Optimal. Leaf size=302 \[ \frac {f^2 \left (4 a e^2-b^2 f^2\right ) \sqrt {d+e x+f \sqrt {a+b x+\frac {e^2 x^2}{f^2}}}}{4 e^3}+\frac {\left (d+e x+f \sqrt {a+b x+\frac {e^2 x^2}{f^2}}\right )^{5/2}}{5 e}-\frac {f^2 \left (2 d e-b f^2\right ) \left (4 a e^2-b^2 f^2\right ) \sqrt {d+e x+f \sqrt {a+b x+\frac {e^2 x^2}{f^2}}}}{8 e^3 \left (b f^2+2 e \left (e x+f \sqrt {a+\frac {x \left (b f^2+e^2 x\right )}{f^2}}\right )\right )}-\frac {3 f^2 \sqrt {2 d e-b f^2} \left (4 a e^2-b^2 f^2\right ) \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {e} \sqrt {d+e x+f \sqrt {a+b x+\frac {e^2 x^2}{f^2}}}}{\sqrt {2 d e-b f^2}}\right )}{8 \sqrt {2} e^{7/2}} \]
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Rubi [A]
time = 0.30, antiderivative size = 302, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 5, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {2141, 911,
1271, 1824, 214} \begin {gather*} -\frac {3 f^2 \left (4 a e^2-b^2 f^2\right ) \sqrt {2 d e-b f^2} \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {e} \sqrt {f \sqrt {a+b x+\frac {e^2 x^2}{f^2}}+d+e x}}{\sqrt {2 d e-b f^2}}\right )}{8 \sqrt {2} e^{7/2}}+\frac {f^2 \left (4 a e^2-b^2 f^2\right ) \sqrt {f \sqrt {a+b x+\frac {e^2 x^2}{f^2}}+d+e x}}{4 e^3}-\frac {f^2 \left (4 a e^2-b^2 f^2\right ) \left (2 d e-b f^2\right ) \sqrt {f \sqrt {a+b x+\frac {e^2 x^2}{f^2}}+d+e x}}{8 e^3 \left (2 e \left (f \sqrt {a+\frac {x \left (b f^2+e^2 x\right )}{f^2}}+e x\right )+b f^2\right )}+\frac {\left (f \sqrt {a+b x+\frac {e^2 x^2}{f^2}}+d+e x\right )^{5/2}}{5 e} \end {gather*}
Antiderivative was successfully verified.
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Rule 214
Rule 911
Rule 1271
Rule 1824
Rule 2141
Rubi steps
\begin {align*} \int \left (d+e x+f \sqrt {a+b x+\frac {e^2 x^2}{f^2}}\right )^{3/2} \, dx &=2 \text {Subst}\left (\int \frac {x^{3/2} \left (d^2 e-(b d-a e) f^2-\left (2 d e-b f^2\right ) x+e x^2\right )}{\left (-2 d e+b f^2+2 e x\right )^2} \, dx,x,d+e x+f \sqrt {a+b x+\frac {e^2 x^2}{f^2}}\right )\\ &=4 \text {Subst}\left (\int \frac {x^4 \left (d^2 e-(b d-a e) f^2+\left (-2 d e+b f^2\right ) x^2+e x^4\right )}{\left (-2 d e+b f^2+2 e x^2\right )^2} \, dx,x,\sqrt {d+e x+f \sqrt {a+b x+\frac {e^2 x^2}{f^2}}}\right )\\ &=-\frac {f^2 \left (2 d e-b f^2\right ) \left (4 a e^2-b^2 f^2\right ) \sqrt {d+e x+f \sqrt {a+b x+\frac {e^2 x^2}{f^2}}}}{8 e^3 \left (b f^2+2 e \left (e x+f \sqrt {a+\frac {x \left (b f^2+e^2 x\right )}{f^2}}\right )\right )}-\frac {\text {Subst}\left (\int \frac {-e f^2 \left (2 d e-b f^2\right ) \left (4 a e^2-b^2 f^2\right )-4 e^2 f^2 \left (4 a e^2-b^2 f^2\right ) x^2+8 e^3 \left (2 d e-b f^2\right ) x^4-16 e^4 x^6}{-2 d e+b f^2+2 e x^2} \, dx,x,\sqrt {d+e x+f \sqrt {a+b x+\frac {e^2 x^2}{f^2}}}\right )}{8 e^4}\\ &=-\frac {f^2 \left (2 d e-b f^2\right ) \left (4 a e^2-b^2 f^2\right ) \sqrt {d+e x+f \sqrt {a+b x+\frac {e^2 x^2}{f^2}}}}{8 e^3 \left (b f^2+2 e \left (e x+f \sqrt {a+\frac {x \left (b f^2+e^2 x\right )}{f^2}}\right )\right )}-\frac {\text {Subst}\left (\int \left (-2 e f^2 \left (4 a