Optimal. Leaf size=413 \[ \frac{d^2 \left (d+e x^2\right )^{3/2} \left (a+b \text{csch}^{-1}(c x)\right )}{3 e^3}-\frac{2 d \left (d+e x^2\right )^{5/2} \left (a+b \text{csch}^{-1}(c x)\right )}{5 e^3}+\frac{\left (d+e x^2\right )^{7/2} \left (a+b \text{csch}^{-1}(c x)\right )}{7 e^3}-\frac{b x \sqrt{-c^2 x^2-1} \left (23 c^4 d^2-12 c^2 d e-75 e^2\right ) \sqrt{d+e x^2}}{1680 c^5 e^2 \sqrt{-c^2 x^2}}+\frac{8 b c d^{7/2} x \tan ^{-1}\left (\frac{\sqrt{d+e x^2}}{\sqrt{d} \sqrt{-c^2 x^2-1}}\right )}{105 e^3 \sqrt{-c^2 x^2}}+\frac{b x \left (35 c^4 d^2 e+105 c^6 d^3+63 c^2 d e^2-75 e^3\right ) \tan ^{-1}\left (\frac{\sqrt{e} \sqrt{-c^2 x^2-1}}{c \sqrt{d+e x^2}}\right )}{1680 c^6 e^{5/2} \sqrt{-c^2 x^2}}+\frac{b x \sqrt{-c^2 x^2-1} \left (d+e x^2\right )^{5/2}}{42 c e^2 \sqrt{-c^2 x^2}}-\frac{b x \sqrt{-c^2 x^2-1} \left (29 c^2 d+25 e\right ) \left (d+e x^2\right )^{3/2}}{840 c^3 e^2 \sqrt{-c^2 x^2}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 1.4122, antiderivative size = 413, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 12, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.522, Rules used = {266, 43, 6302, 12, 1615, 154, 157, 63, 217, 203, 93, 204} \[ \frac{d^2 \left (d+e x^2\right )^{3/2} \left (a+b \text{csch}^{-1}(c x)\right )}{3 e^3}-\frac{2 d \left (d+e x^2\right )^{5/2} \left (a+b \text{csch}^{-1}(c x)\right )}{5 e^3}+\frac{\left (d+e x^2\right )^{7/2} \left (a+b \text{csch}^{-1}(c x)\right )}{7 e^3}-\frac{b x \sqrt{-c^2 x^2-1} \left (23 c^4 d^2-12 c^2 d e-75 e^2\right ) \sqrt{d+e x^2}}{1680 c^5 e^2 \sqrt{-c^2 x^2}}+\frac{8 b c d^{7/2} x \tan ^{-1}\left (\frac{\sqrt{d+e x^2}}{\sqrt{d} \sqrt{-c^2 x^2-1}}\right )}{105 e^3 \sqrt{-c^2 x^2}}+\frac{b x \left (35 c^4 d^2 e+105 c^6 d^3+63 c^2 d e^2-75 e^3\right ) \tan ^{-1}\left (\frac{\sqrt{e} \sqrt{-c^2 x^2-1}}{c \sqrt{d+e x^2}}\right )}{1680 c^6 e^{5/2} \sqrt{-c^2 x^2}}+\frac{b x \sqrt{-c^2 x^2-1} \left (d+e x^2\right )^{5/2}}{42 c e^2 \sqrt{-c^2 x^2}}-\frac{b x \sqrt{-c^2 x^2-1} \left (29 c^2 d+25 e\right ) \left (d+e x^2\right )^{3/2}}{840 c^3 e^2 \sqrt{-c^2 x^2}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 266
Rule 43
Rule 6302
Rule 12
Rule 1615
Rule 154
Rule 157
Rule 63
Rule 217
Rule 203
Rule 93
Rule 204
Rubi steps
\begin{align*} \int x^5 \sqrt{d+e x^2} \left (a+b \text{csch}^{-1}(c x)\right ) \, dx &=\frac{d^2 \left (d+e x^2\right )^{3/2} \left (a+b \text{csch}^{-1}(c x)\right )}{3 e^3}-\frac{2 d \left (d+e x^2\right )^{5/2} \left (a+b \text{csch}^{-1}(c x)\right )}{5 e^3}+\frac{\left (d+e x^2\right )^{7/2} \left (a+b \text{csch}^{-1}(c x)\right )}{7 e^3}-\frac{(b c x) \int \frac{\left (d+e x^2\right )^{3/2} \left (8 d^2-12 d e x^2+15 e^2 x^4\right )}{105 e^3 x \sqrt{-1-c^2 x^2}} \, dx}{\sqrt{-c^2 x^2}}\\ &=\frac{d^2 \left (d+e x^2\right )^{3/2} \left (a+b \text{csch}^{-1}(c x)\right )}{3 e^3}-\frac{2 d \left (d+e x^2\right )^{5/2} \left (a+b \text{csch}^{-1}(c x)\right )}{5 e^3}+\frac{\left (d+e x^2\right )^{7/2} \left (a+b \text{csch}^{-1}(c x)\right )}{7 e^3}-\frac{(b c x) \int \frac{\left (d+e x^2\right )^{3/2} \left (8 d^2-12 d e x^2+15 e^2 x^4\right )}{x \sqrt{-1-c^2 x^2}} \, dx}{105 e^3 \sqrt{-c^2 x^2}}\\ &=\frac{d^2 \left (d+e x^2\right )^{3/2} \left (a+b \text{csch}^{-1}(c x)\right )}{3 e^3}-\frac{2 d \left (d+e x^2\right )^{5/2} \left (a+b \text{csch}^{-1}(c x)\right )}{5 e^3}+\frac{\left (d+e x^2\right )^{7/2} \left (a+b \text{csch}^{-1}(c x)\right )}{7 e^3}-\frac{(b c x) \operatorname{Subst}\left (\int \frac{(d+e x)^{3/2} \left (8 d^2-12 d e x+15 e^2 x^2\right )}{x \sqrt{-1-c^2 x}} \, dx,x,x^2\right )}{210 e^3 \sqrt{-c^2 x^2}}\\ &=\frac{b x \sqrt{-1-c^2 x^2} \left (d+e x^2\right )^{5/2}}{42 c e^2 \sqrt{-c^2 x^2}}+\frac{d^2 \left (d+e x^2\right )^{3/2} \left (a+b \text{csch}^{-1}(c x)\right )}{3 e^3}-\frac{2 d \left (d+e x^2\right )^{5/2} \left (a+b \text{csch}^{-1}(c x)\right )}{5 e^3}+\frac{\left (d+e x^2\right )^{7/2} \left (a+b \text{csch}^{-1}(c x)\right )}{7 e^3}+\frac{(b x) \operatorname{Subst}\left (\int \frac{(d+e x)^{3/2} \left (-24 c^2 d^2 e+\frac{3}{2} e^2 \left (29 c^2 d+25 e\right ) x\right )}{x \sqrt{-1-c^2 x}} \, dx,x,x^2\right )}{630 c e^4 \sqrt{-c^2 x^2}}\\ &=-\frac{b \left (29 c^2 d+25 e\right ) x \sqrt{-1-c^2 x^2} \left (d+e x^2\right )^{3/2}}{840 c^3 e^2 \sqrt{-c^2 x^2}}+\frac{b x \sqrt{-1-c^2 x^2} \left (d+e x^2\right )^{5/2}}{42 c e^2 \sqrt{-c^2 x^2}}+\frac{d^2 \left (d+e x^2\right )^{3/2} \left (a+b \text{csch}^{-1}(c x)\right )}{3 e^3}-\frac{2 d \left (d+e x^2\right )^{5/2} \left (a+b \text{csch}^{-1}(c x)\right )}{5 e^3}+\frac{\left (d+e x^2\right )^{7/2} \left (a+b \text{csch}^{-1}(c x)\right )}{7 e^3}-\frac{(b x) \operatorname{Subst}\left (\int \frac{\sqrt{d+e x} \left (48 c^4 d^3 e-\frac{3}{4} e^2 \left (23 c^4 d^2-12 c^2 d e-75 e^2\right ) x\right )}{x \sqrt{-1-c^2 x}} \, dx,x,x^2\right )}{1260 c^3 e^4 \sqrt{-c^2 x^2}}\\ &=-\frac{b \left (23 c^4 d^2-12 c^2 d e-75 e^2\right ) x \sqrt{-1-c^2 x^2} \sqrt{d+e x^2}}{1680 c^5 e^2 \sqrt{-c^2 x^2}}-\frac{b \left (29 c^2 d+25 e\right ) x \sqrt{-1-c^2 x^2} \left (d+e x^2\right )^{3/2}}{840 c^3 e^2 \sqrt{-c^2 x^2}}+\frac{b x \sqrt{-1-c^2 x^2} \left (d+e x^2\right )^{5/2}}{42 c e^2 \sqrt{-c^2 x^2}}+\frac{d^2 \left (d+e x^2\right )^{3/2} \left (a+b \text{csch}^{-1}(c x)\right )}{3 e^3}-\frac{2 d \left (d+e x^2\right )^{5/2} \left (a+b \text{csch}^{-1}(c x)\right )}{5 e^3}+\frac{\left (d+e x^2\right )^{7/2} \left (a+b \text{csch}^{-1}(c x)\right )}{7 e^3}+\frac{(b x) \operatorname{Subst}\left (\int \frac{-48 c^6 d^4 e-\frac{3}{8} e^2 \left (105 c^6 d^3+35 c^4 d^2 e+63 c^2 d e^2-75 e^3\right ) x}{x \sqrt{-1-c^2 x} \sqrt{d+e x}} \, dx,x,x^2\right )}{1260 c^5 e^4 \sqrt{-c^2 x^2}}\\ &=-\frac{b \left (23 c^4 d^2-12 c^2 d e-75 e^2\right ) x \sqrt{-1-c^2 x^2} \sqrt{d+e x^2}}{1680 c^5 e^2 \sqrt{-c^2 x^2}}-\frac{b \left (29 c^2 d+25 e\right ) x \sqrt{-1-c^2 x^2} \left (d+e x^2\right )^{3/2}}{840 c^3 e^2 \sqrt{-c^2 x^2}}+\frac{b x \sqrt{-1-c^2 x^2} \left (d+e x^2\right )^{5/2}}{42 c e^2 \sqrt{-c^2 x^2}}+\frac{d^2 \left (d+e x^2\right )^{3/2} \left (a+b \text{csch}^{-1}(c x)\right )}{3 e^3}-\frac{2 d \left (d+e x^2\right )^{5/2} \left (a+b \text{csch}^{-1}(c x)\right )}{5 e^3}+\frac{\left (d+e x^2\right )^{7/2} \left (a+b \text{csch}^{-1}(c x)\right )}{7 e^3}-\frac{\left (4 b c d^4 x\right ) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{-1-c^2 x} \sqrt{d+e x}} \, dx,x,x^2\right )}{105 e^3 \sqrt{-c^2 x^2}}-\frac{\left (b \left (105 c^6 d^3+35 c^4 d^2 e+63 c^2 d e^2-75 e^3\right ) x\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{-1-c^2 x} \sqrt{d+e x}} \, dx,x,x^2\right )}{3360 c^5 e^2 \sqrt{-c^2 x^2}}\\ &=-\frac{b \left (23 c^4 d^2-12 c^2 d e-75 e^2\right ) x \sqrt{-1-c^2 x^2} \sqrt{d+e x^2}}{1680 c^5 e^2 \sqrt{-c^2 x^2}}-\frac{b \left (29 c^2 d+25 e\right ) x \sqrt{-1-c^2 x^2} \left (d+e x^2\right )^{3/2}}{840 c^3 e^2 \sqrt{-c^2 x^2}}+\frac{b x \sqrt{-1-c^2 x^2} \left (d+e x^2\right )^{5/2}}{42 c e^2 \sqrt{-c^2 x^2}}+\frac{d^2 \left (d+e x^2\right )^{3/2} \left (a+b \text{csch}^{-1}(c x)\right )}{3 e^3}-\frac{2 d \left (d+e x^2\right )^{5/2} \left (a+b \text{csch}^{-1}(c x)\right )}{5 e^3}+\frac{\left (d+e x^2\right )^{7/2} \left (a+b \text{csch}^{-1}(c x)\right )}{7 e^3}-\frac{\left (8 b c d^4 x\right ) \operatorname{Subst}\left (\int \frac{1}{-d-x^2} \, dx,x,\frac{\sqrt{d+e x^2}}{\sqrt{-1-c^2 x^2}}\right )}{105 e^3 \sqrt{-c^2 x^2}}+\frac{\left (b \left (105 c^6 d^3+35 c^4 d^2 e+63 c^2 d e^2-75 e^3\right ) x\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{d-\frac{e}{c^2}-\frac{e x^2}{c^2}}} \, dx,x,\sqrt{-1-c^2 x^2}\right )}{1680 c^7 e^2 \sqrt{-c^2 x^2}}\\ &=-\frac{b \left (23 c^4 d^2-12 c^2 d e-75 e^2\right ) x \sqrt{-1-c^2 x^2} \sqrt{d+e x^2}}{1680 c^5 e^2 \sqrt{-c^2 x^2}}-\frac{b \left (29 c^2 d+25 e\right ) x \sqrt{-1-c^2 x^2} \left (d+e x^2\right )^{3/2}}{840 c^3 e^2 \sqrt{-c^2 x^2}}+\frac{b x \sqrt{-1-c^2 x^2} \left (d+e x^2\right )^{5/2}}{42 c e^2 \sqrt{-c^2 x^2}}+\frac{d^2 \left (d+e x^2\right )^{3/2} \left (a+b \text{csch}^{-1}(c x)\right )}{3 e^3}-\frac{2 d \left (d+e x^2\right )^{5/2} \left (a+b \text{csch}^{-1}(c x)\right )}{5 e^3}+\frac{\left (d+e x^2\right )^{7/2} \left (a+b \text{csch}^{-1}(c x)\right )}{7 e^3}+\frac{8 b c d^{7/2} x \tan ^{-1}\left (\frac{\sqrt{d+e x^2}}{\sqrt{d} \sqrt{-1-c^2 x^2}}\right )}{105 e^3 \sqrt{-c^2 x^2}}+\frac{\left (b \left (105 c^6 d^3+35 c^4 d^2 e+63 c^2 d e^2-75 e^3\right ) x\right ) \operatorname{Subst}\left (\int \frac{1}{1+\frac{e x^2}{c^2}} \, dx,x,\frac{\sqrt{-1-c^2 x^2}}{\sqrt{d+e x^2}}\right )}{1680 c^7 e^2 \sqrt{-c^2 x^2}}\\ &=-\frac{b \left (23 c^4 d^2-12 c^2 d e-75 e^2\right ) x \sqrt{-1-c^2 x^2} \sqrt{d+e x^2}}{1680 c^5 e^2 \sqrt{-c^2 x^2}}-\frac{b \left (29 c^2 d+25 e\right ) x \sqrt{-1-c^2 x^2} \left (d+e x^2\right )^{3/2}}{840 c^3 e^2 \sqrt{-c^2 x^2}}+\frac{b x \sqrt{-1-c^2 x^2} \left (d+e x^2\right )^{5/2}}{42 c e^2 \sqrt{-c^2 x^2}}+\frac{d^2 \left (d+e x^2\right )^{3/2} \left (a+b \text{csch}^{-1}(c x)\right )}{3 e^3}-\frac{2 d \left (d+e x^2\right )^{5/2} \left (a+b \text{csch}^{-1}(c x)\right )}{5 e^3}+\frac{\left (d+e x^2\right )^{7/2} \left (a+b \text{csch}^{-1}(c x)\right )}{7 e^3}+\frac{b \left (105 c^6 d^3+35 c^4 d^2 e+63 c^2 d e^2-75 e^3\right ) x \tan ^{-1}\left (\frac{\sqrt{e} \sqrt{-1-c^2 x^2}}{c \sqrt{d+e x^2}}\right )}{1680 c^6 e^{5/2} \sqrt{-c^2 x^2}}+\frac{8 b c d^{7/2} x \tan ^{-1}\left (\frac{\sqrt{d+e x^2}}{\sqrt{d} \sqrt{-1-c^2 x^2}}\right )}{105 e^3 \sqrt{-c^2 x^2}}\\ \end{align*}
Mathematica [C] time = 0.678312, size = 345, normalized size = 0.84 \[ \frac{\sqrt{d+e x^2} \left (16 a c^5 \left (-4 d^2 e x^2+8 d^3+3 d e^2 x^4+15 e^3 x^6\right )+b e x \sqrt{\frac{1}{c^2 x^2}+1} \left (c^4 \left (-41 d^2+22 d e x^2+40 e^2 x^4\right )-2 c^2 e \left (19 d+25 e x^2\right )+75 e^2\right )+16 b c^5 \text{csch}^{-1}(c x) \left (-4 d^2 e x^2+8 d^3+3 d e^2 x^4+15 e^3 x^6\right )\right )}{1680 c^5 e^3}-\frac{b \left (128 c^4 d^4 \sqrt{\frac{d}{e x^2}+1} F_1\left (1;\frac{1}{2},\frac{1}{2};2;-\frac{1}{c^2 x^2},-\frac{d}{e x^2}\right )-\frac{e x^4 \sqrt{\frac{1}{c^2 x^2}+1} \left (35 c^4 d^2 e+105 c^6 d^3+63 c^2 d e^2-75 e^3\right ) \sqrt{\frac{e x^2}{d}+1} F_1\left (1;\frac{1}{2},\frac{1}{2};2;-c^2 x^2,-\frac{e x^2}{d}\right )}{\sqrt{c^2 x^2+1}}\right )}{3360 c^5 e^3 x \sqrt{d+e x^2}} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
Maple [F] time = 0.457, size = 0, normalized size = 0. \begin{align*} \int{x}^{5} \left ( a+b{\rm arccsch} \left (cx\right ) \right ) \sqrt{e{x}^{2}+d}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 20.5525, size = 4346, normalized size = 10.52 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{e x^{2} + d}{\left (b \operatorname{arcsch}\left (c x\right ) + a\right )} x^{5}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]