Optimal. Leaf size=183 \[ \frac {\left (f \sqrt {a+\frac {e^2 x^2}{f^2}}+d+e x\right )^{5/2}}{5 e}+\frac {a f^2 \sqrt {f \sqrt {a+\frac {e^2 x^2}{f^2}}+d+e x}}{e}-\frac {a d f^2 \sqrt {f \sqrt {a+\frac {e^2 x^2}{f^2}}+d+e x}}{2 e \left (f \sqrt {a+\frac {e^2 x^2}{f^2}}+e x\right )}-\frac {3 a \sqrt {d} f^2 \tanh ^{-1}\left (\frac {\sqrt {f \sqrt {a+\frac {e^2 x^2}{f^2}}+d+e x}}{\sqrt {d}}\right )}{2 e} \]
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Rubi [A] time = 0.15, antiderivative size = 183, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.185, Rules used = {2117, 897, 1257, 1810, 206} \[ \frac {\left (f \sqrt {a+\frac {e^2 x^2}{f^2}}+d+e x\right )^{5/2}}{5 e}+\frac {a f^2 \sqrt {f \sqrt {a+\frac {e^2 x^2}{f^2}}+d+e x}}{e}-\frac {a d f^2 \sqrt {f \sqrt {a+\frac {e^2 x^2}{f^2}}+d+e x}}{2 e \left (f \sqrt {a+\frac {e^2 x^2}{f^2}}+e x\right )}-\frac {3 a \sqrt {d} f^2 \tanh ^{-1}\left (\frac {\sqrt {f \sqrt {a+\frac {e^2 x^2}{f^2}}+d+e x}}{\sqrt {d}}\right )}{2 e} \]
Antiderivative was successfully verified.
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Rule 206
Rule 897
Rule 1257
Rule 1810
Rule 2117
Rubi steps
\begin {align*} \int \left (d+e x+f \sqrt {a+\frac {e^2 x^2}{f^2}}\right )^{3/2} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {x^{3/2} \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 \frac {x^4 \left (d^2+a f^2-2 d x^2+x^4\right )}{\left (d-x^2\right )^2} \, dx,x,\sqrt {d+e x+f \sqrt {a+\frac {e^2 x^2}{f^2}}}\right )}{e}\\ &=-\frac {a d f^2 \sqrt {d+e x+f \sqrt {a+\frac {e^2 x^2}{f^2}}}}{2 e \left (e x+f \sqrt {a+\frac {e^2 x^2}{f^2}}\right )}-\frac {\operatorname {Subst}\left (\int \frac {a d f^2+2 a f^2 x^2-2 d x^4+2 x^6}{d-x^2} \, dx,x,\sqrt {d+e x+f \sqrt {a+\frac {e^2 x^2}{f^2}}}\right )}{2 e}\\ &=-\frac {a d f^2 \sqrt {d+e x+f \sqrt {a+\frac {e^2 x^2}{f^2}}}}{2 e \left (e x+f \sqrt {a+\frac {e^2 x^2}{f^2}}\right )}-\frac {\operatorname {Subst}\left (\int \left (-2 a f^2-2 x^4+\frac {3 a d f^2}{d-x^2}\right ) \, dx,x,\sqrt {d+e x+f \sqrt {a+\frac {e^2 x^2}{f^2}}}\right )}{2 e}\\ &=\frac {a f^2 \sqrt {d+e x+f \sqrt {a+\frac {e^2 x^2}{f^2}}}}{e}-\frac {a d f^2 \sqrt {d+e x+f \sqrt {a+\frac {e^2 x^2}{f^2}}}}{2 e \left (e x+f \sqrt {a+\frac {e^2 x^2}{f^2}}\right )}+\frac {\left (d+e x+f \sqrt {a+\frac {e^2 x^2}{f^2}}\right )^{5/2}}{5 e}-\frac {\left (3 a d f^2\right ) \operatorname {Subst}\left (\int \frac {1}{d-x^2} \, dx,x,\sqrt {d+e x+f \sqrt {a+\frac {e^2 x^2}{f^2}}}\right )}{2 e}\\ &=\frac {a f^2 \sqrt {d+e x+f \sqrt {a+\frac {e^2 x^2}{f^2}}}}{e}-\frac {a d f^2 \sqrt {d+e x+f \sqrt {a+\frac {e^2 x^2}{f^2}}}}{2 e \left (e x+f \sqrt {a+\frac {e^2 x^2}{f^2}}\right )}+\frac {\left (d+e x+f \sqrt {a+\frac {e^2 x^2}{f^2}}\right )^{5/2}}{5 e}-\frac {3 a \sqrt {d} f^2 \tanh ^{-1}\left (\frac {\sqrt {d+e x+f \sqrt {a+\frac {e^2 x^2}{f^2}}}}{\sqrt {d}}\right )}{2 e}\\ \end {align*}