e^2-b^2 f^2\right )-8 e^3 x^4-\frac {3 \left (8 a d e^4 f^2-2 b^2 d e^2 f^4-4 a b e^3 f^4+b^3 e f^6\right )}{-2 d e+b f^2+2 e x^2}\right ) \, dx,x,\sqrt {d+e x+f \sqrt {a+b x+\frac {e^2 x^2}{f^2}}}\right )}{8 e^4}\\ &=\frac {f^2 \left (4 a e^2-b^2 f^2\right ) \sqrt {d+e x+f \sqrt {a+b x+\frac {e^2 x^2}{f^2}}}}{4 e^3}+\frac {\left (d+e x+f \sqrt {a+b x+\frac {e^2 x^2}{f^2}}\right )^{5/2}}{5 e}-\frac {f^2 \left (2 d e-b f^2\right ) \left (4 a e^2-b^2 f^2\right ) \sqrt {d+e x+f \sqrt {a+b x+\frac {e^2 x^2}{f^2}}}}{8 e^3 \left (b f^2+2 e \left (e x+f \sqrt {a+\frac {x \left (b f^2+e^2 x\right )}{f^2}}\right )\right )}+\frac {\left (3 f^2 \left (2 d e-b f^2\right ) \left (4 a e^2-b^2 f^2\right )\right ) \text {Subst}\left (\int \frac {1}{-2 d e+b f^2+2 e x^2} \, dx,x,\sqrt {d+e x+f \sqrt {a+b x+\frac {e^2 x^2}{f^2}}}\right )}{8 e^3}\\ &=\frac {f^2 \left (4 a e^2-b^2 f^2\right ) \sqrt {d+e x+f \sqrt {a+b x+\frac {e^2 x^2}{f^2}}}}{4 e^3}+\frac {\left (d+e x+f \sqrt {a+b x+\frac {e^2 x^2}{f^2}}\right )^{5/2}}{5 e}-\frac {f^2 \left (2 d e-b f^2\right ) \left (4 a e^2-b^2 f^2\right ) \sqrt {d+e x+f \sqrt {a+b x+\frac {e^2 x^2}{f^2}}}}{8 e^3 \left (b f^2+2 e \left (e x+f \sqrt {a+\frac {x \left (b f^2+e^2 x\right )}{f^2}}\right )\right )}-\frac {3 f^2 \sqrt {2 d e-b f^2} \left (4 a e^2-b^2 f^2\right ) \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {e} \sqrt {d+e x+f \sqrt {a+b x+\frac {e^2 x^2}{f^2}}}}{\sqrt {2 d e-b f^2}}\right )}{8 \sqrt {2} e^{7/2}}\\ \end {align*}
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Mathematica [A]
time = 2.84, size = 443, normalized size = 1.47 \begin {gather*} \frac {\sqrt {d+e x+f \sqrt {a+x \left (b+\frac {e^2 x}{f^2}\right )}} \left (-15 b^3 f^6-2 b^2 e f^4 \left (-5 d+6 e x+10 f \sqrt {a+x \left (b+\frac {e^2 x}{f^2}\right )}\right )+4 b e^2 f^2 \left (2 d^2+17 a f^2+8 e x \left (2 e x+f \sqrt {a+x \left (b+\frac {e^2 x}{f^2}\right )}\right )+4 d \left (3 e x+f \sqrt {a+x \left (b+\frac {e^2 x}{f^2}\right )}\right )\right )+8 e^3 \left (2 (d+2 e x)^2 \left (e x+f \sqrt {a+x \left (b+\frac {e^2 x}{f^2}\right )}\right )+a f^2 \left (-d+16 e x+12 f \sqrt {a+x \left (b+\frac {e^2 x}{f^2}\right )}\right )\right )\right )}{40 e^3 \left (b f^2+2 e \left (e x+f \sqrt {a+x \left (b+\frac {e^2 x}{f^2}\right )}\right )\right )}-\frac {3 a f^2 \sqrt {-d e+\frac {b f^2}{2}} \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {e} \sqrt {d+e x+f \sqrt {a+x \left (b+\frac {e^2 x}{f^2}\right )}}}{\sqrt {-2 d e+b f^2}}\right )}{2 e^{3/2}}+\frac {3 b^2 f^4 \sqrt {-d e+\frac {b f^2}{2}} \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {e} \sqrt {d+e x+f \sqrt {a+x \left (b+\frac {e^2 x}{f^2}\right )}}}{\sqrt {-2 d e+b f^2}}\right )}{8 e^{7/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.01, size = 0, normalized size = 0.00 \[\int \left (d +e x +f \sqrt {a +b x +\frac {e^{2} x^{2}}{f^{2}}}\right )^{\frac {3}{2}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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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.
