Optimal. Leaf size=279 \[ \frac {b \left (3 c^4 d^2+54 c^2 d e-40 e^2\right ) x \sqrt {d+e x^2}}{560 c^5 e}-\frac {b \left (13 c^2 d-30 e\right ) x \left (d+e x^2\right )^{3/2}}{840 c^3 e}-\frac {b x \left (d+e x^2\right )^{5/2}}{42 c e}-\frac {d \left (d+e x^2\right )^{5/2} (a+b \text {ArcTan}(c x))}{5 e^2}+\frac {\left (d+e x^2\right )^{7/2} (a+b \text {ArcTan}(c x))}{7 e^2}+\frac {b \left (c^2 d-e\right )^{5/2} \left (2 c^2 d+5 e\right ) \text {ArcTan}\left (\frac {\sqrt {c^2 d-e} x}{\sqrt {d+e x^2}}\right )}{35 c^7 e^2}+\frac {b \left (35 c^6 d^3+70 c^4 d^2 e-168 c^2 d e^2+80 e^3\right ) \tanh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{560 c^7 e^{3/2}} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.33, antiderivative size = 279, normalized size of antiderivative = 1.00, number of steps
used = 10, number of rules used = 10, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.435, Rules used = {272, 45,
5096, 12, 542, 537, 223, 212, 385, 209} \begin {gather*} \frac {\left (d+e x^2\right )^{7/2} (a+b \text {ArcTan}(c x))}{7 e^2}-\frac {d \left (d+e x^2\right )^{5/2} (a+b \text {ArcTan}(c x))}{5 e^2}+\frac {b \left (c^2 d-e\right )^{5/2} \left (2 c^2 d+5 e\right ) \text {ArcTan}\left (\frac {x \sqrt {c^2 d-e}}{\sqrt {d+e x^2}}\right )}{35 c^7 e^2}-\frac {b x \left (13 c^2 d-30 e\right ) \left (d+e x^2\right )^{3/2}}{840 c^3 e}+\frac {b x \left (3 c^4 d^2+54 c^2 d e-40 e^2\right ) \sqrt {d+e x^2}}{560 c^5 e}+\frac {b \left (35 c^6 d^3+70 c^4 d^2 e-168 c^2 d e^2+80 e^3\right ) \tanh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{560 c^7 e^{3/2}}-\frac {b x \left (d+e x^2\right )^{5/2}}{42 c e} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 12
Rule 45
Rule 209
Rule 212
Rule 223
Rule 272
Rule 385
Rule 537
Rule 542
Rule 5096
Rubi steps
\begin {align*} \int x^3 \left (d+e x^2\right )^{3/2} \left (a+b \tan ^{-1}(c x)\right ) \, dx &=-\frac {d \left (d+e x^2\right )^{5/2} \left (a+b \tan ^{-1}(c x)\right )}{5 e^2}+\frac {\left (d+e x^2\right )^{7/2} \left (a+b \tan ^{-1}(c x)\right )}{7 e^2}-(b c) \int \frac {\left (d+e x^2\right )^{5/2} \left (-2 d+5 e x^2\right )}{35 e^2 \left (1+c^2 x^2\right )} \, dx\\ &=-\frac {d \left (d+e x^2\right )^{5/2} \left (a+b \tan ^{-1}(c x)\right )}{5 e^2}+\frac {\left (d+e x^2\right )^{7/2} \left (a+b \tan ^{-1}(c x)\right )}{7 e^2}-\frac {(b c) \int \frac {\left (d+e x^2\right )^{5/2} \left (-2 d+5 e x^2\right )}{1+c^2 x^2} \, dx}{35 e^2}\\ &=-\frac {b x \left (d+e x^2\right )^{5/2}}{42 c e}-\frac {d \left (d+e x^2\right )^{5/2} \left (a+b \tan ^{-1}(c x)\right )}{5 e^2}+\frac {\left (d+e x^2\right )^{7/2} \left (a+b \tan ^{-1}(c x)\right )}{7 e^2}-\frac {b \int \frac {\left (d+e x^2\right )^{3/2} \left (-d \left (12 c^2 d+5 e\right )+\left (13 c^2 d-30 e\right ) e x^2\right )}{1+c^2 x^2} \, dx}{210 c e^2}\\ &=-\frac {b \left (13 c^2 d-30 e\right ) x \left (d+e x^2\right )^{3/2}}{840 c^3 e}-\frac {b x \left (d+e x^2\right )^{5/2}}{42 c e}-\frac {d \left (d+e x^2\right )^{5/2} \left (a+b \tan ^{-1}(c x)\right )}{5 e^2}+\frac {\left (d+e x^2\right )^{7/2} \left (a+b \tan ^{-1}(c x)\right )}{7 e^2}-\frac {b \int \frac {\sqrt {d+e x^2} \left (-3 d \left (16 c^4 d^2+11 c^2 d e-10 e^2\right )-3 e \left (3 c^4 d^2+54 c^2 d e-40 e^2\right ) x^2\right )}{1+c^2 x^2} \, dx}{840 c^3 e^2}\\ &=\frac {b \left (3 c^4 d^2+54 c^2 d e-40 e^2\right ) x \sqrt {d+e x^2}}{560 c^5 e}-\frac {b \left (13 c^2 d-30 e\right ) x \left (d+e x^2\right )^{3/2}}{840 c^3 e}-\frac {b x \left (d+e x^2\right )^{5/2}}{42 c e}-\frac {d \left (d+e x^2\right )^{5/2} \left (a+b \tan ^{-1}(c x)\right )}{5 e^2}+\frac {\left (d+e x^2\right )^{7/2} \left (a+b \tan ^{-1}(c x)\right )}{7 e^2}-\frac {b \int \frac {-3 d \left (32 c^6 d^3+19 c^4 d^2 e-74 c^2 d e^2+40 e^3\right )-3 e \left (35 c^6 d^3+70 c^4 d^2 e-168 c^2 d e^2+80 e^3\right ) x^2}{\left (1+c^2 x^2\right ) \sqrt {d+e x^2}} \, dx}{1680 c^5 e^2}\\ &=\frac {b \left (3 c^4 d^2+54 c^2 d e-40 e^2\right ) x \sqrt {d+e x^2}}{560 c^5 e}-\frac {b \left (13 c^2 d-30 e\right ) x \left (d+e x^2\right )^{3/2}}{840 c^3 e}-\frac {b x \left (d+e x^2\right )^{5/2}}{42 c e}-\frac {d \left (d+e x^2\right )^{5/2} \left (a+b \tan ^{-1}(c x)\right )}{5 e^2}+\frac {\left (d+e x^2\right )^{7/2} \left (a+b \tan ^{-1}(c x)\right )}{7 e^2}+\frac {\left (b \left (c^2 d-e\right )^3 \left (2 c^2 d+5 e\right )\right ) \int \frac {1}{\left (1+c^2 x^2\right ) \sqrt {d+e x^2}} \, dx}{35 c^7 e^2}+\frac {\left (b \left (35 c^6 d^3+70 c^4 d^2 e-168 c^2 d e^2+80 e^3\right )\right ) \int \frac {1}{\sqrt {d+e x^2}} \, dx}{560 c^7 e}\\ &=\frac {b \left (3 c^4 d^2+54 c^2 d e-40 e^2\right ) x \sqrt {d+e x^2}}{560 c^5 e}-\frac {b \left (13 c^2 d-30 e\right ) x \left (d+e x^2\right )^{3/2}}{840 c^3 e}-\frac {b x \left (d+e x^2\right )^{5/2}}{42 c e}-\frac {d \left (d+e x^2\right )^{5/2} \left (a+b \tan ^{-1}(c x)\right )}{5 e^2}+\frac {\left (d+e x^2\right )^{7/2} \left (a+b \tan ^{-1}(c x)\right )}{7 e^2}+\frac {\left (b \left (c^2 d-e\right )^3 \left (2 c^2 d+5 e\right )\right ) \text {Subst}\left (\int \frac {1}{1-\left (-c^2 d+e\right ) x^2} \, dx,x,\frac {x}{\sqrt {d+e x^2}}\right )}{35 c^7 e^2}+\frac {\left (b \left (35 c^6 d^3+70 c^4 d^2 e-168 c^2 d e^2+80 e^3\right )\right ) \text {Subst}\left (\int \frac {1}{1-e x^2} \, dx,x,\frac {x}{\sqrt {d+e x^2}}\right )}{560 c^7 e}\\ &=\frac {b \left (3 c^4 d^2+54 c^2 d e-40 e^2\right ) x \sqrt {d+e x^2}}{560 c^5 e}-\frac {b \left (13 c^2 d-30 e\right ) x \left (d+e x^2\right )^{3/2}}{840 c^3 e}-\frac {b x \left (d+e x^2\right )^{5/2}}{42 c e}-\frac {d \left (d+e x^2\right )^{5/2} \left (a+b \tan ^{-1}(c x)\right )}{5 e^2}+\frac {\left (d+e x^2\right )^{7/2} \left (a+b \tan ^{-1}(c x)\right )}{7 e^2}+\frac {b \left (c^2 d-e\right )^{5/2} \left (2 c^2 d+5 e\right ) \tan ^{-1}\left (\frac {\sqrt {c^2 d-e} x}{\sqrt {d+e x^2}}\right )}{35 c^7 e^2}+\frac {b \left (35 c^6 d^3+70 c^4 d^2 e-168 c^2 d e^2+80 e^3\right ) \tanh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{560 c^7 e^{3/2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] Result contains complex when optimal does not.
time = 0.47, size = 418, normalized size = 1.50 \begin {gather*} -\frac {c^2 \sqrt {d+e x^2} \left (48 a c^5 \left (2 d-5 e x^2\right ) \left (d+e x^2\right )^2+b e x \left (120 e^2-6 c^2 e \left (37 d+10 e x^2\right )+c^4 \left (57 d^2+106 d e x^2+40 e^2 x^4\right )\right )\right )+48 b c^7 \left (2 d-5 e x^2\right ) \left (d+e x^2\right )^{5/2} \text {ArcTan}(c x)+24 i b \left (c^2 d-e\right )^{5/2} \left (2 c^2 d+5 e\right ) \log \left (-\frac {140 i c^8 e^2 \left (c d-i e x+\sqrt {c^2 d-e} \sqrt {d+e x^2}\right )}{b \left (c^2 d-e\right )^{7/2} \left (2 c^2 d+5 e\right ) (i+c x)}\right )-24 i b \left (c^2 d-e\right )^{5/2} \left (2 c^2 d+5 e\right ) \log \left (\frac {140 i c^8 e^2 \left (c d+i e x+\sqrt {c^2 d-e} \sqrt {d+e x^2}\right )}{b \left (c^2 d-e\right )^{7/2} \left (2 c^2 d+5 e\right ) (-i+c x)}\right )-3 b \sqrt {e} \left (35 c^6 d^3+70 c^4 d^2 e-168 c^2 d e^2+80 e^3\right ) \log \left (e x+\sqrt {e} \sqrt {d+e x^2}\right )}{1680 c^7 e^2} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F]
time = 0.13, size = 0, normalized size = 0.00 \[\int x^{3} \left (e \,x^{2}+d \right )^{\frac {3}{2}} \left (a +b \arctan \left (c x \right )\right )\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F]
