Optimal. Leaf size=326 \[ -\frac {14 d^{9/4} \sqrt {-e} \left (\sqrt {d}+\sqrt {e} x\right ) \sqrt {\frac {d+e x^2}{\left (\sqrt {d}+\sqrt {e} x\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{e} \sqrt {x}}{\sqrt [4]{d}}\right )|\frac {1}{2}\right )}{135 e^{11/4} \sqrt {d+e x^2}}+\frac {28 d^{9/4} \sqrt {-e} \left (\sqrt {d}+\sqrt {e} x\right ) \sqrt {\frac {d+e x^2}{\left (\sqrt {d}+\sqrt {e} x\right )^2}} E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{e} \sqrt {x}}{\sqrt [4]{d}}\right )|\frac {1}{2}\right )}{135 e^{11/4} \sqrt {d+e x^2}}-\frac {28 d^2 \sqrt {-e} \sqrt {x} \sqrt {d+e x^2}}{135 e^{5/2} \left (\sqrt {d}+\sqrt {e} x\right )}+\frac {4 x^{7/2} \sqrt {d+e x^2}}{81 \sqrt {-e}}+\frac {28 d x^{3/2} \sqrt {d+e x^2}}{405 (-e)^{3/2}}+\frac {2}{9} x^{9/2} \tan ^{-1}\left (\frac {\sqrt {-e} x}{\sqrt {d+e x^2}}\right ) \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.19, antiderivative size = 326, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {5151, 321, 329, 305, 220, 1196} \[ -\frac {28 d^2 \sqrt {-e} \sqrt {x} \sqrt {d+e x^2}}{135 e^{5/2} \left (\sqrt {d}+\sqrt {e} x\right )}-\frac {14 d^{9/4} \sqrt {-e} \left (\sqrt {d}+\sqrt {e} x\right ) \sqrt {\frac {d+e x^2}{\left (\sqrt {d}+\sqrt {e} x\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{e} \sqrt {x}}{\sqrt [4]{d}}\right )|\frac {1}{2}\right )}{135 e^{11/4} \sqrt {d+e x^2}}+\frac {28 d^{9/4} \sqrt {-e} \left (\sqrt {d}+\sqrt {e} x\right ) \sqrt {\frac {d+e x^2}{\left (\sqrt {d}+\sqrt {e} x\right )^2}} E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{e} \sqrt {x}}{\sqrt [4]{d}}\right )|\frac {1}{2}\right )}{135 e^{11/4} \sqrt {d+e x^2}}+\frac {4 x^{7/2} \sqrt {d+e x^2}}{81 \sqrt {-e}}+\frac {28 d x^{3/2} \sqrt {d+e x^2}}{405 (-e)^{3/2}}+\frac {2}{9} x^{9/2} \tan ^{-1}\left (\frac {\sqrt {-e} x}{\sqrt {d+e x^2}}\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 220
Rule 305
Rule 321
Rule 329
Rule 1196
Rule 5151
Rubi steps
\begin {align*} \int x^{7/2} \tan ^{-1}\left (\frac {\sqrt {-e} x}{\sqrt {d+e x^2}}\right ) \, dx &=\frac {2}{9} x^{9/2} \tan ^{-1}\left (\frac {\sqrt {-e} x}{\sqrt {d+e x^2}}\right )-\frac {1}{9} \left (2 \sqrt {-e}\right ) \int \frac {x^{9/2}}{\sqrt {d+e x^2}} \, dx\\ &=\frac {4 x^{7/2} \sqrt {d+e x^2}}{81 \sqrt {-e}}+\frac {2}{9} x^{9/2} \tan ^{-1}\left (\frac {\sqrt {-e} x}{\sqrt {d+e x^2}}\right )-\frac {(14 d) \int \frac {x^{5/2}}{\sqrt {d+e x^2}} \, dx}{81 \sqrt {-e}}\\ &=\frac {28 d x^{3/2} \sqrt {d+e x^2}}{405 (-e)^{3/2}}+\frac {4 x^{7/2} \sqrt {d+e x^2}}{81 \sqrt {-e}}+\frac {2}{9} x^{9/2} \tan ^{-1}\left (\frac {\sqrt {-e} x}{\sqrt {d+e x^2}}\right )-\frac {\left (14 d^2\right ) \int \frac {\sqrt {x}}{\sqrt {d+e x^2}} \, dx}{135 (-e)^{3/2}}\\ &=\frac {28 d x^{3/2} \sqrt {d+e x^2}}{405 (-e)^{3/2}}+\frac {4 x^{7/2} \sqrt {d+e x^2}}{81 \sqrt {-e}}+\frac {2}{9} x^{9/2} \tan ^{-1}\left (\frac {\sqrt {-e} x}{\sqrt {d+e x^2}}\right )-\frac {\left (28 d^2\right ) \operatorname {Subst}\left (\int \frac {x^2}{\sqrt {d+e x^4}} \, dx,x,\sqrt {x}\right )}{135 (-e)^{3/2}}\\ &=\frac {28 d x^{3/2} \sqrt {d+e x^2}}{405 (-e)^{3/2}}+\frac {4 x^{7/2} \sqrt {d+e x^2}}{81 \sqrt {-e}}+\frac {2}{9} x^{9/2} \tan ^{-1}\left (\frac {\sqrt {-e} x}{\sqrt {d+e x^2}}\right )-\frac {\left (28 d^{5/2} \sqrt {-e}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {d+e x^4}} \, dx,x,\sqrt {x}\right )}{135 e^{5/2}}+\frac {\left (28 d^{5/2} \sqrt {-e}\right ) \operatorname {Subst}\left (\int \frac {1-\frac {\sqrt {e} x^2}{\sqrt {d}}}{\sqrt {d+e x^4}} \, dx,x,\sqrt {x}\right )}{135 e^{5/2}}\\ &=\frac {28 d x^{3/2} \sqrt {d+e x^2}}{405 (-e)^{3/2}}+\frac {4 x^{7/2} \sqrt {d+e x^2}}{81 \sqrt {-e}}-\frac {28 d^2 \sqrt {-e} \sqrt {x} \sqrt {d+e x^2}}{135 e^{5/2} \left (\sqrt {d}+\sqrt {e} x\right )}+\frac {2}{9} x^{9/2} \tan ^{-1}\left (\frac {\sqrt {-e} x}{\sqrt {d+e x^2}}\right )+\frac {28 d^{9/4} \sqrt {-e} \left (\sqrt {d}+\sqrt {e} x\right ) \sqrt {\frac {d+e x^2}{\left (\sqrt {d}+\sqrt {e} x\right )^2}} E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{e} \sqrt {x}}{\sqrt [4]{d}}\right )|\frac {1}{2}\right )}{135 e^{11/4} \sqrt {d+e x^2}}-\frac {14 d^{9/4} \sqrt {-e} \left (\sqrt {d}+\sqrt {e} x\right ) \sqrt {\frac {d+e x^2}{\left (\sqrt {d}+\sqrt {e} x\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{e} \sqrt {x}}{\sqrt [4]{d}}\right )|\frac {1}{2}\right )}{135 e^{11/4} \sqrt {d+e x^2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] time = 0.16, size = 139, normalized size = 0.43 \[ \frac {2 x^{3/2} \left (2 \sqrt {-e} \left (7 d^2+2 d e x^2-5 e^2 x^4\right )-14 d^2 \sqrt {-e} \sqrt {\frac {e x^2}{d}+1} \, _2F_1\left (\frac {1}{2},\frac {3}{4};\frac {7}{4};-\frac {e x^2}{d}\right )+45 e^2 x^3 \sqrt {d+e x^2} \tan ^{-1}\left (\frac {\sqrt {-e} x}{\sqrt {d+e x^2}}\right )\right )}{405 e^2 \sqrt {d+e x^2}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 0.55, size = 0, normalized size = 0.00 \[ {\rm integral}\left (x^{\frac {7}{2}} \arctan \left (\frac {\sqrt {-e} x}{\sqrt {e x^{2} + d}}\right ), x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [F] time = 0.23, size = 0, normalized size = 0.00 \[ \int x^{\frac {7}{2}} \arctan \left (\frac {x \sqrt {-e}}{\sqrt {e \,x^{2}+d}}\right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: RuntimeError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int x^{7/2}\,\mathrm {atan}\left (\frac {\sqrt {-e}\,x}{\sqrt {e\,x^2+d}}\right ) \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________