Optimal. Leaf size=297 \[ \frac{16 c^4 x \left (a+b \sinh ^{-1}(c x)\right )}{3 d^2 \sqrt{c^2 d x^2+d}}+\frac{8 c^4 x \left (a+b \sinh ^{-1}(c x)\right )}{3 d \left (c^2 d x^2+d\right )^{3/2}}+\frac{2 c^2 \left (a+b \sinh ^{-1}(c x)\right )}{d x \left (c^2 d x^2+d\right )^{3/2}}-\frac{a+b \sinh ^{-1}(c x)}{3 d x^3 \left (c^2 d x^2+d\right )^{3/2}}+\frac{b c^3 \sqrt{c^2 d x^2+d}}{6 d^3 \left (c^2 x^2+1\right )^{3/2}}-\frac{b c \sqrt{c^2 d x^2+d}}{6 d^3 x^2 \sqrt{c^2 x^2+1}}-\frac{8 b c^3 \log (x) \sqrt{c^2 d x^2+d}}{3 d^3 \sqrt{c^2 x^2+1}}-\frac{4 b c^3 \sqrt{c^2 d x^2+d} \log \left (c^2 x^2+1\right )}{3 d^3 \sqrt{c^2 x^2+1}} \]
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Rubi [A] time = 0.372379, antiderivative size = 297, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 7, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.269, Rules used = {5747, 5690, 5687, 260, 261, 266, 44} \[ \frac{16 c^4 x \left (a+b \sinh ^{-1}(c x)\right )}{3 d^2 \sqrt{c^2 d x^2+d}}+\frac{8 c^4 x \left (a+b \sinh ^{-1}(c x)\right )}{3 d \left (c^2 d x^2+d\right )^{3/2}}+\frac{2 c^2 \left (a+b \sinh ^{-1}(c x)\right )}{d x \left (c^2 d x^2+d\right )^{3/2}}-\frac{a+b \sinh ^{-1}(c x)}{3 d x^3 \left (c^2 d x^2+d\right )^{3/2}}+\frac{b c^3}{6 d^2 \sqrt{c^2 x^2+1} \sqrt{c^2 d x^2+d}}-\frac{b c \sqrt{c^2 x^2+1}}{6 d^2 x^2 \sqrt{c^2 d x^2+d}}-\frac{8 b c^3 \sqrt{c^2 x^2+1} \log (x)}{3 d^2 \sqrt{c^2 d x^2+d}}-\frac{4 b c^3 \sqrt{c^2 x^2+1} \log \left (c^2 x^2+1\right )}{3 d^2 \sqrt{c^2 d x^2+d}} \]
Antiderivative was successfully verified.
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Rule 5747
Rule 5690
Rule 5687
Rule 260
Rule 261
Rule 266
Rule 44
Rubi steps
\begin{align*} \int \frac{a+b \sinh ^{-1}(c x)}{x^4 \left (d+c^2 d x^2\right )^{5/2}} \, dx &=-\frac{a+b \sinh ^{-1}(c x)}{3 d x^3 \left (d+c^2 d x^2\right )^{3/2}}-\left (2 c^2\right ) \int \frac{a+b \sinh ^{-1}(c x)}{x^2 \left (d+c^2 d x^2\right )^{5/2}} \, dx+\frac{\left (b c \sqrt{1+c^2 x^2}\right ) \int \frac{1}{x^3 \left (1+c^2 x^2\right )^2} \, dx}{3 d^2 \sqrt{d+c^2 d x^2}}\\ &=-\frac{a+b \sinh ^{-1}(c x)}{3 d x^3 \left (d+c^2 d x^2\right )^{3/2}}+\frac{2 c^2 \left (a+b \sinh ^{-1}(c x)\right )}{d x \left (d+c^2 d x^2\right )^{3/2}}+\left (8 c^4\right ) \int \frac{a+b \sinh ^{-1}(c x)}{\left (d+c^2 d x^2\right )^{5/2}} \, 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 )^2} \, dx,x,x^2\right )}{6 d^2 \sqrt{d+c^2 d x^2}}-\frac{\left (2 b c^3 \sqrt{1+c^2 x^2}\right ) \int \frac{1}{x \left (1+c^2 x^2\right )^2} \, dx}{d^2 \sqrt{d+c^2 d x^2}}\\ &=-\frac{a+b \sinh ^{-1}(c x)}{3 d x^3 \left (d+c^2 d x^2\right )^{3/2}}+\frac{2 c^2 \left (a+b \sinh ^{-1}(c x)\right )}{d x \left (d+c^2 d x^2\right )^{3/2}}+\frac{8 c^4 x \left (a+b \sinh ^{-1}(c x)\right )}{3 d \left (d+c^2 d x^2\right )^{3/2}}+\frac{\left (16 c^4\right ) \int \frac{a+b \sinh ^{-1}(c x)}{\left (d+c^2 d x^2\right )^{3/2}} \, dx}{3 d}+\frac{\left (b c \sqrt{1+c^2 x^2}\right ) \operatorname{Subst}\left (\int \left (\frac{1}{x^2}-\frac{2 c^2}{x}+\frac{c^4}{\left (1+c^2 x\right )^2}+\frac{2 c^4}{1+c^2 x}\right ) \, dx,x,x^2\right )}{6 d^2 \sqrt{d+c^2 d x^2}}-\frac{\left (b c^3 \sqrt{1+c^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{1}{x \left (1+c^2 x\right )^2} \, dx,x,x^2\right )}{d^2 \sqrt{d+c^2 d x^2}}-\frac{\left (8 b c^5 \sqrt{1+c^2 x^2}\right ) \int \frac{x}{\left (1+c^2 x^2\right )^2} \, dx}{3 d^2 \sqrt{d+c^2 d x^2}}\\ &=\frac{7 b c^3}{6 d^2 \sqrt{1+c^2 x^2} \sqrt{d+c^2 d x^2}}-\frac{b c \sqrt{1+c^2 x^2}}{6 d^2 x^2 \sqrt{d+c^2 d x^2}}-\frac{a+b \sinh ^{-1}(c x)}{3 d x^3 \left (d+c^2 d x^2\right )^{3/2}}+\frac{2 c^2 \left (a+b \sinh ^{-1}(c x)\right )}{d x \left (d+c^2 d x^2\right )^{3/2}}+\frac{8 c^4 x \left (a+b \sinh ^{-1}(c x)\right )}{3 d \left (d+c^2 d x^2\right )^{3/2}}+\frac{16 c^4 x \left (a+b \sinh ^{-1}(c x)\right )}{3 d^2 \sqrt{d+c^2 d x^2}}-\frac{2 b c^3 \sqrt{1+c^2 x^2} \log (x)}{3 d^2 \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 )}{3 d^2 \sqrt{d+c^2 d x^2}}-\frac{\left (b c^3 \sqrt{1+c^2 x^2}\right ) \operatorname{Subst}\left (\int \left (\frac{1}{x}-\frac{c^2}{\left (1+c^2 x\right )^2}-\frac{c^2}{1+c^2 x}\right ) \, dx,x,x^2\right )}{d^2 \sqrt{d+c^2 d x^2}}-\frac{\left (16 b c^5 \sqrt{1+c^2 x^2}\right ) \int \frac{x}{1+c^2 x^2} \, dx}{3 d^2 \sqrt{d+c^2 d x^2}}\\ &=\frac{b c^3}{6 d^2 \sqrt{1+c^2 x^2} \sqrt{d+c^2 d x^2}}-\frac{b c \sqrt{1+c^2 x^2}}{6 d^2 x^2 \sqrt{d+c^2 d x^2}}-\frac{a+b \sinh ^{-1}(c x)}{3 d x^3 \left (d+c^2 d x^2\right )^{3/2}}+\frac{2 c^2 \left (a+b \sinh ^{-1}(c x)\right )}{d x \left (d+c^2 d x^2\right )^{3/2}}+\frac{8 c^4 x \left (a+b \sinh ^{-1}(c x)\right )}{3 d \left (d+c^2 d x^2\right )^{3/2}}+\frac{16 c^4 x \left (a+b \sinh ^{-1}(c x)\right )}{3 d^2 \sqrt{d+c^2 d x^2}}-\frac{8 b c^3 \sqrt{1+c^2 x^2} \log (x)}{3 d^2 \sqrt{d+c^2 d x^2}}-\frac{4 b c^3 \sqrt{1+c^2 x^2} \log \left (1+c^2 x^2\right )}{3 d^2 \sqrt{d+c^2 d x^2}}\\ \end{align*}
Mathematica [A] time = 0.356522, size = 267, normalized size = 0.9 \[ \frac{\sqrt{c^2 d x^2+d} \left (32 a c^6 x^6 \sqrt{c^2 x^2+1}+48 a c^4 x^4 \sqrt{c^2 x^2+1}+12 a c^2 x^2 \sqrt{c^2 x^2+1}-2 a \sqrt{c^2 x^2+1}-b c^3 x^3-16 b c^7 x^7 \log \left (c^2 x^2+1\right )-32 b c^5 x^5 \log \left (c^2 x^2+1\right )+8 b c^3 x^3 \left (c^2 x^2+1\right )^2 \log \left (\frac{1}{c^2 x^2}+1\right )-16 b c^3 x^3 \log \left (c^2 x^2+1\right )+2 b \sqrt{c^2 x^2+1} \left (16 c^6 x^6+24 c^4 x^4+6 c^2 x^2-1\right ) \sinh ^{-1}(c x)-b c x\right )}{6 d^3 x^3 \left (c^2 x^2+1\right )^{5/2}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.175, size = 1790, normalized size = 6. \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^{6} d^{3} x^{10} + 3 \, c^{4} d^{3} x^{8} + 3 \, c^{2} d^{3} x^{6} + d^{3} 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{5}{2}} x^{4}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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