Optimal. Leaf size=512 \[ -\frac {x^4 \left (a+b \sinh ^{-1}(c x)\right )^2}{3 c^2 d \left (c^2 d x^2+d\right )^{3/2}}+\frac {8 \sqrt {c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )^2}{3 c^6 d^3}-\frac {22 b \sqrt {c^2 x^2+1} \tan ^{-1}\left (e^{\sinh ^{-1}(c x)}\right ) \left (a+b \sinh ^{-1}(c x)\right )}{3 c^6 d^2 \sqrt {c^2 d x^2+d}}-\frac {16 a b x \sqrt {c^2 x^2+1}}{3 c^5 d^2 \sqrt {c^2 d x^2+d}}+\frac {11 b x \sqrt {c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )}{3 c^5 d^2 \sqrt {c^2 d x^2+d}}-\frac {4 x^2 \left (a+b \sinh ^{-1}(c x)\right )^2}{3 c^4 d^2 \sqrt {c^2 d x^2+d}}-\frac {b x^3 \left (a+b \sinh ^{-1}(c x)\right )}{3 c^3 d^2 \sqrt {c^2 x^2+1} \sqrt {c^2 d x^2+d}}+\frac {11 i b^2 \sqrt {c^2 x^2+1} \text {Li}_2\left (-i e^{\sinh ^{-1}(c x)}\right )}{3 c^6 d^2 \sqrt {c^2 d x^2+d}}-\frac {11 i b^2 \sqrt {c^2 x^2+1} \text {Li}_2\left (i e^{\sinh ^{-1}(c x)}\right )}{3 c^6 d^2 \sqrt {c^2 d x^2+d}}+\frac {2 b^2 \left (c^2 x^2+1\right )}{c^6 d^2 \sqrt {c^2 d x^2+d}}+\frac {b^2}{3 c^6 d^2 \sqrt {c^2 d x^2+d}}-\frac {16 b^2 x \sqrt {c^2 x^2+1} \sinh ^{-1}(c x)}{3 c^5 d^2 \sqrt {c^2 d x^2+d}} \]
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Rubi [A] time = 0.88, antiderivative size = 512, normalized size of antiderivative = 1.00, number of steps used = 26, number of rules used = 11, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.393, Rules used = {5751, 5717, 5653, 261, 5767, 5693, 4180, 2279, 2391, 266, 43} \[ \frac {11 i b^2 \sqrt {c^2 x^2+1} \text {PolyLog}\left (2,-i e^{\sinh ^{-1}(c x)}\right )}{3 c^6 d^2 \sqrt {c^2 d x^2+d}}-\frac {11 i b^2 \sqrt {c^2 x^2+1} \text {PolyLog}\left (2,i e^{\sinh ^{-1}(c x)}\right )}{3 c^6 d^2 \sqrt {c^2 d x^2+d}}-\frac {16 a b x \sqrt {c^2 x^2+1}}{3 c^5 d^2 \sqrt {c^2 d x^2+d}}-\frac {b x^3 \left (a+b \sinh ^{-1}(c x)\right )}{3 c^3 d^2 \sqrt {c^2 x^2+1} \sqrt {c^2 d x^2+d}}-\frac {4 x^2 \left (a+b \sinh ^{-1}(c x)\right )^2}{3 c^4 d^2 \sqrt {c^2 d x^2+d}}+\frac {11 b x \sqrt {c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )}{3 c^5 d^2 \sqrt {c^2 d x^2+d}}+\frac {8 \sqrt {c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )^2}{3 c^6 d^3}-\frac {22 b \sqrt {c^2 x^2+1} \tan ^{-1}\left (e^{\sinh ^{-1}(c x)}\right ) \left (a+b \sinh ^{-1}(c x)\right )}{3 c^6 d^2 \sqrt {c^2 d x^2+d}}-\frac {x^4 \left (a+b \sinh ^{-1}(c x)\right )^2}{3 c^2 d \left (c^2 d x^2+d\right )^{3/2}}+\frac {2 b^2 \left (c^2 x^2+1\right )}{c^6 d^2 \sqrt {c^2 d x^2+d}}+\frac {b^2}{3 c^6 d^2 \sqrt {c^2 d x^2+d}}-\frac {16 b^2 x \sqrt {c^2 x^2+1} \sinh ^{-1}(c x)}{3 c^5 d^2 \sqrt {c^2 d x^2+d}} \]
Antiderivative was successfully verified.
