Optimal. Leaf size=192 \[ \frac {x^3 \sqrt {c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )}{4 c^2 d}+\frac {3 \sqrt {c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )^2}{16 b c^5 \sqrt {c^2 d x^2+d}}-\frac {3 x \sqrt {c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )}{8 c^4 d}-\frac {b x^4 \sqrt {c^2 x^2+1}}{16 c \sqrt {c^2 d x^2+d}}+\frac {3 b x^2 \sqrt {c^2 x^2+1}}{16 c^3 \sqrt {c^2 d x^2+d}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.25, antiderivative size = 192, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {5758, 5677, 5675, 30} \[ \frac {x^3 \sqrt {c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )}{4 c^2 d}-\frac {3 x \sqrt {c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )}{8 c^4 d}+\frac {3 \sqrt {c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )^2}{16 b c^5 \sqrt {c^2 d x^2+d}}-\frac {b x^4 \sqrt {c^2 x^2+1}}{16 c \sqrt {c^2 d x^2+d}}+\frac {3 b x^2 \sqrt {c^2 x^2+1}}{16 c^3 \sqrt {c^2 d x^2+d}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 30
Rule 5675
Rule 5677
Rule 5758
Rubi steps
\begin {align*} \int \frac {x^4 \left (a+b \sinh ^{-1}(c x)\right )}{\sqrt {d+c^2 d x^2}} \, dx &=\frac {x^3 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{4 c^2 d}-\frac {3 \int \frac {x^2 \left (a+b \sinh ^{-1}(c x)\right )}{\sqrt {d+c^2 d x^2}} \, dx}{4 c^2}-\frac {\left (b \sqrt {1+c^2 x^2}\right ) \int x^3 \, dx}{4 c \sqrt {d+c^2 d x^2}}\\ &=-\frac {b x^4 \sqrt {1+c^2 x^2}}{16 c \sqrt {d+c^2 d x^2}}-\frac {3 x \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{8 c^4 d}+\frac {x^3 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{4 c^2 d}+\frac {3 \int \frac {a+b \sinh ^{-1}(c x)}{\sqrt {d+c^2 d x^2}} \, dx}{8 c^4}+\frac {\left (3 b \sqrt {1+c^2 x^2}\right ) \int x \, dx}{8 c^3 \sqrt {d+c^2 d x^2}}\\ &=\frac {3 b x^2 \sqrt {1+c^2 x^2}}{16 c^3 \sqrt {d+c^2 d x^2}}-\frac {b x^4 \sqrt {1+c^2 x^2}}{16 c \sqrt {d+c^2 d x^2}}-\frac {3 x \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{8 c^4 d}+\frac {x^3 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{4 c^2 d}+\frac {\left (3 \sqrt {1+c^2 x^2}\right ) \int \frac {a+b \sinh ^{-1}(c x)}{\sqrt {1+c^2 x^2}} \, dx}{8 c^4 \sqrt {d+c^2 d x^2}}\\ &=\frac {3 b x^2 \sqrt {1+c^2 x^2}}{16 c^3 \sqrt {d+c^2 d x^2}}-\frac {b x^4 \sqrt {1+c^2 x^2}}{16 c \sqrt {d+c^2 d x^2}}-\frac {3 x \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{8 c^4 d}+\frac {x^3 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{4 c^2 d}+\frac {3 \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{16 b c^5 \sqrt {d+c^2 d x^2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.62, size = 151, normalized size = 0.79 \[ \frac {\frac {16 a c x \left (2 c^2 x^2-3\right ) \sqrt {c^2 d x^2+d}}{d}+\frac {48 a \log \left (\sqrt {d} \sqrt {c^2 d x^2+d}+c d x\right )}{\sqrt {d}}+\frac {b \sqrt {c^2 x^2+1} \left (4 \sinh ^{-1}(c x) \left (6 \sinh ^{-1}(c x)-8 \sinh \left (2 \sinh ^{-1}(c x)\right )+\sinh \left (4 \sinh ^{-1}(c x)\right )\right )+16 \cosh \left (2 \sinh ^{-1}(c x)\right )-\cosh \left (4 \sinh ^{-1}(c x)\right )\right )}{\sqrt {c^2 d x^2+d}}}{128 c^5} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 0.60, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {b x^{4} \operatorname {arsinh}\left (c x\right ) + a x^{4}}{\sqrt {c^{2} d x^{2} + d}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (b \operatorname {arsinh}\left (c x\right ) + a\right )} x^{4}}{\sqrt {c^{2} d x^{2} + d}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [B] time = 0.34, size = 347, normalized size = 1.81 \[ \frac {a \,x^{3} \sqrt {c^{2} d \,x^{2}+d}}{4 c^{2} d}-\frac {3 a x \sqrt {c^{2} d \,x^{2}+d}}{8 c^{4} d}+\frac {3 a \ln \left (\frac {x \,c^{2} d}{\sqrt {c^{2} d}}+\sqrt {c^{2} d \,x^{2}+d}\right )}{8 c^{4} \sqrt {c^{2} d}}+\frac {b \sqrt {d \left (c^{2} x^{2}+1\right )}\, \arcsinh \left (c x \right ) x^{5}}{4 d \left (c^{2} x^{2}+1\right )}-\frac {b \sqrt {d \left (c^{2} x^{2}+1\right )}\, x^{4}}{16 c d \sqrt {c^{2} x^{2}+1}}-\frac {b \sqrt {d \left (c^{2} x^{2}+1\right )}\, \arcsinh \left (c x \right ) x^{3}}{8 c^{2} d \left (c^{2} x^{2}+1\right )}+\frac {3 b \sqrt {d \left (c^{2} x^{2}+1\right )}\, x^{2}}{16 c^{3} d \sqrt {c^{2} x^{2}+1}}-\frac {3 b \sqrt {d \left (c^{2} x^{2}+1\right )}\, \arcsinh \left (c x \right ) x}{8 c^{4} d \left (c^{2} x^{2}+1\right )}+\frac {3 b \sqrt {d \left (c^{2} x^{2}+1\right )}\, \arcsinh \left (c x \right )^{2}}{16 \sqrt {c^{2} x^{2}+1}\, c^{5} d}+\frac {15 b \sqrt {d \left (c^{2} x^{2}+1\right )}}{128 c^{5} d \sqrt {c^{2} x^{2}+1}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [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]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {x^4\,\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}{\sqrt {d\,c^2\,x^2+d}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{4} \left (a + b \operatorname {asinh}{\left (c x \right )}\right )}{\sqrt {d \left (c^{2} x^{2} + 1\right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________