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Mathematica [A] time = 0.24, size = 175, normalized size = 0.96 \[ \frac {\frac {2}{5} \left (f \sqrt {a+\frac {e^2 x^2}{f^2}}+d+e x\right )^{5/2}+2 a f^2 \sqrt {f \sqrt {a+\frac {e^2 x^2}{f^2}}+d+e x}-\frac {a d f^2 \sqrt {f \sqrt {a+\frac {e^2 x^2}{f^2}}+d+e x}}{f \sqrt {a+\frac {e^2 x^2}{f^2}}+e x}-3 a \sqrt {d} f^2 \tanh ^{-1}\left (\frac {\sqrt {f \sqrt {a+\frac {e^2 x^2}{f^2}}+d+e x}}{\sqrt {d}}\right )}{2 e} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.55, size = 337, normalized size = 1.84 \[ \left [\frac {15 \, a \sqrt {d} f^{2} \log \left (a f^{2} - 2 \, d e x + 2 \, d f \sqrt {\frac {e^{2} x^{2} + a f^{2}}{f^{2}}} + 2 \, {\left (\sqrt {d} e x - \sqrt {d} f \sqrt {\frac {e^{2} x^{2} + a f^{2}}{f^{2}}}\right )} \sqrt {e x + f \sqrt {\frac {e^{2} x^{2} + a f^{2}}{f^{2}}} + d}\right ) + 2 \, {\left (4 \, e^{2} x^{2} + 12 \, a f^{2} + 9 \, d e x + 2 \, d^{2} + {\left (4 \, e f x - d f\right )} \sqrt {\frac {e^{2} x^{2} + a f^{2}}{f^{2}}}\right )} \sqrt {e x + f \sqrt {\frac {e^{2} x^{2} + a f^{2}}{f^{2}}} + d}}{20 \, e}, \frac {15 \, a \sqrt {-d} f^{2} \arctan \left (\frac {\sqrt {e x + f \sqrt {\frac {e^{2} x^{2} + a f^{2}}{f^{2}}} + d} \sqrt {-d}}{d}\right ) + {\left (4 \, e^{2} x^{2} + 12 \, a f^{2} + 9 \, d e x + 2 \, d^{2} + {\left (4 \, e f x - d f\right )} \sqrt {\frac {e^{2} x^{2} + a f^{2}}{f^{2}}}\right )} \sqrt {e x + f \sqrt {\frac {e^{2} x^{2} + a f^{2}}{f^{2}}} + d}}{10 \, e}\right ] \]
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 \[ \int {\left (e x + \sqrt {\frac {e^{2} x^{2}}{f^{2}} + a} f + d\right )}^{\frac {3}{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.02, size = 0, normalized size = 0.00 \[ \int \left (e x +d +\sqrt {\frac {e^{2} x^{2}}{f^{2}}+a}\, f \right )^{\frac {3}{2}}\, dx \]
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 \[ \int {\left (e x + \sqrt {\frac {e^{2} x^{2}}{f^{2}} + a} f + d\right )}^{\frac {3}{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int {\left (d+e\,x+f\,\sqrt {a+\frac {e^2\,x^2}{f^2}}\right )}^{3/2} \,d x \]
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 \[ \int \left (d + e x + f \sqrt {a + \frac {e^{2} x^{2}}{f^{2}}}\right )^{\frac {3}{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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