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Rule 43
Rule 261
Rule 266
Rule 2279
Rule 2391
Rule 4180
Rule 5653
Rule 5693
Rule 5717
Rule 5751
Rule 5767
Rubi steps
\begin {align*} \int \frac {x^5 \left (a+b \sinh ^{-1}(c x)\right )^2}{\left (d+c^2 d x^2\right )^{5/2}} \, dx &=-\frac {x^4 \left (a+b \sinh ^{-1}(c x)\right )^2}{3 c^2 d \left (d+c^2 d x^2\right )^{3/2}}+\frac {4 \int \frac {x^3 \left (a+b \sinh ^{-1}(c x)\right )^2}{\left (d+c^2 d x^2\right )^{3/2}} \, dx}{3 c^2 d}+\frac {\left (2 b \sqrt {1+c^2 x^2}\right ) \int \frac {x^4 \left (a+b \sinh ^{-1}(c x)\right )}{\left (1+c^2 x^2\right )^2} \, dx}{3 c d^2 \sqrt {d+c^2 d x^2}}\\ &=-\frac {b x^3 \left (a+b \sinh ^{-1}(c x)\right )}{3 c^3 d^2 \sqrt {1+c^2 x^2} \sqrt {d+c^2 d x^2}}-\frac {x^4 \left (a+b \sinh ^{-1}(c x)\right )^2}{3 c^2 d \left (d+c^2 d x^2\right )^{3/2}}-\frac {4 x^2 \left (a+b \sinh ^{-1}(c x)\right )^2}{3 c^4 d^2 \sqrt {d+c^2 d x^2}}+\frac {8 \int \frac {x \left (a+b \sinh ^{-1}(c x)\right )^2}{\sqrt {d+c^2 d x^2}} \, dx}{3 c^4 d^2}+\frac {\left (b \sqrt {1+c^2 x^2}\right ) \int \frac {x^2 \left (a+b \sinh ^{-1}(c x)\right )}{1+c^2 x^2} \, dx}{c^3 d^2 \sqrt {d+c^2 d x^2}}+\frac {\left (8 b \sqrt {1+c^2 x^2}\right ) \int \frac {x^2 \left (a+b \sinh ^{-1}(c x)\right )}{1+c^2 x^2} \, dx}{3 c^3 d^2 \sqrt {d+c^2 d x^2}}+\frac {\left (b^2 \sqrt {1+c^2 x^2}\right ) \int \frac {x^3}{\left (1+c^2 x^2\right )^{3/2}} \, dx}{3 c^2 d^2 \sqrt {d+c^2 d x^2}}\\ &=-\frac {b x^3 \left (a+b \sinh ^{-1}(c x)\right )}{3 c^3 d^2 \sqrt {1+c^2 x^2} \sqrt {d+c^2 d x^2}}+\frac {11 b x \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{3 c^5 d^2 \sqrt {d+c^2 d x^2}}-\frac {x^4 \left (a+b \sinh ^{-1}(c x)\right )^2}{3 c^2 d \left (d+c^2 d x^2\right )^{3/2}}-\frac {4 x^2 \left (a+b \sinh ^{-1}(c x)\right )^2}{3 c^4 d^2 \sqrt {d+c^2 d x^2}}+\frac {8 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{3 c^6 d^3}-\frac {\left (b \sqrt {1+c^2 x^2}\right ) \int \frac {a+b \sinh ^{-1}(c x)}{1+c^2 x^2} \, dx}{c^5 d^2 \sqrt {d+c^2 d x^2}}-\frac {\left (8 b \sqrt {1+c^2 x^2}\right ) \int \frac {a+b \sinh ^{-1}(c x)}{1+c^2 x^2} \, dx}{3 c^5 d^2 \sqrt {d+c^2 d x^2}}-\frac {\left (16 b \sqrt {1+c^2 x^2}\right ) \int \left (a+b \sinh ^{-1}(c x)\right ) \, dx}{3 c^5 d^2 \sqrt {d+c^2 d x^2}}-\frac {\left (b^2 \sqrt {1+c^2 x^2}\right ) \int \frac {x}{\sqrt {1+c^2 x^2}} \, dx}{c^4 d^2 \sqrt {d+c^2 d x^2}}-\frac {\left (8 b^2 \sqrt {1+c^2 x^2}\right ) \int \frac {x}{\sqrt {1+c^2 x^2}} \, dx}{3 c^4 d^2 \sqrt {d+c^2 d x^2}}+\frac {\left (b^2 \sqrt {1+c^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {x}{\left (1+c^2 x\right )^{3/2}} \, dx,x,x^2\right )}{6 c^2 d^2 \sqrt {d+c^2 d x^2}}\\ &=-\frac {16 a b x \sqrt {1+c^2 x^2}}{3 c^5 d^2 \sqrt {d+c^2 d x^2}}-\frac {11 b^2 \left (1+c^2 x^2\right )}{3 c^6 d^2 \sqrt {d+c^2 d x^2}}-\frac {b x^3 \left (a+b \sinh ^{-1}(c x)\right )}{3 c^3 d^2 \sqrt {1+c^2 x^2} \sqrt {d+c^2 d x^2}}+\frac {11 b x \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{3 c^5 d^2 \sqrt {d+c^2 d x^2}}-\frac {x^4 \left (a+b \sinh ^{-1}(c x)\right )^2}{3 c^2 d \left (d+c^2 d x^2\right )^{3/2}}-\frac {4 x^2 \left (a+b \sinh ^{-1}(c x)\right )^2}{3 c^4 d^2 \sqrt {d+c^2 d x^2}}+\frac {8 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{3 c^6 d^3}-\frac {\left (b \sqrt {1+c^2 x^2}\right ) \operatorname {Subst}\left (\int (a+b x) \text {sech}(x) \, dx,x,\sinh ^{-1}(c x)\right )}{c^6 d^2 \sqrt {d+c^2 d x^2}}-\frac {\left (8 b \sqrt {1+c^2 x^2}\right ) \operatorname {Subst}\left (\int (a+b x) \text {sech}(x) \, dx,x,\sinh ^{-1}(c x)\right )}{3 c^6 d^2 \sqrt {d+c^2 d x^2}}-\frac {\left (16 b^2 \sqrt {1+c^2 x^2}\right ) \int \sinh ^{-1}(c x) \, dx}{3 c^5 d^2 \sqrt {d+c^2 d x^2}}+\frac {\left (b^2 \sqrt {1+c^2 x^2}\right ) \operatorname {Subst}\left (\int \left (-\frac {1}{c^2 \left (1+c^2 x\right )^{3/2}}+\frac {1}{c^2 \sqrt {1+c^2 x}}\right ) \, dx,x,x^2\right )}{6 c^2 d^2 \sqrt {d+c^2 d x^2}}\\ &=\frac {b^2}{3 c^6 d^2 \sqrt {d+c^2 d x^2}}-\frac {16 a b x \sqrt {1+c^2 x^2}}{3 c^5 d^2 \sqrt {d+c^2 d x^2}}-\frac {10 b^2 \left (1+c^2 x^2\right )}{3 c^6 d^2 \sqrt {d+c^2 d x^2}}-\frac {16 b^2 x \sqrt {1+c^2 x^2} \sinh ^{-1}(c x)}{3 c^5 d^2 \sqrt {d+c^2 d x^2}}-\frac {b x^3 \left (a+b \sinh ^{-1}(c x)\right )}{3 c^3 d^2 \sqrt {1+c^2 x^2} \sqrt {d+c^2 d x^2}}+\frac {11 b x \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{3 c^5 d^2 \sqrt {d+c^2 d x^2}}-\frac {x^4 \left (a+b \sinh ^{-1}(c x)\right )^2}{3 c^2 d \left (d+c^2 d x^2\right )^{3/2}}-\frac {4 x^2 \left (a+b \sinh ^{-1}(c x)\right )^2}{3 c^4 d^2 \sqrt {d+c^2 d x^2}}+\frac {8 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{3 c^6 d^3}-\frac {22 b \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right ) \tan ^{-1}\left (e^{\sinh ^{-1}(c x)}\right )}{3 c^6 d^2 \sqrt {d+c^2 d x^2}}+\frac {\left (i b^2 \sqrt {1+c^2 x^2}\right ) \operatorname {Subst}\left (\int \log \left (1-i e^x\right ) \, dx,x,\sinh ^{-1}(c x)\right )}{c^6 d^2 \sqrt {d+c^2 d x^2}}-\frac {\left (i b^2 \sqrt {1+c^2 x^2}\right ) \operatorname {Subst}\left (\int \log \left (1+i e^x\right ) \, dx,x,\sinh ^{-1}(c x)\right )}{c^6 d^2 \sqrt {d+c^2 d x^2}}+\frac {\left (8 i b^2 \sqrt {1+c^2 x^2}\right ) \operatorname {Subst}\left (\int \log \left (1-i e^x\right ) \, dx,x,\sinh ^{-1}(c x)\right )}{3 c^6 d^2 \sqrt {d+c^2 d x^2}}-\frac {\left (8 i b^2 \sqrt {1+c^2 x^2}\right ) \operatorname {Subst}\left (\int \log \left (1+i e^x\right ) \, dx,x,\sinh ^{-1}(c x)\right )}{3 c^6 d^2 \sqrt {d+c^2 d x^2}}+\frac {\left (16 b^2 \sqrt {1+c^2 x^2}\right ) \int \frac {x}{\sqrt {1+c^2 x^2}} \, dx}{3 c^4 d^2 \sqrt {d+c^2 d x^2}}\\ &=\frac {b^2}{3 c^6 d^2 \sqrt {d+c^2 d x^2}}-\frac {16 a b x \sqrt {1+c^2 x^2}}{3 c^5 d^2 \sqrt {d+c^2 d x^2}}+\frac {2 b^2 \left (1+c^2 x^2\right )}{c^6 d^2 \sqrt {d+c^2 d x^2}}-\frac {16 b^2 x \sqrt {1+c^2 x^2} \sinh ^{-1}(c x)}{3 c^5 d^2 \sqrt {d+c^2 d x^2}}-\frac {b x^3 \left (a+b \sinh ^{-1}(c x)\right )}{3 c^3 d^2 \sqrt {1+c^2 x^2} \sqrt {d+c^2 d x^2}}+\frac {11 b x \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{3 c^5 d^2 \sqrt {d+c^2 d x^2}}-\frac {x^4 \left (a+b \sinh ^{-1}(c x)\right )^2}{3 c^2 d \left (d+c^2 d x^2\right )^{3/2}}-\frac {4 x^2 \left (a+b \sinh ^{-1}(c x)\right )^2}{3 c^4 d^2 \sqrt {d+c^2 d x^2}}+\frac {8 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{3 c^6 d^3}-\frac {22 b \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right ) \tan ^{-1}\left (e^{\sinh ^{-1}(c x)}\right )}{3 c^6 d^2 \sqrt {d+c^2 d x^2}}+\frac {\left (i b^2 \sqrt {1+c^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {\log (1-i x)}{x} \, dx,x,e^{\sinh ^{-1}(c x)}\right )}{c^6 d^2 \sqrt {d+c^2 d x^2}}-\frac {\left (i b^2 \sqrt {1+c^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {\log (1+i x)}{x} \, dx,x,e^{\sinh ^{-1}(c x)}\right )}{c^6 d^2 \sqrt {d+c^2 d x^2}}+\frac {\left (8 i b^2 \sqrt {1+c^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {\log (1-i x)}{x} \, dx,x,e^{\sinh ^{-1}(c x)}\right )}{3 c^6 d^2 \sqrt {d+c^2 d x^2}}-\frac {\left (8 i b^2 \sqrt {1+c^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {\log (1+i x)}{x} \, dx,x,e^{\sinh ^{-1}(c x)}\right )}{3 c^6 d^2 \sqrt {d+c^2 d x^2}}\\ &=\frac {b^2}{3 c^6 d^2 \sqrt {d+c^2 d x^2}}-\frac {16 a b x \sqrt {1+c^2 x^2}}{3 c^5 d^2 \sqrt {d+c^2 d x^2}}+\frac {2 b^2 \left (1+c^2 x^2\right )}{c^6 d^2 \sqrt {d+c^2 d x^2}}-\frac {16 b^2 x \sqrt {1+c^2 x^2} \sinh ^{-1}(c x)}{3 c^5 d^2 \sqrt {d+c^2 d x^2}}-\frac {b x^3 \left (a+b \sinh ^{-1}(c x)\right )}{3 c^3 d^2 \sqrt {1+c^2 x^2} \sqrt {d+c^2 d x^2}}+\frac {11 b x \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{3 c^5 d^2 \sqrt {d+c^2 d x^2}}-\frac {x^4 \left (a+b \sinh ^{-1}(c x)\right )^2}{3 c^2 d \left (d+c^2 d x^2\right )^{3/2}}-\frac {4 x^2 \left (a+b \sinh ^{-1}(c x)\right )^2}{3 c^4 d^2 \sqrt {d+c^2 d x^2}}+\frac {8 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{3 c^6 d^3}-\frac {22 b \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right ) \tan ^{-1}\left (e^{\sinh ^{-1}(c x)}\right )}{3 c^6 d^2 \sqrt {d+c^2 d x^2}}+\frac {11 i b^2 \sqrt {1+c^2 x^2} \text {Li}_2\left (-i e^{\sinh ^{-1}(c x)}\right )}{3 c^6 d^2 \sqrt {d+c^2 d x^2}}-\frac {11 i b^2 \sqrt {1+c^2 x^2} \text {Li}_2\left (i e^{\sinh ^{-1}(c x)}\right )}{3 c^6 d^2 \sqrt {d+c^2 d x^2}}\\ \end {align*}
