Optimal. Leaf size=81 \[ -\frac {b \left (3 c^2 d-e\right ) \sqrt {1+c^2 x^2}}{3 c^3}-\frac {b e \left (1+c^2 x^2\right )^{3/2}}{9 c^3}+d x \left (a+b \sinh ^{-1}(c x)\right )+\frac {1}{3} e x^3 \left (a+b \sinh ^{-1}(c x)\right ) \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.05, antiderivative size = 81, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 3, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {5792, 455, 45}
\begin {gather*} d x \left (a+b \sinh ^{-1}(c x)\right )+\frac {1}{3} e x^3 \left (a+b \sinh ^{-1}(c x)\right )-\frac {b \sqrt {c^2 x^2+1} \left (3 c^2 d-e\right )}{3 c^3}-\frac {b e \left (c^2 x^2+1\right )^{3/2}}{9 c^3} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 45
Rule 455
Rule 5792
Rubi steps
\begin {align*} \int \left (d+e x^2\right ) \left (a+b \sinh ^{-1}(c x)\right ) \, dx &=d x \left (a+b \sinh ^{-1}(c x)\right )+\frac {1}{3} e x^3 \left (a+b \sinh ^{-1}(c x)\right )-(b c) \int \frac {x \left (d+\frac {e x^2}{3}\right )}{\sqrt {1+c^2 x^2}} \, dx\\ &=d x \left (a+b \sinh ^{-1}(c x)\right )+\frac {1}{3} e x^3 \left (a+b \sinh ^{-1}(c x)\right )-\frac {1}{2} (b c) \text {Subst}\left (\int \frac {d+\frac {e x}{3}}{\sqrt {1+c^2 x}} \, dx,x,x^2\right )\\ &=d x \left (a+b \sinh ^{-1}(c x)\right )+\frac {1}{3} e x^3 \left (a+b \sinh ^{-1}(c x)\right )-\frac {1}{2} (b c) \text {Subst}\left (\int \left (\frac {3 c^2 d-e}{3 c^2 \sqrt {1+c^2 x}}+\frac {e \sqrt {1+c^2 x}}{3 c^2}\right ) \, dx,x,x^2\right )\\ &=-\frac {b \left (3 c^2 d-e\right ) \sqrt {1+c^2 x^2}}{3 c^3}-\frac {b e \left (1+c^2 x^2\right )^{3/2}}{9 c^3}+d x \left (a+b \sinh ^{-1}(c x)\right )+\frac {1}{3} e x^3 \left (a+b \sinh ^{-1}(c x)\right )\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.05, size = 71, normalized size = 0.88 \begin {gather*} \frac {1}{9} \left (3 a x \left (3 d+e x^2\right )-\frac {b \sqrt {1+c^2 x^2} \left (-2 e+c^2 \left (9 d+e x^2\right )\right )}{c^3}+3 b x \left (3 d+e x^2\right ) \sinh ^{-1}(c x)\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.60, size = 109, normalized size = 1.35
method | result | size |
derivativedivides | \(\frac {\frac {a \left (d \,c^{3} x +\frac {1}{3} e \,c^{3} x^{3}\right )}{c^{2}}+\frac {b \left (\arcsinh \left (c x \right ) d \,c^{3} x +\frac {\arcsinh \left (c x \right ) e \,c^{3} x^{3}}{3}-d \,c^{2} \sqrt {c^{2} x^{2}+1}-\frac {e \left (\frac {c^{2} x^{2} \sqrt {c^{2} x^{2}+1}}{3}-\frac {2 \sqrt {c^{2} x^{2}+1}}{3}\right )}{3}\right )}{c^{2}}}{c}\) | \(109\) |
default | \(\frac {\frac {a \left (d \,c^{3} x +\frac {1}{3} e \,c^{3} x^{3}\right )}{c^{2}}+\frac {b \left (\arcsinh \left (c x \right ) d \,c^{3} x +\frac {\arcsinh \left (c x \right ) e \,c^{3} x^{3}}{3}-d \,c^{2} \sqrt {c^{2} x^{2}+1}-\frac {e \left (\frac {c^{2} x^{2} \sqrt {c^{2} x^{2}+1}}{3}-\frac {2 \sqrt {c^{2} x^{2}+1}}{3}\right )}{3}\right )}{c^{2}}}{c}\) | \(109\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.26, size = 93, normalized size = 1.15 \begin {gather*} \frac {1}{3} \, a x^{3} e + a d x + \frac {1}{9} \, {\left (3 \, x^{3} \operatorname {arsinh}\left (c x\right ) - c {\left (\frac {\sqrt {c^{2} x^{2} + 1} x^{2}}{c^{2}} - \frac {2 \, \sqrt {c^{2} x^{2} + 1}}{c^{4}}\right )}\right )} b e + \frac {{\left (c x \operatorname {arsinh}\left (c x\right ) - \sqrt {c^{2} x^{2} + 1}\right )} b d}{c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 0.35, size = 134, normalized size = 1.65 \begin {gather*} \frac {3 \, a c^{3} x^{3} \cosh \left (1\right ) + 3 \, a c^{3} x^{3} \sinh \left (1\right ) + 9 \, a c^{3} d x + 3 \, {\left (b c^{3} x^{3} \cosh \left (1\right ) + b c^{3} x^{3} \sinh \left (1\right ) + 3 \, b c^{3} d x\right )} \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right ) - {\left (9 \, b c^{2} d + {\left (b c^{2} x^{2} - 2 \, b\right )} \cosh \left (1\right ) + {\left (b c^{2} x^{2} - 2 \, b\right )} \sinh \left (1\right )\right )} \sqrt {c^{2} x^{2} + 1}}{9 \, c^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A]
time = 0.15, size = 109, normalized size = 1.35 \begin {gather*} \begin {cases} a d x + \frac {a e x^{3}}{3} + b d x \operatorname {asinh}{\left (c x \right )} + \frac {b e x^{3} \operatorname {asinh}{\left (c x \right )}}{3} - \frac {b d \sqrt {c^{2} x^{2} + 1}}{c} - \frac {b e x^{2} \sqrt {c^{2} x^{2} + 1}}{9 c} + \frac {2 b e \sqrt {c^{2} x^{2} + 1}}{9 c^{3}} & \text {for}\: c \neq 0 \\a \left (d x + \frac {e x^{3}}{3}\right ) & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: RuntimeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )\,\left (e\,x^2+d\right ) \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________