Optimal. Leaf size=196 \[ d^2 x \left (a+b \cosh ^{-1}(c x)\right )+\frac {2}{3} d e x^3 \left (a+b \cosh ^{-1}(c x)\right )+\frac {1}{5} e^2 x^5 \left (a+b \cosh ^{-1}(c x)\right )-\frac {2 b e \left (1-c^2 x^2\right )^2 \left (5 c^2 d+3 e\right )}{45 c^5 \sqrt {c x-1} \sqrt {c x+1}}+\frac {b e^2 \left (1-c^2 x^2\right )^3}{25 c^5 \sqrt {c x-1} \sqrt {c x+1}}+\frac {b \left (1-c^2 x^2\right ) \left (15 c^4 d^2+10 c^2 d e+3 e^2\right )}{15 c^5 \sqrt {c x-1} \sqrt {c x+1}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.20, antiderivative size = 196, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {194, 5705, 12, 520, 1247, 698} \[ d^2 x \left (a+b \cosh ^{-1}(c x)\right )+\frac {2}{3} d e x^3 \left (a+b \cosh ^{-1}(c x)\right )+\frac {1}{5} e^2 x^5 \left (a+b \cosh ^{-1}(c x)\right )+\frac {b \left (1-c^2 x^2\right ) \left (15 c^4 d^2+10 c^2 d e+3 e^2\right )}{15 c^5 \sqrt {c x-1} \sqrt {c x+1}}-\frac {2 b e \left (1-c^2 x^2\right )^2 \left (5 c^2 d+3 e\right )}{45 c^5 \sqrt {c x-1} \sqrt {c x+1}}+\frac {b e^2 \left (1-c^2 x^2\right )^3}{25 c^5 \sqrt {c x-1} \sqrt {c x+1}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 12
Rule 194
Rule 520
Rule 698
Rule 1247
Rule 5705
Rubi steps
\begin {align*} \int \left (d+e x^2\right )^2 \left (a+b \cosh ^{-1}(c x)\right ) \, dx &=d^2 x \left (a+b \cosh ^{-1}(c x)\right )+\frac {2}{3} d e x^3 \left (a+b \cosh ^{-1}(c x)\right )+\frac {1}{5} e^2 x^5 \left (a+b \cosh ^{-1}(c x)\right )-(b c) \int \frac {x \left (15 d^2+10 d e x^2+3 e^2 x^4\right )}{15 \sqrt {-1+c x} \sqrt {1+c x}} \, dx\\ &=d^2 x \left (a+b \cosh ^{-1}(c x)\right )+\frac {2}{3} d e x^3 \left (a+b \cosh ^{-1}(c x)\right )+\frac {1}{5} e^2 x^5 \left (a+b \cosh ^{-1}(c x)\right )-\frac {1}{15} (b c) \int \frac {x \left (15 d^2+10 d e x^2+3 e^2 x^4\right )}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx\\ &=d^2 x \left (a+b \cosh ^{-1}(c x)\right )+\frac {2}{3} d e x^3 \left (a+b \cosh ^{-1}(c x)\right )+\frac {1}{5} e^2 x^5 \left (a+b \cosh ^{-1}(c x)\right )-\frac {\left (b c \sqrt {-1+c^2 x^2}\right ) \int \frac {x \left (15 d^2+10 d e x^2+3 e^2 x^4\right )}{\sqrt {-1+c^2 x^2}} \, dx}{15 \sqrt {-1+c x} \sqrt {1+c x}}\\ &=d^2 x \left (a+b \cosh ^{-1}(c x)\right )+\frac {2}{3} d e x^3 \left (a+b \cosh ^{-1}(c x)\right )+\frac {1}{5} e^2 x^5 \left (a+b \cosh ^{-1}(c x)\right )-\frac {\left (b c \sqrt {-1+c^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {15 d^2+10 d e x+3 e^2 x^2}{\sqrt {-1+c^2 x}} \, dx,x,x^2\right )}{30 \sqrt {-1+c x} \sqrt {1+c x}}\\ &=d^2 x \left (a+b \cosh ^{-1}(c x)\right )+\frac {2}{3} d e x^3 \left (a+b \cosh ^{-1}(c x)\right )+\frac {1}{5} e^2 x^5 \left (a+b \cosh ^{-1}(c x)\right )-\frac {\left (b c \sqrt {-1+c^2 x^2}\right ) \operatorname {Subst}\left (\int \left (\frac {15 c^4 d^2+10 c^2 d e+3 e^2}{c^4 \sqrt {-1+c^2 x}}+\frac {2 e \left (5 c^2 d+3 e\right ) \sqrt {-1+c^2 x}}{c^4}+\frac {3 e^2 \left (-1+c^2 x\right )^{3/2}}{c^4}\right ) \, dx,x,x^2\right )}{30 \sqrt {-1+c x} \sqrt {1+c x}}\\ &=\frac {b \left (15 c^4 d^2+10 c^2 d e+3 e^2\right ) \left (1-c^2 x^2\right )}{15 c^5 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {2 b e \left (5 c^2 d+3 e\right ) \left (1-c^2 x^2\right )^2}{45 c^5 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b e^2 \left (1-c^2 x^2\right )^3}{25 c^5 \sqrt {-1+c x} \sqrt {1+c x}}+d^2 x \left (a+b \cosh ^{-1}(c x)\right )+\frac {2}{3} d e x^3 \left (a+b \cosh ^{-1}(c x)\right )+\frac {1}{5} e^2 x^5 \left (a+b \cosh ^{-1}(c x)\right )\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.21, size = 130, normalized size = 0.66 \[ \frac {1}{225} \left (15 a x \left (15 d^2+10 d e x^2+3 e^2 x^4\right )-\frac {b \sqrt {c x-1} \sqrt {c x+1} \left (c^4 \left (225 d^2+50 d e x^2+9 e^2 x^4\right )+4 c^2 e \left (25 d+3 e x^2\right )+24 e^2\right )}{c^5}+15 b x \cosh ^{-1}(c x) \left (15 d^2+10 d e x^2+3 e^2 x^4\right )\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.45, size = 163, normalized size = 0.83 \[ \frac {45 \, a c^{5} e^{2} x^{5} + 150 \, a c^{5} d e x^{3} + 225 \, a c^{5} d^{2} x + 15 \, {\left (3 \, b c^{5} e^{2} x^{5} + 10 \, b c^{5} d e x^{3} + 15 \, b c^{5} d^{2} x\right )} \log \left (c x + \sqrt {c^{2} x^{2} - 1}\right ) - {\left (9 \, b c^{4} e^{2} x^{4} + 225 \, b c^{4} d^{2} + 100 \, b c^{2} d e + 24 \, b e^{2} + 2 \, {\left (25 \, b c^{4} d e + 6 \, b c^{2} e^{2}\right )} x^{2}\right )} \sqrt {c^{2} x^{2} - 1}}{225 \, c^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: RuntimeError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.01, size = 157, normalized size = 0.80 \[ \frac {\frac {a \left (\frac {1}{5} e^{2} c^{5} x^{5}+\frac {2}{3} c^{5} d e \,x^{3}+x \,c^{5} d^{2}\right )}{c^{4}}+\frac {b \left (\frac {\mathrm {arccosh}\left (c x \right ) e^{2} c^{5} x^{5}}{5}+\frac {2 \,\mathrm {arccosh}\left (c x \right ) c^{5} d e \,x^{3}}{3}+\mathrm {arccosh}\left (c x \right ) c^{5} x \,d^{2}-\frac {\sqrt {c x -1}\, \sqrt {c x +1}\, \left (9 c^{4} e^{2} x^{4}+50 c^{4} d e \,x^{2}+225 d^{2} c^{4}+12 c^{2} e^{2} x^{2}+100 c^{2} d e +24 e^{2}\right )}{225}\right )}{c^{4}}}{c} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.47, size = 180, normalized size = 0.92 \[ \frac {1}{5} \, a e^{2} x^{5} + \frac {2}{3} \, a d e x^{3} + \frac {2}{9} \, {\left (3 \, x^{3} \operatorname {arcosh}\left (c x\right ) - c {\left (\frac {\sqrt {c^{2} x^{2} - 1} x^{2}}{c^{2}} + \frac {2 \, \sqrt {c^{2} x^{2} - 1}}{c^{4}}\right )}\right )} b d e + \frac {1}{75} \, {\left (15 \, x^{5} \operatorname {arcosh}\left (c x\right ) - {\left (\frac {3 \, \sqrt {c^{2} x^{2} - 1} x^{4}}{c^{2}} + \frac {4 \, \sqrt {c^{2} x^{2} - 1} x^{2}}{c^{4}} + \frac {8 \, \sqrt {c^{2} x^{2} - 1}}{c^{6}}\right )} c\right )} b e^{2} + a d^{2} x + \frac {{\left (c x \operatorname {arcosh}\left (c x\right ) - \sqrt {c^{2} x^{2} - 1}\right )} b d^{2}}{c} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )\,{\left (e\,x^2+d\right )}^2 \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 2.22, size = 246, normalized size = 1.26 \[ \begin {cases} a d^{2} x + \frac {2 a d e x^{3}}{3} + \frac {a e^{2} x^{5}}{5} + b d^{2} x \operatorname {acosh}{\left (c x \right )} + \frac {2 b d e x^{3} \operatorname {acosh}{\left (c x \right )}}{3} + \frac {b e^{2} x^{5} \operatorname {acosh}{\left (c x \right )}}{5} - \frac {b d^{2} \sqrt {c^{2} x^{2} - 1}}{c} - \frac {2 b d e x^{2} \sqrt {c^{2} x^{2} - 1}}{9 c} - \frac {b e^{2} x^{4} \sqrt {c^{2} x^{2} - 1}}{25 c} - \frac {4 b d e \sqrt {c^{2} x^{2} - 1}}{9 c^{3}} - \frac {4 b e^{2} x^{2} \sqrt {c^{2} x^{2} - 1}}{75 c^{3}} - \frac {8 b e^{2} \sqrt {c^{2} x^{2} - 1}}{75 c^{5}} & \text {for}\: c \neq 0 \\\left (a + \frac {i \pi b}{2}\right ) \left (d^{2} x + \frac {2 d e x^{3}}{3} + \frac {e^{2} x^{5}}{5}\right ) & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________