Optimal. Leaf size=223 \[ \frac {\left (d+e x^2\right )^{5/2} \left (a+b \tan ^{-1}(c x)\right )}{5 e^2}-\frac {d \left (d+e x^2\right )^{3/2} \left (a+b \tan ^{-1}(c x)\right )}{3 e^2}+\frac {b \left (c^2 d-e\right )^{3/2} \left (2 c^2 d+3 e\right ) \tan ^{-1}\left (\frac {x \sqrt {c^2 d-e}}{\sqrt {d+e x^2}}\right )}{15 c^5 e^2}-\frac {b x \left (c^2 d-12 e\right ) \sqrt {d+e x^2}}{120 c^3 e}+\frac {b \left (15 c^4 d^2+20 c^2 d e-24 e^2\right ) \tanh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{120 c^5 e^{3/2}}-\frac {b x \left (d+e x^2\right )^{3/2}}{20 c e} \]
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Rubi [A] time = 0.37, antiderivative size = 223, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 10, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.435, Rules used = {266, 43, 4976, 12, 528, 523, 217, 206, 377, 203} \[ \frac {\left (d+e x^2\right )^{5/2} \left (a+b \tan ^{-1}(c x)\right )}{5 e^2}-\frac {d \left (d+e x^2\right )^{3/2} \left (a+b \tan ^{-1}(c x)\right )}{3 e^2}+\frac {b \left (15 c^4 d^2+20 c^2 d e-24 e^2\right ) \tanh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{120 c^5 e^{3/2}}+\frac {b \left (c^2 d-e\right )^{3/2} \left (2 c^2 d+3 e\right ) \tan ^{-1}\left (\frac {x \sqrt {c^2 d-e}}{\sqrt {d+e x^2}}\right )}{15 c^5 e^2}-\frac {b x \left (c^2 d-12 e\right ) \sqrt {d+e x^2}}{120 c^3 e}-\frac {b x \left (d+e x^2\right )^{3/2}}{20 c e} \]
Antiderivative was successfully verified.
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Rule 12
Rule 43
Rule 203
Rule 206
Rule 217
Rule 266
Rule 377
Rule 523
Rule 528
Rule 4976
Rubi steps
\begin {align*} \int x^3 \sqrt {d+e x^2} \left (a+b \tan ^{-1}(c x)\right ) \, dx &=-\frac {d \left (d+e x^2\right )^{3/2} \left (a+b \tan ^{-1}(c x)\right )}{3 e^2}+\frac {\left (d+e x^2\right )^{5/2} \left (a+b \tan ^{-1}(c x)\right )}{5 e^2}-(b c) \int \frac {\left (d+e x^2\right )^{3/2} \left (-2 d+3 e x^2\right )}{15 e^2 \left (1+c^2 x^2\right )} \, dx\\ &=-\frac {d \left (d+e x^2\right )^{3/2} \left (a+b \tan ^{-1}(c x)\right )}{3 e^2}+\frac {\left (d+e x^2\right )^{5/2} \left (a+b \tan ^{-1}(c x)\right )}{5 e^2}-\frac {(b c) \int \frac {\left (d+e x^2\right )^{3/2} \left (-2 d+3 e x^2\right )}{1+c^2 x^2} \, dx}{15 e^2}\\ &=-\frac {b x \left (d+e x^2\right )^{3/2}}{20 c e}-\frac {d \left (d+e x^2\right )^{3/2} \left (a+b \tan ^{-1}(c x)\right )}{3 e^2}+\frac {\left (d+e x^2\right )^{5/2} \left (a+b \tan ^{-1}(c x)\right )}{5 e^2}-\frac {b \int \frac {\sqrt {d+e x^2} \left (-d \left (8 c^2 d+3 e\right )+\left (c^2 d-12 e\right ) e x^2\right )}{1+c^2 x^2} \, dx}{60 c e^2}\\ &=-\frac {b \left (c^2 d-12 e\right ) x \sqrt {d+e x^2}}{120 c^3 e}-\frac {b x \left (d+e x^2\right )^{3/2}}{20 c e}-\frac {d \left (d+e x^2\right )^{3/2} \left (a+b \tan ^{-1}(c x)\right )}{3 e^2}+\frac {\left (d+e x^2\right )^{5/2} \left (a+b \tan ^{-1}(c x)\right )}{5 e^2}-\frac {b \int \frac {-d \left (16 c^4 d^2+7 c^2 d e-12 e^2\right )-e \left (15 c^4 d^2+20 c^2 d e-24 e^2\right ) x^2}{\left (1+c^2 x^2\right ) \sqrt {d+e x^2}} \, dx}{120 c^3 e^2}\\ &=-\frac {b \left (c^2 d-12 e\right ) x \sqrt {d+e x^2}}{120 c^3 e}-\frac {b x \left (d+e x^2\right )^{3/2}}{20 c e}-\frac {d \left (d+e x^2\right )^{3/2} \left (a+b \tan ^{-1}(c x)\right )}{3 e^2}+\frac {\left (d+e x^2\right )^{5/2} \left (a+b \tan ^{-1}(c x)\right )}{5 e^2}+\frac {\left (b \left (c^2 d-e\right )^2 \left (2 c^2 d+3 e\right )\right ) \int \frac {1}{\left (1+c^2 x^2\right ) \sqrt {d+e x^2}} \, dx}{15 c^5 e^2}+\frac {\left (b \left (15 c^4 d^2+20 c^2 d e-24 e^2\right )\right ) \int \frac {1}{\sqrt {d+e x^2}} \, dx}{120 c^5 e}\\ &=-\frac {b \left (c^2 d-12 e\right ) x \sqrt {d+e x^2}}{120 c^3 e}-\frac {b x \left (d+e x^2\right )^{3/2}}{20 c e}-\frac {d \left (d+e x^2\right )^{3/2} \left (a+b \tan ^{-1}(c x)\right )}{3 e^2}+\frac {\left (d+e x^2\right )^{5/2} \left (a+b \tan ^{-1}(c x)\right )}{5 e^2}+\frac {\left (b \left (c^2 d-e\right )^2 \left (2 c^2 d+3 e\right )\right ) \operatorname {Subst}\left (\int \frac {1}{1-\left (-c^2 d+e\right ) x^2} \, dx,x,\frac {x}{\sqrt {d+e x^2}}\right )}{15 c^5 e^2}+\frac {\left (b \left (15 c^4 d^2+20 c^2 d e-24 e^2\right )\right ) \operatorname {Subst}\left (\int \frac {1}{1-e x^2} \, dx,x,\frac {x}{\sqrt {d+e x^2}}\right )}{120 c^5 e}\\ &=-\frac {b \left (c^2 d-12 e\right ) x \sqrt {d+e x^2}}{120 c^3 e}-\frac {b x \left (d+e x^2\right )^{3/2}}{20 c e}-\frac {d \left (d+e x^2\right )^{3/2} \left (a+b \tan ^{-1}(c x)\right )}{3 e^2}+\frac {\left (d+e x^2\right )^{5/2} \left (a+b \tan ^{-1}(c x)\right )}{5 e^2}+\frac {b \left (c^2 d-e\right )^{3/2} \left (2 c^2 d+3 e\right ) \tan ^{-1}\left (\frac {\sqrt {c^2 d-e} x}{\sqrt {d+e x^2}}\right )}{15 c^5 e^2}+\frac {b \left (15 c^4 d^2+20 c^2 d e-24 e^2\right ) \tanh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{120 c^5 e^{3/2}}\\ \end {align*}
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Mathematica [C] time = 0.55, size = 391, normalized size = 1.75 \[ \frac {-c^2 \sqrt {d+e x^2} \left (8 a c^3 \left (2 d^2-d e x^2-3 e^2 x^4\right )+b e x \left (c^2 \left (7 d+6 e x^2\right )-12 e\right )\right )-8 b c^5 \tan ^{-1}(c x) \sqrt {d+e x^2} \left (2 d^2-d e x^2-3 e^2 x^4\right )-4 i b \left (c^2 d-e\right )^{3/2} \left (2 c^2 d+3 e\right ) \log \left (-\frac {60 i c^6 e^2 \left (\sqrt {c^2 d-e} \sqrt {d+e x^2}+c d-i e x\right )}{b (c x+i) \left (c^2 d-e\right )^{5/2} \left (2 c^2 d+3 e\right )}\right )+4 i b \left (c^2 d-e\right )^{3/2} \left (2 c^2 d+3 e\right ) \log \left (\frac {60 i c^6 e^2 \left (\sqrt {c^2 d-e} \sqrt {d+e x^2}+c d+i e x\right )}{b (c x-i) \left (c^2 d-e\right )^{5/2} \left (2 c^2 d+3 e\right )}\right )+b \sqrt {e} \left (15 c^4 d^2+20 c^2 d e-24 e^2\right ) \log \left (\sqrt {e} \sqrt {d+e x^2}+e x\right )}{120 c^5 e^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 3.80, size = 1200, normalized size = 5.38 \[ \text {result too large to display} \]
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 \[ \mathit {sage}_{0} x \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 1.34, size = 0, normalized size = 0.00 \[ \int x^{3} \sqrt {e \,x^{2}+d}\, \left (a +b \arctan \left (c x \right )\right )\, dx \]
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 \[ \frac {1}{15} \, {\left (\frac {3 \, {\left (e x^{2} + d\right )}^{\frac {3}{2}} x^{2}}{e} - \frac {2 \, {\left (e x^{2} + d\right )}^{\frac {3}{2}} d}{e^{2}}\right )} a + b \int \sqrt {e x^{2} + d} x^{3} \arctan \left (c x\right )\,{d x} \]
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 \[ \int x^3\,\left (a+b\,\mathrm {atan}\left (c\,x\right )\right )\,\sqrt {e\,x^2+d} \,d x \]
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 \[ \int x^{3} \left (a + b \operatorname {atan}{\left (c x \right )}\right ) \sqrt {d + e x^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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