Optimal. Leaf size=262 \[ \frac {\left (d+e x^2\right )^{5/2} \left (a+b \sec ^{-1}(c x)\right )}{5 e}+\frac {b c d^{5/2} x \tan ^{-1}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d} \sqrt {c^2 x^2-1}}\right )}{5 e \sqrt {c^2 x^2}}-\frac {b x \sqrt {c^2 x^2-1} \left (d+e x^2\right )^{3/2}}{20 c \sqrt {c^2 x^2}}-\frac {b x \left (15 c^4 d^2+10 c^2 d e+3 e^2\right ) \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {c^2 x^2-1}}{c \sqrt {d+e x^2}}\right )}{40 c^4 \sqrt {e} \sqrt {c^2 x^2}}-\frac {b x \sqrt {c^2 x^2-1} \left (7 c^2 d+3 e\right ) \sqrt {d+e x^2}}{40 c^3 \sqrt {c^2 x^2}} \]
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Rubi [A] time = 0.27, antiderivative size = 262, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 10, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.476, Rules used = {5236, 446, 102, 154, 157, 63, 217, 206, 93, 204} \[ \frac {\left (d+e x^2\right )^{5/2} \left (a+b \sec ^{-1}(c x)\right )}{5 e}-\frac {b x \left (15 c^4 d^2+10 c^2 d e+3 e^2\right ) \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {c^2 x^2-1}}{c \sqrt {d+e x^2}}\right )}{40 c^4 \sqrt {e} \sqrt {c^2 x^2}}+\frac {b c d^{5/2} x \tan ^{-1}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d} \sqrt {c^2 x^2-1}}\right )}{5 e \sqrt {c^2 x^2}}-\frac {b x \sqrt {c^2 x^2-1} \left (d+e x^2\right )^{3/2}}{20 c \sqrt {c^2 x^2}}-\frac {b x \sqrt {c^2 x^2-1} \left (7 c^2 d+3 e\right ) \sqrt {d+e x^2}}{40 c^3 \sqrt {c^2 x^2}} \]
Antiderivative was successfully verified.
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Rule 63
Rule 93
Rule 102
Rule 154
Rule 157
Rule 204
Rule 206
Rule 217
Rule 446
Rule 5236
Rubi steps
\begin {align*} \int x \left (d+e x^2\right )^{3/2} \left (a+b \sec ^{-1}(c x)\right ) \, dx &=\frac {\left (d+e x^2\right )^{5/2} \left (a+b \sec ^{-1}(c x)\right )}{5 e}-\frac {(b c x) \int \frac {\left (d+e x^2\right )^{5/2}}{x \sqrt {-1+c^2 x^2}} \, dx}{5 e \sqrt {c^2 x^2}}\\ &=\frac {\left (d+e x^2\right )^{5/2} \left (a+b \sec ^{-1}(c x)\right )}{5 e}-\frac {(b c x) \operatorname {Subst}\left (\int \frac {(d+e x)^{5/2}}{x \sqrt {-1+c^2 x}} \, dx,x,x^2\right )}{10 e \sqrt {c^2 x^2}}\\ &=-\frac {b x \sqrt {-1+c^2 x^2} \left (d+e x^2\right )^{3/2}}{20 c \sqrt {c^2 x^2}}+\frac {\left (d+e x^2\right )^{5/2} \left (a+b \sec ^{-1}(c x)\right )}{5 e}-\frac {(b x) \operatorname {Subst}\left (\int \frac {\sqrt {d+e x} \left (2 c^2 d^2+\frac {1}{2} e \left (7 c^2 d+3 e\right ) x\right )}{x \sqrt {-1+c^2 x}} \, dx,x,x^2\right )}{20 c e \sqrt {c^2 x^2}}\\ &=-\frac {b \left (7 c^2 d+3 e\right ) x \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}}{40 c^3 \sqrt {c^2 x^2}}-\frac {b x \sqrt {-1+c^2 x^2} \left (d+e x^2\right )^{3/2}}{20 c \sqrt {c^2 x^2}}+\frac {\left (d+e x^2\right )^{5/2} \left (a+b \sec ^{-1}(c x)\right )}{5 e}-\frac {(b x) \operatorname {Subst}\left (\int \frac {2 c^4 d^3+\frac {1}{4} e \left (15 c^4 d^2+10 c^2 d e+3 e^2\right ) x}{x \sqrt {-1+c^2 x} \sqrt {d+e x}} \, dx,x,x^2\right )}{20 c^3 e \sqrt {c^2 x^2}}\\ &=-\frac {b \left (7 c^2 d+3 e\right ) x \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}}{40 c^3 \sqrt {c^2 x^2}}-\frac {b x \sqrt {-1+c^2 x^2} \left (d+e x^2\right )^{3/2}}{20 c \sqrt {c^2 x^2}}+\frac {\left (d+e x^2\right )^{5/2} \left (a+b \sec ^{-1}(c x)\right )}{5 e}-\frac {\left (b c d^3 x\right ) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {-1+c^2 x} \sqrt {d+e x}} \, dx,x,x^2\right )}{10 e \sqrt {c^2 x^2}}-\frac {\left (b \left (15 c^4 d^2+10 c^2 d