Optimal. Leaf size=59 \[ -\frac {a+b \sinh ^{-1}(c+d x)}{2 d e^3 (c+d x)^2}-\frac {b \sqrt {(c+d x)^2+1}}{2 d e^3 (c+d x)} \]
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Rubi [A] time = 0.05, antiderivative size = 59, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {5865, 12, 5661, 264} \[ -\frac {a+b \sinh ^{-1}(c+d x)}{2 d e^3 (c+d x)^2}-\frac {b \sqrt {(c+d x)^2+1}}{2 d e^3 (c+d x)} \]
Antiderivative was successfully verified.
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Rule 12
Rule 264
Rule 5661
Rule 5865
Rubi steps
\begin {align*} \int \frac {a+b \sinh ^{-1}(c+d x)}{(c e+d e x)^3} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {a+b \sinh ^{-1}(x)}{e^3 x^3} \, dx,x,c+d x\right )}{d}\\ &=\frac {\operatorname {Subst}\left (\int \frac {a+b \sinh ^{-1}(x)}{x^3} \, dx,x,c+d x\right )}{d e^3}\\ &=-\frac {a+b \sinh ^{-1}(c+d x)}{2 d e^3 (c+d x)^2}+\frac {b \operatorname {Subst}\left (\int \frac {1}{x^2 \sqrt {1+x^2}} \, dx,x,c+d x\right )}{2 d e^3}\\ &=-\frac {b \sqrt {1+(c+d x)^2}}{2 d e^3 (c+d x)}-\frac {a+b \sinh ^{-1}(c+d x)}{2 d e^3 (c+d x)^2}\\ \end {align*}
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Mathematica [A] time = 0.06, size = 57, normalized size = 0.97 \[ \frac {\frac {-a-b \sinh ^{-1}(c+d x)}{2 (c+d x)^2}-\frac {b \sqrt {(c+d x)^2+1}}{2 (c+d x)}}{d e^3} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.55, size = 118, normalized size = 2.00 \[ \frac {a d^{2} x^{2} + 2 \, a c d x - b c^{2} \log \left (d x + c + \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} + 1}\right ) - {\left (b c^{2} d x + b c^{3}\right )} \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} + 1}}{2 \, {\left (c^{2} d^{3} e^{3} x^{2} + 2 \, c^{3} d^{2} e^{3} x + c^{4} d e^{3}\right )}} \]
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 \[ \int \frac {b \operatorname {arsinh}\left (d x + c\right ) + a}{{\left (d e x + c e\right )}^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 60, normalized size = 1.02 \[ \frac {-\frac {a}{2 e^{3} \left (d x +c \right )^{2}}+\frac {b \left (-\frac {\arcsinh \left (d x +c \right )}{2 \left (d x +c \right )^{2}}-\frac {\sqrt {1+\left (d x +c \right )^{2}}}{2 \left (d x +c \right )}\right )}{e^{3}}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.41, size = 117, normalized size = 1.98 \[ -\frac {1}{2} \, b {\left (\frac {\sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} + 1} d}{d^{3} e^{3} x + c d^{2} e^{3}} + \frac {\operatorname {arsinh}\left (d x + c\right )}{d^{3} e^{3} x^{2} + 2 \, c d^{2} e^{3} x + c^{2} d e^{3}}\right )} - \frac {a}{2 \, {\left (d^{3} e^{3} x^{2} + 2 \, c d^{2} e^{3} x + c^{2} d e^{3}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {a+b\,\mathrm {asinh}\left (c+d\,x\right )}{{\left (c\,e+d\,e\,x\right )}^3} \,d x \]
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 \[ \frac {\int \frac {a}{c^{3} + 3 c^{2} d x + 3 c d^{2} x^{2} + d^{3} x^{3}}\, dx + \int \frac {b \operatorname {asinh}{\left (c + d x \right )}}{c^{3} + 3 c^{2} d x + 3 c d^{2} x^{2} + d^{3} x^{3}}\, dx}{e^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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