Optimal. Leaf size=181 \[ \frac {\left (d+e x^2\right )^{5/2} \left (a+b \tan ^{-1}(c x)\right )}{5 e}-\frac {b \left (c^2 d-e\right )^{5/2} \tan ^{-1}\left (\frac {x \sqrt {c^2 d-e}}{\sqrt {d+e x^2}}\right )}{5 c^5 e}-\frac {b x \left (7 c^2 d-4 e\right ) \sqrt {d+e x^2}}{40 c^3}-\frac {b \left (15 c^4 d^2-20 c^2 d e+8 e^2\right ) \tanh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{40 c^5 \sqrt {e}}-\frac {b x \left (d+e x^2\right )^{3/2}}{20 c} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.23, antiderivative size = 181, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 8, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.381, Rules used = {4974, 416, 528, 523, 217, 206, 377, 203} \[ \frac {\left (d+e x^2\right )^{5/2} \left (a+b \tan ^{-1}(c x)\right )}{5 e}-\frac {b \left (15 c^4 d^2-20 c^2 d e+8 e^2\right ) \tanh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{40 c^5 \sqrt {e}}-\frac {b x \left (7 c^2 d-4 e\right ) \sqrt {d+e x^2}}{40 c^3}-\frac {b \left (c^2 d-e\right )^{5/2} \tan ^{-1}\left (\frac {x \sqrt {c^2 d-e}}{\sqrt {d+e x^2}}\right )}{5 c^5 e}-\frac {b x \left (d+e x^2\right )^{3/2}}{20 c} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 203
Rule 206
Rule 217
Rule 377
Rule 416
Rule 523
Rule 528
Rule 4974
Rubi steps
\begin {align*} \int x \left (d+e x^2\right )^{3/2} \left (a+b \tan ^{-1}(c x)\right ) \, dx &=\frac {\left (d+e x^2\right )^{5/2} \left (a+b \tan ^{-1}(c x)\right )}{5 e}-\frac {(b c) \int \frac {\left (d+e x^2\right )^{5/2}}{1+c^2 x^2} \, dx}{5 e}\\ &=-\frac {b x \left (d+e x^2\right )^{3/2}}{20 c}+\frac {\left (d+e x^2\right )^{5/2} \left (a+b \tan ^{-1}(c x)\right )}{5 e}-\frac {b \int \frac {\sqrt {d+e x^2} \left (d \left (4 c^2 d-e\right )+\left (7 c^2 d-4 e\right ) e x^2\right )}{1+c^2 x^2} \, dx}{20 c e}\\ &=-\frac {b \left (7 c^2 d-4 e\right ) x \sqrt {d+e x^2}}{40 c^3}-\frac {b x \left (d+e x^2\right )^{3/2}}{20 c}+\frac {\left (d+e x^2\right )^{5/2} \left (a+b \tan ^{-1}(c x)\right )}{5 e}-\frac {b \int \frac {d \left (8 c^4 d^2-9 c^2 d e+4 e^2\right )+e \left (15 c^4 d^2-20 c^2 d e+8 e^2\right ) x^2}{\left (1+c^2 x^2\right ) \sqrt {d+e x^2}} \, dx}{40 c^3 e}\\ &=-\frac {b \left (7 c^2 d-4 e\right ) x \sqrt {d+e x^2}}{40 c^3}-\frac {b x \left (d+e x^2\right )^{3/2}}{20 c}+\frac {\left (d+e x^2\right )^{5/2} \left (a+b \tan ^{-1}(c x)\right )}{5 e}-\frac {\left (b \left (c^2 d-e\right )^3\right ) \int \frac {1}{\left (1+c^2 x^2\right ) \sqrt {d+e x^2}} \, dx}{5 c^5 e}-\frac {\left (b \left (15 c^4 d^2-20 c^2 d e+8 e^2\right )\right ) \int \frac {1}{\sqrt {d+e x^2}} \, dx}{40 c^5}\\ &=-\frac {b \left (7 c^2 d-4 e\right ) x \sqrt {d+e x^2}}{40 c^3}-\frac {b x \left (d+e x^2\right )^{3/2}}{20 c}+\frac {\left (d+e x^2\right )^{5/2} \left (a+b \tan ^{-1}(c x)\right )}{5 e}-\frac {\left (b \left (c^2 d-e\right )^3\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 )}{5 c^5 e}-\frac {\left (b \left (15 c^4 d^2-20 c^2 d e+8 e^2\right )\right ) \operatorname {Subst}\left (\int \frac {1}{1-e x^2} \, dx,x,\frac {x}{\sqrt {d+e x^2}}\right )}{40 c^5}\\ &=-\frac {b \left (7 c^2 d-4 e\right ) x \sqrt {d+e x^2}}{40 c^3}-\frac {b x \left (d+e x^2\right )^{3/2}}{20 c}+\frac {\left (d+e x^2\right )^{5/2} \left (a+b \tan ^{-1}(c x)\right )}{5 e}-\frac {b \left (c^2 d-e\right )^{5/2} \tan ^{-1}\left (\frac {\sqrt {c^2 d-e} x}{\sqrt {d+e x^2}}\right )}{5 c^5 e}-\frac {b \left (15 c^4 d^2-20 c^2 d e+8 e^2\right ) \tanh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{40 c^5 \sqrt {e}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] time = 0.46, size = 313, normalized size = 1.73 \[ \frac {c^2 \sqrt {d+e x^2} \left (8 a c^3 \left (d+e x^2\right )^2+b e x \left (4 e-c^2 \left (9 d+2 e x^2\right )\right )\right )+8 b c^5 \tan ^{-1}(c x) \left (d+e x^2\right )^{5/2}-4 i b \left (c^2 d-e\right )^{5/2} \log \left (\frac {20 c^6 e \left (-i \sqrt {c^2 d-e} \sqrt {d+e x^2}-i c d+e x\right )}{b (c x-i) \left (c^2 d-e\right )^{7/2}}\right )+4 i b \left (c^2 d-e\right )^{5/2} \log \left (\frac {20 c^6 e \left (i \sqrt {c^2 d-e} \sqrt {d+e x^2}+i c d+e x\right )}{b (c x+i) \left (c^2 d-e\right )^{7/2}}\right )-b \sqrt {e} \left (15 c^4 d^2-20 c^2 d e+8 e^2\right ) \log \left (\sqrt {e} \sqrt {d+e x^2}+e x\right )}{40 c^5 e} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 3.63, size = 1192, normalized size = 6.59 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
maple [F] time = 0.97, size = 0, normalized size = 0.00 \[ \int x \left (e \,x^{2}+d \right )^{\frac {3}{2}} \left (a +b \arctan \left (c x \right )\right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int x\,\left (a+b\,\mathrm {atan}\left (c\,x\right )\right )\,{\left (e\,x^2+d\right )}^{3/2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int x \left (a + b \operatorname {atan}{\left (c x \right )}\right ) \left (d + e x^{2}\right )^{\frac {3}{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________