e+3 e^2\right ) x\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {-1+c^2 x} \sqrt {d+e x}} \, dx,x,x^2\right )}{80 c^3 \sqrt {c^2 x^2}}\\ &=-\frac {b \left (7 c^2 d+3 e\right ) x \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}}{40 c^3 \sqrt {c^2 x^2}}-\frac {b x \sqrt {-1+c^2 x^2} \left (d+e x^2\right )^{3/2}}{20 c \sqrt {c^2 x^2}}+\frac {\left (d+e x^2\right )^{5/2} \left (a+b \sec ^{-1}(c x)\right )}{5 e}-\frac {\left (b c d^3 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 )}{5 e \sqrt {c^2 x^2}}-\frac {\left (b \left (15 c^4 d^2+10 c^2 d e+3 e^2\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 )}{40 c^5 \sqrt {c^2 x^2}}\\ &=-\frac {b \left (7 c^2 d+3 e\right ) x \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}}{40 c^3 \sqrt {c^2 x^2}}-\frac {b x \sqrt {-1+c^2 x^2} \left (d+e x^2\right )^{3/2}}{20 c \sqrt {c^2 x^2}}+\frac {\left (d+e x^2\right )^{5/2} \left (a+b \sec ^{-1}(c x)\right )}{5 e}+\frac {b c d^{5/2} x \tan ^{-1}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d} \sqrt {-1+c^2 x^2}}\right )}{5 e \sqrt {c^2 x^2}}-\frac {\left (b \left (15 c^4 d^2+10 c^2 d e+3 e^2\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 )}{40 c^5 \sqrt {c^2 x^2}}\\ &=-\frac {b \left (7 c^2 d+3 e\right ) x \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}}{40 c^3 \sqrt {c^2 x^2}}-\frac {b x \sqrt {-1+c^2 x^2} \left (d+e x^2\right )^{3/2}}{20 c \sqrt {c^2 x^2}}+\frac {\left (d+e x^2\right )^{5/2} \left (a+b \sec ^{-1}(c x)\right )}{5 e}+\frac {b c d^{5/2} x \tan ^{-1}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d} \sqrt {-1+c^2 x^2}}\right )}{5 e \sqrt {c^2 x^2}}-\frac {b \left (15 c^4 d^2+10 c^2 d e+3 e^2\right ) x \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {-1+c^2 x^2}}{c \sqrt {d+e x^2}}\right )}{40 c^4 \sqrt {e} \sqrt {c^2 x^2}}\\ \end {align*}
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Mathematica [A] time = 0.77, size = 305, normalized size = 1.16 \[ \frac {\sqrt {d+e x^2} \left (8 a c^3 \left (d+e x^2\right )^2+8 b c^3 \sec ^{-1}(c x) \left (d+e x^2\right )^2-b e x \sqrt {1-\frac {1}{c^2 x^2}} \left (c^2 \left (9 d+2 e x^2\right )+3 e\right )\right )}{40 c^3 e}-\frac {b x \sqrt {1-\frac {1}{c^2 x^2}} \left (8 c^7 d^{5/2} \sqrt {d+e x^2} \tan ^{-1}\left (\frac {\sqrt {d} \sqrt {c^2 x^2-1}}{\sqrt {d+e x^2}}\right )+\sqrt {c^2} \sqrt {e} \sqrt {c^2 d+e} \left (15 c^4 d^2+10 c^2 d e+3 e^2\right ) \sqrt {\frac {c^2 \left (d+e x^2\right )}{c^2 d+e}} \sinh ^{-1}\left (\frac {c \sqrt {e} \sqrt {c^2 x^2-1}}{\sqrt {c^2} \sqrt {c^2 d+e}}\right )\right )}{40 c^6 e \sqrt {c^2 x^2-1} \sqrt {d+e x^2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 3.17, size = 1377, normalized size = 5.26 \[ \text {result too large to display} \]
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 \[ \int {\left (e x^{2} + d\right )}^{\frac {3}{2}} {\left (b \operatorname {arcsec}\left (c x\right ) + a\right )} x\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 3.27, size = 0, normalized size = 0.00 \[ \int x \left (e \,x^{2}+d \right )^{\frac {3}{2}} \left (a +b \,\mathrm {arcsec}\left (c x \right )\right )\, dx \]
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 \[ \text {result too large to display} \]
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 \[ \int x\,{\left (e\,x^2+d\right )}^{3/2}\,\left (a+b\,\mathrm {acos}\left (\frac {1}{c\,x}\right )\right ) \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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