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.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 17.15, size = 783, normalized size = 2.81 \begin {gather*} \left [\frac {{\left (3 \, {\left (35 \, b c^{6} d^{3} + 70 \, b c^{4} d^{2} e - 168 \, b c^{2} d e^{2} + 80 \, b e^{3}\right )} e^{\frac {1}{2}} \log \left (-2 \, x^{2} e - 2 \, \sqrt {x^{2} e + d} x e^{\frac {1}{2}} - d\right ) + 24 \, {\left (2 \, b c^{6} d^{3} + b c^{4} d^{2} e - 8 \, b c^{2} d e^{2} + 5 \, b e^{3}\right )} \sqrt {-c^{2} d + e} \log \left (\frac {c^{4} d^{2} x^{4} - 6 \, c^{2} d^{2} x^{2} + 8 \, x^{4} e^{2} + 4 \, {\left (c^{2} d x^{3} - 2 \, x^{3} e - d x\right )} \sqrt {-c^{2} d + e} \sqrt {x^{2} e + d} + d^{2} - 8 \, {\left (c^{2} d x^{4} - d x^{2}\right )} e}{c^{4} x^{4} + 2 \, c^{2} x^{2} + 1}\right ) - 2 \, {\left (96 \, a c^{7} d^{3} - 48 \, {\left (5 \, b c^{7} x^{6} e^{3} + 8 \, b c^{7} d x^{4} e^{2} + b c^{7} d^{2} x^{2} e - 2 \, b c^{7} d^{3}\right )} \arctan \left (c x\right ) - 20 \, {\left (12 \, a c^{7} x^{6} - 2 \, b c^{6} x^{5} + 3 \, b c^{4} x^{3} - 6 \, b c^{2} x\right )} e^{3} - 2 \, {\left (192 \, a c^{7} d x^{4} - 53 \, b c^{6} d x^{3} + 111 \, b c^{4} d x\right )} e^{2} - 3 \, {\left (16 \, a c^{7} d^{2} x^{2} - 19 \, b c^{6} d^{2} x\right )} e\right )} \sqrt {x^{2} e + d}\right )} e^{\left (-2\right )}}{3360 \, c^{7}}, \frac {{\left (3 \, {\left (35 \, b c^{6} d^{3} + 70 \, b c^{4} d^{2} e - 168 \, b c^{2} d e^{2} + 80 \, b e^{3}\right )} e^{\frac {1}{2}} \log \left (-2 \, x^{2} e - 2 \, \sqrt {x^{2} e + d} x e^{\frac {1}{2}} - d\right ) + 48 \, {\left (2 \, b c^{6} d^{3} + b c^{4} d^{2} e - 8 \, b c^{2} d e^{2} + 5 \, b e^{3}\right )} \sqrt {c^{2} d - e} \arctan \left (\frac {{\left (c^{2} d x^{2} - 2 \, x^{2} e - d\right )} \sqrt {c^{2} d - e} \sqrt {x^{2} e + d}}{2 \, {\left (c^{2} d^{2} x - x^{3} e^{2} + {\left (c^{2} d x^{3} - d x\right )} e\right )}}\right ) - 2 \, {\left (96 \, a c^{7} d^{3} - 48 \, {\left (5 \, b c^{7} x^{6} e^{3} + 8 \, b c^{7} d x^{4} e^{2} + b c^{7} d^{2} x^{2} e - 2 \, b c^{7} d^{3}\right )} \arctan \left (c x\right ) - 20 \, {\left (12 \, a c^{7} x^{6} - 2 \, b c^{6} x^{5} + 3 \, b c^{4} x^{3} - 6 \, b c^{2} x\right )} e^{3} - 2 \, {\left (192 \, a c^{7} d x^{4} - 53 \, b c^{6} d x^{3} + 111 \, b c^{4} d x\right )} e^{2} - 3 \, {\left (16 \, a c^{7} d^{2} x^{2} - 19 \, b c^{6} d^{2} x\right )} e\right )} \sqrt {x^{2} e + d}\right )} e^{\left (-2\right )}}{3360 \, c^{7}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int x^{3} \left (a + b \operatorname {atan}{\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.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int x^3\,\left (a+b\,\mathrm {atan}\left (c\,x\right )\right )\,{\left (e\,x^2+d\right )}^{3/2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________