Optimal. Leaf size=175 \[ -\frac {a d^3 f^2}{2 e \left (f \sqrt {a+\frac {e^2 x^2}{f^2}}+e x\right )}+\frac {3 a d^2 f^2 \log \left (f \sqrt {a+\frac {e^2 x^2}{f^2}}+e x\right )}{2 e}+\frac {\left (f \sqrt {a+\frac {e^2 x^2}{f^2}}+d+e x\right )^4}{8 e}+\frac {a f^2 \left (f \sqrt {a+\frac {e^2 x^2}{f^2}}+d+e x\right )^2}{4 e}+\frac {a d f^2 \left (f \sqrt {a+\frac {e^2 x^2}{f^2}}+e x\right )}{e} \]
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Rubi [A] time = 0.13, antiderivative size = 175, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.080, Rules used = {2117, 893} \[ -\frac {a d^3 f^2}{2 e \left (f \sqrt {a+\frac {e^2 x^2}{f^2}}+e x\right )}+\frac {3 a d^2 f^2 \log \left (f \sqrt {a+\frac {e^2 x^2}{f^2}}+e x\right )}{2 e}+\frac {\left (f \sqrt {a+\frac {e^2 x^2}{f^2}}+d+e x\right )^4}{8 e}+\frac {a f^2 \left (f \sqrt {a+\frac {e^2 x^2}{f^2}}+d+e x\right )^2}{4 e}+\frac {a d f^2 \left (f \sqrt {a+\frac {e^2 x^2}{f^2}}+e x\right )}{e} \]
Antiderivative was successfully verified.
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Rule 893
Rule 2117
Rubi steps
\begin {align*} \int \left (d+e x+f \sqrt {a+\frac {e^2 x^2}{f^2}}\right )^3 \, dx &=\frac {\operatorname {Subst}\left (\int \frac {x^3 \left (d^2+a f^2-2 d x+x^2\right )}{(d-x)^2} \, dx,x,d+e x+f \sqrt {a+\frac {e^2 x^2}{f^2}}\right )}{2 e}\\ &=\frac {\operatorname {Subst}\left (\int \left (2 a d f^2+\frac {a d^3 f^2}{(d-x)^2}-\frac {3 a d^2 f^2}{d-x}+a f^2 x+x^3\right ) \, dx,x,d+e x+f \sqrt {a+\frac {e^2 x^2}{f^2}}\right )}{2 e}\\ &=-\frac {a d^3 f^2}{2 e \left (e x+f \sqrt {a+\frac {e^2 x^2}{f^2}}\right )}+\frac {a d f^2 \left (e x+f \sqrt {a+\frac {e^2 x^2}{f^2}}\right )}{e}+\frac {a f^2 \left (d+e x+f \sqrt {a+\frac {e^2 x^2}{f^2}}\right )^2}{4 e}+\frac {\left (d+e x+f \sqrt {a+\frac {e^2 x^2}{f^2}}\right )^4}{8 e}+\frac {3 a d^2 f^2 \log \left (e x+f \sqrt {a+\frac {e^2 x^2}{f^2}}\right )}{2 e}\\ \end {align*}
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Mathematica [A] time = 0.32, size = 158, normalized size = 0.90 \[ \frac {-\frac {4 a d^3 f^2}{f \sqrt {a+\frac {e^2 x^2}{f^2}}+e x}+12 a d^2 f^2 \log \left (f \sqrt {a+\frac {e^2 x^2}{f^2}}+e x\right )+\left (f \sqrt {a+\frac {e^2 x^2}{f^2}}+d+e x\right )^4+2 a f^2 \left (f \sqrt {a+\frac {e^2 x^2}{f^2}}+d+e x\right )^2+8 a d f^2 \left (f \sqrt {a+\frac {e^2 x^2}{f^2}}+e x\right )}{8 e} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.42, size = 161, normalized size = 0.92 \[ \frac {2 \, e^{4} x^{4} + 4 \, d e^{3} x^{3} - 3 \, a d^{2} f^{2} \log \left (-e x + f \sqrt {\frac {e^{2} x^{2} + a f^{2}}{f^{2}}}\right ) + 3 \, {\left (a e^{2} f^{2} + d^{2} e^{2}\right )} x^{2} + 2 \, {\left (3 \, a d e f^{2} + d^{3} e\right )} x + {\left (2 \, e^{3} f x^{3} + 4 \, d e^{2} f x^{2} + 4 \, a d f^{3} + {\left (2 \, a e f^{3} + 3 \, d^{2} e f\right )} x\right )} \sqrt {\frac {e^{2} x^{2} + a f^{2}}{f^{2}}}}{2 \, e} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.45, size = 163, normalized size = 0.93 \[ -\frac {3}{2} \, a d^{2} f {\left | f \right |} e^{\left (-1\right )} \log \left ({\left | -x e + \sqrt {a f^{2} + x^{2} e^{2}} \right |}\right ) + \frac {3}{2} \, a f^{2} x^{2} e + 3 \, a d f^{2} x + x^{4} e^{3} + 2 \, d x^{3} e^{2} + \frac {3}{2} \, d^{2} x^{2} e + d^{3} x + \frac {1}{2} \, {\left (4 \, a d f {\left | f \right |} e^{\left (-1\right )} + {\left (2 \, {\left (\frac {x {\left | f \right |} e^{2}}{f} + \frac {2 \, d {\left | f \right |} e}{f}\right )} x + \frac {{\left (2 \, a f^{4} {\left | f \right |} e^{4} + 3 \, d^{2} f^{2} {\left | f \right |} e^{4}\right )} e^{\left (-4\right )}}{f^{3}}\right )} x\right )} \sqrt {a f^{2} + x^{2} e^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 175, normalized size = 1.00 \[ e^{3} x^{4}+\frac {3 a e \,f^{2} x^{2}}{2}+2 d \,e^{2} x^{3}+\frac {3 a \,d^{2} f \ln \left (\frac {e^{2} x}{\sqrt {\frac {e^{2}}{f^{2}}}\, f^{2}}+\sqrt {\frac {e^{2} x^{2}}{f^{2}}+a}\right )}{2 \sqrt {\frac {e^{2}}{f^{2}}}}+3 a d \,f^{2} x +\frac {3 d^{2} e \,x^{2}}{2}+d^{3} x +\frac {3 \sqrt {\frac {e^{2} x^{2}}{f^{2}}+a}\, d^{2} f x}{2}+\left (\frac {e^{2} x^{2}}{f^{2}}+a \right )^{\frac {3}{2}} f^{3} x +\frac {d^{4}}{4 e}+\frac {2 \left (\frac {e^{2} x^{2}+a \,f^{2}}{f^{2}}\right )^{\frac {3}{2}} d \,f^{3}}{e} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.94, size = 266, normalized size = 1.52 \[ \frac {1}{4} \, e^{3} x^{4} + \frac {3 \, {\left (\frac {e^{2} x^{2}}{f^{2}} + a\right )}^{2} f^{4}}{4 \, e} - \frac {3}{8} \, {\left (\frac {a^{2} f^{3} \operatorname {arsinh}\left (\frac {e x}{\sqrt {a} f}\right )}{e^{3}} - \frac {2 \, {\left (\frac {e^{2} x^{2}}{f^{2}} + a\right )}^{\frac {3}{2}} f^{2} x}{e^{2}} + \frac {\sqrt {\frac {e^{2} x^{2}}{f^{2}} + a} a f^{2} x}{e^{2}}\right )} e^{2} f + \frac {1}{8} \, {\left (\frac {3 \, a^{2} f \operatorname {arsinh}\left (\frac {e x}{\sqrt {a} f}\right )}{e} + 2 \, {\left (\frac {e^{2} x^{2}}{f^{2}} + a\right )}^{\frac {3}{2}} x + 3 \, \sqrt {\frac {e^{2} x^{2}}{f^{2}} + a} a x\right )} f^{3} + d^{3} x + \frac {3}{2} \, {\left (e x^{2} + {\left (\frac {a f \operatorname {arsinh}\left (\frac {e x}{\sqrt {a} f}\right )}{e} + \sqrt {\frac {e^{2} x^{2}}{f^{2}} + a} x\right )} f\right )} d^{2} + {\left (e^{2} x^{3} + \frac {2 \, {\left (\frac {e^{2} x^{2}}{f^{2}} + a\right )}^{\frac {3}{2}} f^{3}}{e} + {\left (\frac {e^{2} x^{3}}{f^{2}} + 3 \, a x\right )} f^{2}\right )} d \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int {\left (d+e\,x+f\,\sqrt {a+\frac {e^2\,x^2}{f^2}}\right )}^3 \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 10.46, size = 279, normalized size = 1.59 \[ \frac {a^{\frac {3}{2}} f^{3} x \sqrt {1 + \frac {e^{2} x^{2}}{a f^{2}}}}{2} + \frac {a^{\frac {3}{2}} f^{3} x}{2 \sqrt {1 + \frac {e^{2} x^{2}}{a f^{2}}}} + \frac {3 \sqrt {a} d^{2} f x \sqrt {1 + \frac {e^{2} x^{2}}{a f^{2}}}}{2} + \frac {3 \sqrt {a} e^{2} f x^{3}}{2 \sqrt {1 + \frac {e^{2} x^{2}}{a f^{2}}}} + \frac {3 a d^{2} f^{2} \operatorname {asinh}{\left (\frac {e x}{\sqrt {a} f} \right )}}{2 e} + 3 a d f^{2} x + \frac {3 a e f^{2} x^{2}}{2} + d^{3} x + \frac {3 d^{2} e x^{2}}{2} + 2 d e^{2} x^{3} + 6 d e f \left (\begin {cases} \frac {\sqrt {a} x^{2}}{2} & \text {for}\: e^{2} = 0 \\\frac {f^{2} \left (a + \frac {e^{2} x^{2}}{f^{2}}\right )^{\frac {3}{2}}}{3 e^{2}} & \text {otherwise} \end {cases}\right ) + e^{3} x^{4} + \frac {e^{4} x^{5}}{\sqrt {a} f \sqrt {1 + \frac {e^{2} x^{2}}{a f^{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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