Optimal. Leaf size=85 \[ -\frac {b \sqrt {(c+d x)^2+1} \left (a+b \sinh ^{-1}(c+d x)\right )}{d e^3 (c+d x)}-\frac {\left (a+b \sinh ^{-1}(c+d x)\right )^2}{2 d e^3 (c+d x)^2}+\frac {b^2 \log (c+d x)}{d e^3} \]
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Rubi [A] time = 0.14, antiderivative size = 85, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {5865, 12, 5661, 5723, 29} \[ -\frac {b \sqrt {(c+d x)^2+1} \left (a+b \sinh ^{-1}(c+d x)\right )}{d e^3 (c+d x)}-\frac {\left (a+b \sinh ^{-1}(c+d x)\right )^2}{2 d e^3 (c+d x)^2}+\frac {b^2 \log (c+d x)}{d e^3} \]
Antiderivative was successfully verified.
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Rule 12
Rule 29
Rule 5661
Rule 5723
Rule 5865
Rubi steps
\begin {align*} \int \frac {\left (a+b \sinh ^{-1}(c+d x)\right )^2}{(c e+d e x)^3} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {\left (a+b \sinh ^{-1}(x)\right )^2}{e^3 x^3} \, dx,x,c+d x\right )}{d}\\ &=\frac {\operatorname {Subst}\left (\int \frac {\left (a+b \sinh ^{-1}(x)\right )^2}{x^3} \, dx,x,c+d x\right )}{d e^3}\\ &=-\frac {\left (a+b \sinh ^{-1}(c+d x)\right )^2}{2 d e^3 (c+d x)^2}+\frac {b \operatorname {Subst}\left (\int \frac {a+b \sinh ^{-1}(x)}{x^2 \sqrt {1+x^2}} \, dx,x,c+d x\right )}{d e^3}\\ &=-\frac {b \sqrt {1+(c+d x)^2} \left (a+b \sinh ^{-1}(c+d x)\right )}{d e^3 (c+d x)}-\frac {\left (a+b \sinh ^{-1}(c+d x)\right )^2}{2 d e^3 (c+d x)^2}+\frac {b^2 \operatorname {Subst}\left (\int \frac {1}{x} \, dx,x,c+d x\right )}{d e^3}\\ &=-\frac {b \sqrt {1+(c+d x)^2} \left (a+b \sinh ^{-1}(c+d x)\right )}{d e^3 (c+d x)}-\frac {\left (a+b \sinh ^{-1}(c+d x)\right )^2}{2 d e^3 (c+d x)^2}+\frac {b^2 \log (c+d x)}{d e^3}\\ \end {align*}
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Mathematica [A] time = 0.25, size = 120, normalized size = 1.41 \[ -\frac {a \left (a+2 b (c+d x) \sqrt {c^2+2 c d x+d^2 x^2+1}\right )+2 b \sinh ^{-1}(c+d x) \left (a+b (c+d x) \sqrt {c^2+2 c d x+d^2 x^2+1}\right )-2 b^2 (c+d x)^2 \log (c+d x)+b^2 \sinh ^{-1}(c+d x)^2}{2 d e^3 (c+d x)^2} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.66, size = 319, normalized size = 3.75 \[ -\frac {2 \, a b c^{2} d^{2} x^{2} + 4 \, a b c^{3} d x + 2 \, a b c^{4} + b^{2} c^{2} \log \left (d x + c + \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} + 1}\right )^{2} + a^{2} c^{2} - 2 \, {\left (a b d^{2} x^{2} + 2 \, a b c d x - {\left (b^{2} c^{2} d x + b^{2} c^{3}\right )} \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} + 1}\right )} \log \left (d x + c + \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} + 1}\right ) - 2 \, {\left (b^{2} c^{2} d^{2} x^{2} + 2 \, b^{2} c^{3} d x + b^{2} c^{4}\right )} \log \left (d x + c\right ) - 2 \, {\left (a b d^{2} x^{2} + 2 \, a b c d x + a b c^{2}\right )} \log \left (-d x - c + \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} + 1}\right ) + 2 \, {\left (a b c^{2} d x + a b c^{3}\right )} \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} + 1}}{2 \, {\left (c^{2} d^{3} e^{3} x^{2} + 2 \, c^{3} d^{2} e^{3} x + c^{4} d e^{3}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (b \operatorname {arsinh}\left (d x + c\right ) + a\right )}^{2}}{{\left (d e x + c e\right )}^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.20, size = 180, normalized size = 2.12 \[ -\frac {a^{2}}{2 d \,e^{3} \left (d x +c \right )^{2}}-\frac {b^{2} \arcsinh \left (d x +c \right )}{d \,e^{3}}-\frac {b^{2} \arcsinh \left (d x +c \right ) \sqrt {1+\left (d x +c \right )^{2}}}{d \,e^{3} \left (d x +c \right )}-\frac {b^{2} \arcsinh \left (d x +c \right )^{2}}{2 d \,e^{3} \left (d x +c \right )^{2}}+\frac {b^{2} \ln \left (\left (d x +c +\sqrt {1+\left (d x +c \right )^{2}}\right )^{2}-1\right )}{d \,e^{3}}-\frac {a b \arcsinh \left (d x +c \right )}{d \,e^{3} \left (d x +c \right )^{2}}-\frac {a b \sqrt {1+\left (d x +c \right )^{2}}}{d \,e^{3} \left (d x +c \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.42, size = 230, normalized size = 2.71 \[ -{\left (\frac {\sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} + 1} d \operatorname {arsinh}\left (d x + c\right )}{d^{3} e^{3} x + c d^{2} e^{3}} - \frac {\log \left (d x + c\right )}{d e^{3}}\right )} b^{2} - a b {\left (\frac {\sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} + 1} d}{d^{3} e^{3} x + c d^{2} e^{3}} + \frac {\operatorname {arsinh}\left (d x + c\right )}{d^{3} e^{3} x^{2} + 2 \, c d^{2} e^{3} x + c^{2} d e^{3}}\right )} - \frac {b^{2} \operatorname {arsinh}\left (d x + c\right )^{2}}{2 \, {\left (d^{3} e^{3} x^{2} + 2 \, c d^{2} e^{3} x + c^{2} d e^{3}\right )}} - \frac {a^{2}}{2 \, {\left (d^{3} e^{3} x^{2} + 2 \, c d^{2} e^{3} x + c^{2} d e^{3}\right )}} \]
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 \[ \int \frac {{\left (a+b\,\mathrm {asinh}\left (c+d\,x\right )\right )}^2}{{\left (c\,e+d\,e\,x\right )}^3} \,d x \]
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 \[ \frac {\int \frac {a^{2}}{c^{3} + 3 c^{2} d x + 3 c d^{2} x^{2} + d^{3} x^{3}}\, dx + \int \frac {b^{2} \operatorname {asinh}^{2}{\left (c + d x \right )}}{c^{3} + 3 c^{2} d x + 3 c d^{2} x^{2} + d^{3} x^{3}}\, dx + \int \frac {2 a b \operatorname {asinh}{\left (c + d x \right )}}{c^{3} + 3 c^{2} d x + 3 c d^{2} x^{2} + d^{3} x^{3}}\, dx}{e^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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