Optimal. Leaf size=418 \[ \frac {b \left (3 c^4 d^2-38 c^2 d e-25 e^2\right ) \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2} \sqrt {d+e x^2}}{560 c^6 e}-\frac {b \left (13 c^2 d+25 e\right ) \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2} \left (d+e x^2\right )^{3/2}}{840 c^4 e}-\frac {b \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2} \left (d+e x^2\right )^{5/2}}{42 c^2 e}-\frac {d \left (d+e x^2\right )^{5/2} \left (a+b \text {sech}^{-1}(c x)\right )}{5 e^2}+\frac {\left (d+e x^2\right )^{7/2} \left (a+b \text {sech}^{-1}(c x)\right )}{7 e^2}+\frac {b \left (35 c^6 d^3-35 c^4 d^2 e-63 c^2 d e^2-25 e^3\right ) \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \text {ArcTan}\left (\frac {\sqrt {e} \sqrt {1-c^2 x^2}}{c \sqrt {d+e x^2}}\right )}{560 c^7 e^{3/2}}+\frac {2 b d^{7/2} \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \tanh ^{-1}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d} \sqrt {1-c^2 x^2}}\right )}{35 e^2} \]
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Rubi [A]
time = 0.36, antiderivative size = 418, normalized size of antiderivative = 1.00, 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 = {272, 45,
6436, 12, 587, 159, 163, 65, 223, 209, 95, 213} \begin {gather*} -\frac {d \left (d+e x^2\right )^{5/2} \left (a+b \text {sech}^{-1}(c x)\right )}{5 e^2}+\frac {\left (d+e x^2\right )^{7/2} \left (a+b \text {sech}^{-1}(c x)\right )}{7 e^2}+\frac {b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \left (35 c^6 d^3-35 c^4 d^2 e-63 c^2 d e^2-25 e^3\right ) \text {ArcTan}\left (\frac {\sqrt {e} \sqrt {1-c^2 x^2}}{c \sqrt {d+e x^2}}\right )}{560 c^7 e^{3/2}}+\frac {2 b d^{7/2} \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \tanh ^{-1}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d} \sqrt {1-c^2 x^2}}\right )}{35 e^2}-\frac {b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \sqrt {1-c^2 x^2} \left (d+e x^2\right )^{5/2}}{42 c^2 e}-\frac {b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \sqrt {1-c^2 x^2} \left (13 c^2 d+25 e\right ) \left (d+e x^2\right )^{3/2}}{840 c^4 e}+\frac {b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \sqrt {1-c^2 x^2} \left (3 c^4 d^2-38 c^2 d e-25 e^2\right ) \sqrt {d+e x^2}}{560 c^6 e} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 45
Rule 65
Rule 95
Rule 159
Rule 163
Rule 209
Rule 213
Rule 223
Rule 272
Rule 587
Rule 6436
Rubi steps
\begin {align*} \int x^3 \left (d+e x^2\right )^{3/2} \left (a+b \text {sech}^{-1}(c x)\right ) \, dx &=-\frac {d \left (d+e x^2\right )^{5/2} \left (a+b \text {sech}^{-1}(c x)\right )}{5 e^2}+\frac {\left (d+e x^2\right )^{7/2} \left (a+b \text {sech}^{-1}(c x)\right )}{7 e^2}+\left (b \sqrt {\frac {1}{1+c x}} \sqrt {1+c x}\right ) \int \frac {\left (d+e x^2\right )^{5/2} \left (-2 d+5 e x^2\right )}{35 e^2 x \sqrt {1-c^2 x^2}} \, dx\\ &=-\frac {d \left (d+e x^2\right )^{5/2} \left (a+b \text {sech}^{-1}(c x)\right )}{5 e^2}+\frac {\left (d+e x^2\right )^{7/2} \left (a+b \text {sech}^{-1}(c x)\right )}{7 e^2}+\frac {\left (b \sqrt {\frac {1}{1+c x}} \sqrt {1+c x}\right ) \int \frac {\left (d+e x^2\right )^{5/2} \left (-2 d+5 e x^2\right )}{x \sqrt {1-c^2 x^2}} \, dx}{35 e^2}\\ &=-\frac {d \left (d+e x^2\right )^{5/2} \left (a+b \text {sech}^{-1}(c x)\right )}{5 e^2}+\frac {\left (d+e x^2\right )^{7/2} \left (a+b \text {sech}^{-1}(c x)\right )}{7 e^2}+\frac {\left (b \sqrt {\frac {1}{1+c x}} \sqrt {1+c x}\right ) \text {Subst}\left (\int \frac {(d+e x)^{5/2} (-2 d+5 e x)}{x \sqrt {1-c^2 x}} \, dx,x,x^2\right )}{70 e^2}\\ &=-\frac {b \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2} \left (d+e x^2\right )^{5/2}}{42 c^2 e}-\frac {d \left (d+e