^2 d x^2}}-\frac{\left (2 b c^3 \sqrt{1+c^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{1}{x} \, dx,x,x^2\right )}{3 d \sqrt{d+c^2 d x^2}}+\frac{\left (2 b c^5 \sqrt{1+c^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{1}{1+c^2 x} \, dx,x,x^2\right )}{3 d \sqrt{d+c^2 d x^2}}\\ &=-\frac{b c \sqrt{1+c^2 x^2}}{6 d x^2 \sqrt{d+c^2 d x^2}}-\frac{a+b \sinh ^{-1}(c x)}{3 d x^3 \sqrt{d+c^2 d x^2}}+\frac{4 c^2 \left (a+b \sinh ^{-1}(c x)\right )}{3 d x \sqrt{d+c^2 d x^2}}+\frac{8 c^4 x \left (a+b \sinh ^{-1}(c x)\right )}{3 d \sqrt{d+c^2 d x^2}}-\frac{5 b c^3 \sqrt{1+c^2 x^2} \log (x)}{3 d \sqrt{d+c^2 d x^2}}-\frac{b c^3 \sqrt{1+c^2 x^2} \log \left (1+c^2 x^2\right )}{2 d \sqrt{d+c^2 d x^2}}\\ \end{align*}
Mathematica [A] time = 0.299031, size = 216, normalized size = 0.95 \[ \frac{\sqrt{c^2 d x^2+d} \left (16 a c^4 x^4 \sqrt{c^2 x^2+1}+8 a c^2 x^2 \sqrt{c^2 x^2+1}-2 a \sqrt{c^2 x^2+1}-b c^3 x^3-8 b c^5 x^5 \log \left (c^2 x^2+1\right )+5 b c^3 x^3 \left (c^2 x^2+1\right ) \log \left (\frac{1}{c^2 x^2}+1\right )-8 b c^3 x^3 \log \left (c^2 x^2+1\right )+2 b \sqrt{c^2 x^2+1} \left (8 c^4 x^4+4 c^2 x^2-1\right ) \sinh ^{-1}(c x)-b c x\right )}{6 d^2 x^3 \left (c^2 x^2+1\right )^{3/2}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.171, size = 965, normalized size = 4.2 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\sqrt{c^{2} d x^{2} + d}{\left (b \operatorname{arsinh}\left (c x\right ) + a\right )}}{c^{4} d^{2} x^{8} + 2 \, c^{2} d^{2} x^{6} + d^{2} x^{4}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{b \operatorname{arsinh}\left (c x\right ) + a}{{\left (c^{2} d x^{2} + d\right )}^{\frac{3}{2}} x^{4}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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