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Fricas [A]
time = 0.50, size = 612, normalized size = 2.03 \begin {gather*} \left [-\frac {1}{80} \, {\left (15 \, \sqrt {\frac {1}{2}} {\left (b^{2} f^{4} - 4 \, a f^{2} e^{2}\right )} \sqrt {-{\left (b f^{2} - 2 \, d e\right )} e^{\left (-1\right )}} \log \left (b^{2} f^{4} - 4 \, b d f^{2} e - 8 \, d x e^{3} + 4 \, {\left (b f^{2} x + a f^{2}\right )} e^{2} - 4 \, {\left (2 \, \sqrt {\frac {1}{2}} \sqrt {-{\left (b f^{2} - 2 \, d e\right )} e^{\left (-1\right )}} f \sqrt {\frac {b f^{2} x + a f^{2} + x^{2} e^{2}}{f^{2}}} e^{2} - \sqrt {\frac {1}{2}} {\left (b f^{2} e + 2 \, x e^{3}\right )} \sqrt {-{\left (b f^{2} - 2 \, d e\right )} e^{\left (-1\right )}}\right )} \sqrt {x e + f \sqrt {\frac {b f^{2} x + a f^{2} + x^{2} e^{2}}{f^{2}}} + d} - 4 \, {\left (b f^{3} e - 2 \, d f e^{2}\right )} \sqrt {\frac {b f^{2} x + a f^{2} + x^{2} e^{2}}{f^{2}}}\right ) + 2 \, {\left (15 \, b^{2} f^{4} - 10 \, b d f^{2} e - 16 \, x^{2} e^{4} - 36 \, d x e^{3} + 2 \, {\left (b f^{2} x - 24 \, a f^{2} - 4 \, d^{2}\right )} e^{2} - 2 \, {\left (5 \, b f^{3} e + 8 \, f x e^{3} - 2 \, d f e^{2}\right )} \sqrt {\frac {b f^{2} x + a f^{2} + x^{2} e^{2}}{f^{2}}}\right )} \sqrt {x e + f \sqrt {\frac {b f^{2} x + a f^{2} + x^{2} e^{2}}{f^{2}}} + d}\right )} e^{\left (-3\right )}, -\frac {1}{40} \, {\left (15 \, \sqrt {\frac {1}{2}} {\left (b^{2} f^{4} - 4 \, a f^{2} e^{2}\right )} \sqrt {b f^{2} - 2 \, d e} \arctan \left (-\frac {2 \, \sqrt {\frac {1}{2}} \sqrt {x e + f \sqrt {\frac {b f^{2} x + a f^{2} + x^{2} e^{2}}{f^{2}}} + d} e^{\frac {1}{2}}}{\sqrt {b f^{2} - 2 \, d e}}\right ) e^{\left (-\frac {1}{2}\right )} + {\left (15 \, b^{2} f^{4} - 10 \, b d f^{2} e - 16 \, x^{2} e^{4} - 36 \, d x e^{3} + 2 \, {\left (b f^{2} x - 24 \, a f^{2} - 4 \, d^{2}\right )} e^{2} - 2 \, {\left (5 \, b f^{3} e + 8 \, f x e^{3} - 2 \, d f e^{2}\right )} \sqrt {\frac {b f^{2} x + a f^{2} + x^{2} e^{2}}{f^{2}}}\right )} \sqrt {x e + f \sqrt {\frac {b f^{2} x + a f^{2} + x^{2} e^{2}}{f^{2}}} + d}\right )} e^{\left (-3\right )}\right ] \end {gather*}
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 \begin {gather*} \int \left (d + e x + f \sqrt {a + b x + \frac {e^{2} x^{2}}{f^{2}}}\right )^{\frac {3}{2}}\, dx \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int {\left (d+e\,x+f\,\sqrt {a+b\,x+\frac {e^2\,x^2}{f^2}}\right )}^{3/2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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