x^2\right )^{5/2} \left (a+b \text {sech}^{-1}(c x)\right )}{5 e^2}+\frac {\left (d+e x^2\right )^{7/2} \left (a+b \text {sech}^{-1}(c x)\right )}{7 e^2}-\frac {\left (b \sqrt {\frac {1}{1+c x}} \sqrt {1+c x}\right ) \text {Subst}\left (\int \frac {(d+e x)^{3/2} \left (6 c^2 d^2-\frac {1}{2} e \left (13 c^2 d+25 e\right ) x\right )}{x \sqrt {1-c^2 x}} \, dx,x,x^2\right )}{210 c^2 e^2}\\ &=-\frac {b \left (13 c^2 d+25 e\right ) \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2} \left (d+e x^2\right )^{3/2}}{840 c^4 e}-\frac {b \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2} \left (d+e x^2\right )^{5/2}}{42 c^2 e}-\frac {d \left (d+e x^2\right )^{5/2} \left (a+b \text {sech}^{-1}(c x)\right )}{5 e^2}+\frac {\left (d+e x^2\right )^{7/2} \left (a+b \text {sech}^{-1}(c x)\right )}{7 e^2}+\frac {\left (b \sqrt {\frac {1}{1+c x}} \sqrt {1+c x}\right ) \text {Subst}\left (\int \frac {\sqrt {d+e x} \left (-12 c^4 d^3-\frac {3}{4} e \left (3 c^4 d^2-38 c^2 d e-25 e^2\right ) x\right )}{x \sqrt {1-c^2 x}} \, dx,x,x^2\right )}{420 c^4 e^2}\\ &=\frac {b \left (3 c^4 d^2-38 c^2 d e-25 e^2\right ) \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2} \sqrt {d+e x^2}}{560 c^6 e}-\frac {b \left (13 c^2 d+25 e\right ) \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2} \left (d+e x^2\right )^{3/2}}{840 c^4 e}-\frac {b \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2} \left (d+e x^2\right )^{5/2}}{42 c^2 e}-\frac {d \left (d+e x^2\right )^{5/2} \left (a+b \text {sech}^{-1}(c x)\right )}{5 e^2}+\frac {\left (d+e x^2\right )^{7/2} \left (a+b \text {sech}^{-1}(c x)\right )}{7 e^2}-\frac {\left (b \sqrt {\frac {1}{1+c x}} \sqrt {1+c x}\right ) \text {Subst}\left (\int \frac {12 c^6 d^4+\frac {3}{8} e \left (35 c^6 d^3-35 c^4 d^2 e-63 c^2 d e^2-25 e^3\right ) x}{x \sqrt {1-c^2 x} \sqrt {d+e x}} \, dx,x,x^2\right )}{420 c^6 e^2}\\ &=\frac {b \left (3 c^4 d^2-38 c^2 d e-25 e^2\right ) \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2} \sqrt {d+e x^2}}{560 c^6 e}-\frac {b \left (13 c^2 d+25 e\right ) \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2} \left (d+e x^2\right )^{3/2}}{840 c^4 e}-\frac {b \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2} \left (d+e x^2\right )^{5/2}}{42 c^2 e}-\frac {d \left (d+e x^2\right )^{5/2} \left (a+b \text {sech}^{-1}(c x)\right )}{5 e^2}+\frac {\left (d+e x^2\right )^{7/2} \left (a+b \text {sech}^{-1}(c x)\right )}{7 e^2}-\frac {\left (b d^4 \sqrt {\frac {1}{1+c x}} \sqrt {1+c x}\right ) \text {Subst}\left (\int \frac {1}{x \sqrt {1-c^2 x} \sqrt {d+e x}} \, dx,x,x^2\right )}{35 e^2}-\frac {\left (b \left (35 c^6 d^3-35 c^4 d^2 e-63 c^2 d e^2-25 e^3\right ) \sqrt {\frac {1}{1+c x}} \sqrt {1+c x}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1-c^2 x} \sqrt {d+e x}} \, dx,x,x^2\right )}{1120 c^6 e}\\ &=\frac {b \left (3 c^4 d^2-38 c^2 d e-25 e^2\right ) \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2} \sqrt {d+e x^2}}{560 c^6 e}-\frac {b \left (13 c^2 d+25 e\right ) \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2} \left (d+e x^2\right )^{3/2}}{840 c^4 e}-\frac {b \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2} \left (d+e x^2\right )^{5/2}}{42 c^2 e}-\frac {d \left (d+e x^2\right )^{5/2} \left (a+b \text {sech}^{-1}(c x)\right )}{5 e^2}+\frac {\left (d+e x^2\right )^{7/2} \left (a+b \text {sech}^{-1}(c x)\right )}{7 e^2}-\frac {\left (2 b d^4 \sqrt {\frac {1}{1+c x}} \sqrt {1+c x}\right ) \text {Subst}\left (\int \frac {1}{-d+x^2} \, dx,x,\frac {\sqrt {d+e x^2}}{\sqrt {1-c^2 x^2}}\right )}{35 e^2}+\frac {\left (b \left (35 c^6 d^3-35 c^4 d^2 e-63 c^2 d e^2-25 e^3\right ) \sqrt {\frac {1}{1+c x}} \sqrt {1+c x}\right ) \text {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 )}{560 c^8 e}\\ &=\frac {b \left (3 c^4 d^2-38 c^2 d e-25 e^2\right ) \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2} \sqrt {d+e x^2}}{560 c^6 e}-\frac {b \left (13 c^2 d+25 e\right ) \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2} \left (d+e x^2\right )^{3/2}}{840 c^4 e}-\frac {b \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2} \left (d+e x^2\right )^{5/2}}{42 c^2 e}-\frac {d \left (d+e x^2\right )^{5/2} \left (a+b \text {sech}^{-1}(c x)\right )}{5 e^2}+\frac {\left (d+e x^2\right )^{7/2} \left (a+b \text {sech}^{-1}(c x)\right )}{7 e^2}+\frac {2 b d^{7/2} \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \tanh ^{-1}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d} \sqrt {1-c^2 x^2}}\right )}{35 e^2}+\frac {\left (b \left (35 c^6 d^3-35 c^4 d^2 e-63 c^2 d e^2-25 e^3\right ) \sqrt {\frac {1}{1+c x}} \sqrt {1+c x}\right ) \text {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 )}{560 c^8 e}\\ &=\frac {b \left (3 c^4 d^2-38 c^2 d e-25 e^2\right ) \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2} \sqrt {d+e x^2}}{560 c^6 e}-\frac {b \left (13 c^2 d+25 e\right ) \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2} \left (d+e x^2\right )^{3/2}}{840 c^4 e}-\frac {b \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2} \left (d+e x^2\right )^{5/2}}{42 c^2 e}-\frac {d \left (d+e x^2\right )^{5/2} \left (a+b \text {sech}^{-1}(c x)\right )}{5 e^2}+\frac {\left (d+e x^2\right )^{7/2} \left (a+b \text {sech}^{-1}(c x)\right )}{7 e^2}+\frac {b \left (35 c^6 d^3-35 c^4 d^2 e-63 c^2 d e^2-25 e^3\right ) \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \tan ^{-1}\left (\frac {\sqrt {e} \sqrt {1-c^2 x^2}}{c \sqrt {d+e x^2}}\right )}{560 c^7 e^{3/2}}+\frac {2 b d^{7/2} \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \tanh ^{-1}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d} \sqrt {1-c^2 x^2}}\right )}{35 e^2}\\ \end {align*}
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Mathematica [A]
time = 41.76, size = 313, normalized size = 0.75 \begin {gather*} -\frac {\sqrt {d+e x^2} \left (48 a c^6 \left (2 d-5 e x^2\right ) \left (d+e x^2\right )^2+b e \sqrt {\frac {1-c x}{1+c x}} (1+c x) \left (75 e^2+2 c^2 e \left (82 d+25 e x^2\right )+c^4 \left (57 d^2+106 d e x^2+40 e^2 x^4\right )\right )+48 b c^6 \left (2 d-5 e x^2\right ) \left (d+e x^2\right )^2 \text {sech}^{-1}(c x)\right )}{1680 c^6 e^2}-\frac {b \sqrt {\frac {1-c x}{1+c x}} \sqrt {-1+c^2 x^2} \left (-32 c^7 d^{7/2} \text {ArcTan}\left (\frac {\sqrt {d} \sqrt {-1+c^2 x^2}}{\sqrt {d+e x^2}}\right )+\sqrt {e} \left (-35 c^6 d^3+35 c^4 d^2 e+63 c^2 d e^2+25 e^3\right ) \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {-1+c^2 x^2}}{c \sqrt {d+e x^2}}\right )\right )}{560 c^7 e^2 (-1+c x)} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.78, size = 0, normalized size = 0.00 \[\int x^{3} \left (e \,x^{2}+d \right )^{\frac {3}{2}} \left (a +b \,\mathrm {arcsech}\left (c x \right )\right )\, dx\]
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 1021 vs.
\(2 (271) = 542\).
time = 1.59, size = 2076, normalized size = 4.97 \begin {gather*} \text {Too large to display} \end {gather*}
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 \begin {gather*} \int x^{3} \left (a + b \operatorname {asech}{\left (c x \right )}\right ) \left (d + e x^{2}\right )^{\frac {3}{2}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int x^3\,{\left (e\,x^2+d\right )}^{3/2}\,\left (a+b\,\mathrm {acosh}\left (\frac {1}{c\,x}\right )\right ) \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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