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Mathematica [A] time = 1.77, size = 333, normalized size = 0.65 \[ \frac {\sqrt {c^2 d x^2+d} \left (a^2 \left (3 c^4 x^4+12 c^2 x^2+8\right )+a b \left (2 \left (3 c^4 x^4+12 c^2 x^2+8\right ) \sinh ^{-1}(c x)-\sqrt {c^2 x^2+1} \left (c x \left (6 c^2 x^2+5\right )+22 \left (c^2 x^2+1\right ) \tan ^{-1}\left (\tanh \left (\frac {1}{2} \sinh ^{-1}(c x)\right )\right )\right )\right )+b^2 \left (11 i \left (c^2 x^2+1\right )^{3/2} \left (\text {Li}_2\left (-i e^{-\sinh ^{-1}(c x)}\right )-\text {Li}_2\left (i e^{-\sinh ^{-1}(c x)}\right )\right )+3 \left (c^2 x^2+1\right )^2 \left (\sinh ^{-1}(c x)^2+2\right )-6 c x \left (c^2 x^2+1\right )^{3/2} \sinh ^{-1}(c x)+\left (c^2 x^2+1\right ) \left (6 \sinh ^{-1}(c x)^2+1\right )+c x \sqrt {c^2 x^2+1} \sinh ^{-1}(c x)+11 i \left (c^2 x^2+1\right )^{3/2} \sinh ^{-1}(c x) \left (\log \left (1-i e^{-\sinh ^{-1}(c x)}\right )-\log \left (1+i e^{-\sinh ^{-1}(c x)}\right )\right )-\sinh ^{-1}(c x)^2\right )\right )}{3 c^6 d^3 \left (c^2 x^2+1\right )^2} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.47, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (b^{2} x^{5} \operatorname {arsinh}\left (c x\right )^{2} + 2 \, a b x^{5} \operatorname {arsinh}\left (c x\right ) + a^{2} x^{5}\right )} \sqrt {c^{2} d x^{2} + d}}{c^{6} d^{3} x^{6} + 3 \, c^{4} d^{3} x^{4} + 3 \, c^{2} d^{3} x^{2} + d^{3}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.50, size = 1040, normalized size = 2.03 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {1}{3} \, a^{2} {\left (\frac {3 \, x^{4}}{{\left (c^{2} d x^{2} + d\right )}^{\frac {3}{2}} c^{2} d} + \frac {12 \, x^{2}}{{\left (c^{2} d x^{2} + d\right )}^{\frac {3}{2}} c^{4} d} + \frac {8}{{\left (c^{2} d x^{2} + d\right )}^{\frac {3}{2}} c^{6} d}\right )} + \frac {{\left (3 \, b^{2} c^{4} \sqrt {d} x^{4} + 12 \, b^{2} c^{2} \sqrt {d} x^{2} + 8 \, b^{2} \sqrt {d}\right )} \sqrt {c^{2} x^{2} + 1} \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right )^{2}}{3 \, {\left (c^{10} d^{3} x^{4} + 2 \, c^{8} d^{3} x^{2} + c^{6} d^{3}\right )}} + \int -\frac {2 \, {\left ({\left (12 \, b^{2} c^{3} x^{3} - 3 \, {\left (a b c^{5} - b^{2} c^{5}\right )} x^{5} + 8 \, b^{2} c x\right )} {\left (c^{2} x^{2} + 1\right )} + {\left (15 \, b^{2} c^{4} x^{4} - 3 \, {\left (a b c^{6} - b^{2} c^{6}\right )} x^{6} + 20 \, b^{2} c^{2} x^{2} + 8 \, b^{2}\right )} \sqrt {c^{2} x^{2} + 1}\right )} \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right )}{3 \, {\left (c^{12} d^{\frac {5}{2}} x^{7} + 3 \, c^{10} d^{\frac {5}{2}} x^{5} + 3 \, c^{8} d^{\frac {5}{2}} x^{3} + c^{6} d^{\frac {5}{2}} x + {\left (c^{11} d^{\frac {5}{2}} x^{6} + 3 \, c^{9} d^{\frac {5}{2}} x^{4} + 3 \, c^{7} d^{\frac {5}{2}} x^{2} + c^{5} d^{\frac {5}{2}}\right )} \sqrt {c^{2} x^{2} + 1}\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {x^5\,{\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}^2}{{\left (d\,c^2\,x^2+d\right )}^{5/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{5} \left (a + b \operatorname {asinh}{\left (c x \right )}\right )^{2}}{\left (d \left (c^{2} x^{2} + 1\right )\right )^{\frac {5}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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