Optimal. Leaf size=203 \[ -\frac {x^3 \left (a+b \sinh ^{-1}(c x)\right )}{3 c^2 d \left (c^2 d x^2+d\right )^{3/2}}+\frac {\sqrt {c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )^2}{2 b c^5 d^2 \sqrt {c^2 d x^2+d}}-\frac {x \left (a+b \sinh ^{-1}(c x)\right )}{c^4 d^2 \sqrt {c^2 d x^2+d}}+\frac {b}{6 c^5 d^2 \sqrt {c^2 x^2+1} \sqrt {c^2 d x^2+d}}+\frac {2 b \sqrt {c^2 x^2+1} \log \left (c^2 x^2+1\right )}{3 c^5 d^2 \sqrt {c^2 d x^2+d}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.30, antiderivative size = 203, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {5751, 5677, 5675, 260, 266, 43} \[ -\frac {x \left (a+b \sinh ^{-1}(c x)\right )}{c^4 d^2 \sqrt {c^2 d x^2+d}}+\frac {\sqrt {c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )^2}{2 b c^5 d^2 \sqrt {c^2 d x^2+d}}-\frac {x^3 \left (a+b \sinh ^{-1}(c x)\right )}{3 c^2 d \left (c^2 d x^2+d\right )^{3/2}}+\frac {b}{6 c^5 d^2 \sqrt {c^2 x^2+1} \sqrt {c^2 d x^2+d}}+\frac {2 b \sqrt {c^2 x^2+1} \log \left (c^2 x^2+1\right )}{3 c^5 d^2 \sqrt {c^2 d x^2+d}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 43
Rule 260
Rule 266
Rule 5675
Rule 5677
Rule 5751
Rubi steps
\begin {align*} \int \frac {x^4 \left (a+b \sinh ^{-1}(c x)\right )}{\left (d+c^2 d x^2\right )^{5/2}} \, dx &=-\frac {x^3 \left (a+b \sinh ^{-1}(c x)\right )}{3 c^2 d \left (d+c^2 d x^2\right )^{3/2}}+\frac {\int \frac {x^2 \left (a+b \sinh ^{-1}(c x)\right )}{\left (d+c^2 d x^2\right )^{3/2}} \, dx}{c^2 d}+\frac {\left (b \sqrt {1+c^2 x^2}\right ) \int \frac {x^3}{\left (1+c^2 x^2\right )^2} \, dx}{3 c d^2 \sqrt {d+c^2 d x^2}}\\ &=-\frac {x^3 \left (a+b \sinh ^{-1}(c x)\right )}{3 c^2 d \left (d+c^2 d x^2\right )^{3/2}}-\frac {x \left (a+b \sinh ^{-1}(c x)\right )}{c^4 d^2 \sqrt {d+c^2 d x^2}}+\frac {\int \frac {a+b \sinh ^{-1}(c x)}{\sqrt {d+c^2 d x^2}} \, dx}{c^4 d^2}+\frac {\left (b \sqrt {1+c^2 x^2}\right ) \int \frac {x}{1+c^2 x^2} \, dx}{c^3 d^2 \sqrt {d+c^2 d x^2}}+\frac {\left (b \sqrt {1+c^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {x}{\left (1+c^2 x\right )^2} \, dx,x,x^2\right )}{6 c d^2 \sqrt {d+c^2 d x^2}}\\ &=-\frac {x^3 \left (a+b \sinh ^{-1}(c x)\right )}{3 c^2 d \left (d+c^2 d x^2\right )^{3/2}}-\frac {x \left (a+b \sinh ^{-1}(c x)\right )}{c^4 d^2 \sqrt {d+c^2 d x^2}}+\frac {b \sqrt {1+c^2 x^2} \log \left (1+c^2 x^2\right )}{2 c^5 d^2 \sqrt {d+c^2 d x^2}}+\frac {\sqrt {1+c^2 x^2} \int \frac {a+b \sinh ^{-1}(c x)}{\sqrt {1+c^2 x^2}} \, dx}{c^4 d^2 \sqrt {d+c^2 d x^2}}+\frac {\left (b \sqrt {1+c^2 x^2}\right ) \operatorname {Subst}\left (\int \left (-\frac {1}{c^2 \left (1+c^2 x\right )^2}+\frac {1}{c^2 \left (1+c^2 x\right )}\right ) \, dx,x,x^2\right )}{6 c d^2 \sqrt {d+c^2 d x^2}}\\ &=\frac {b}{6 c^5 d^2 \sqrt {1+c^2 x^2} \sqrt {d+c^2 d x^2}}-\frac {x^3 \left (a+b \sinh ^{-1}(c x)\right )}{3 c^2 d \left (d+c^2 d x^2\right )^{3/2}}-\frac {x \left (a+b \sinh ^{-1}(c x)\right )}{c^4 d^2 \sqrt {d+c^2 d x^2}}+\frac {\sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{2 b c^5 d^2 \sqrt {d+c^2 d x^2}}+\frac {2 b \sqrt {1+c^2 x^2} \log \left (1+c^2 x^2\right )}{3 c^5 d^2 \sqrt {d+c^2 d x^2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.53, size = 191, normalized size = 0.94 \[ \frac {-2 a c \sqrt {d} x \left (4 c^2 x^2+3\right )+6 a \left (c^2 x^2+1\right ) \sqrt {c^2 d x^2+d} \log \left (\sqrt {d} \sqrt {c^2 d x^2+d}+c d x\right )+b \sqrt {d} \left (\sqrt {c^2 x^2+1}-8 c x \left (c^2 x^2+1\right ) \sinh ^{-1}(c x)+\left (c^2 x^2+1\right )^{3/2} \left (4 \log \left (c^2 x^2+1\right )+3 \sinh ^{-1}(c x)^2\right )+2 c x \sinh ^{-1}(c x)\right )}{6 c^5 d^{5/2} \left (c^2 x^2+1\right ) \sqrt {c^2 d x^2+d}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 0.60, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (b x^{4} \operatorname {arsinh}\left (c x\right ) + a x^{4}\right )} \sqrt {c^{2} d x^{2} + d}}{c^{6} d^{3} x^{6} + 3 \, c^{4} d^{3} x^{4} + 3 \, c^{2} d^{3} x^{2} + d^{3}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (b \operatorname {arsinh}\left (c x\right ) + a\right )} x^{4}}{{\left (c^{2} d x^{2} + d\right )}^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [B] time = 0.37, size = 1430, normalized size = 7.04 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ -\frac {1}{3} \, {\left (x {\left (\frac {3 \, x^{2}}{{\left (c^{2} d x^{2} + d\right )}^{\frac {3}{2}} c^{2} d} + \frac {2}{{\left (c^{2} d x^{2} + d\right )}^{\frac {3}{2}} c^{4} d}\right )} + \frac {x}{\sqrt {c^{2} d x^{2} + d} c^{4} d^{2}} - \frac {3 \, \operatorname {arsinh}\left (c x\right )}{c^{5} d^{\frac {5}{2}}}\right )} a + b \int \frac {x^{4} \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right )}{{\left (c^{2} d x^{2} + d\right )}^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {x^4\,\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}{{\left (d\,c^2\,x^2+d\right )}^{5/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{4} \left (a + b \operatorname {asinh}{\left (c x \right )}\right )}{\left (d \left (c^{2} x^{2} + 1\right )\right )^{\frac {5}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________