Optimal. Leaf size=370 \[ \frac {f^2 \left (2 d e-b f^2\right ) \left (4 a e^2-b^2 f^2\right ) \sqrt {d+e x+f \sqrt {a+b x+\frac {e^2 x^2}{f^2}}}}{4 e^4}+\frac {f^2 \left (4 a e^2-b^2 f^2\right ) \left (d+e x+f \sqrt {a+b x+\frac {e^2 x^2}{f^2}}\right )^{3/2}}{12 e^3}+\frac {\left (d+e x+f \sqrt {a+b x+\frac {e^2 x^2}{f^2}}\right )^{7/2}}{7 e}-\frac {f^2 \left (2 d e-b f^2\right )^2 \left (4 a e^2-b^2 f^2\right ) \sqrt {d+e x+f \sqrt {a+b x+\frac {e^2 x^2}{f^2}}}}{16 e^4 \left (b f^2+2 e \left (e x+f \sqrt {a+\frac {x \left (b f^2+e^2 x\right )}{f^2}}\right )\right )}-\frac {5 f^2 \left (2 d e-b f^2\right )^{3/2} \left (4 a e^2-b^2 f^2\right ) \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {e} \sqrt {d+e x+f \sqrt {a+b x+\frac {e^2 x^2}{f^2}}}}{\sqrt {2 d e-b f^2}}\right )}{16 \sqrt {2} e^{9/2}} \]
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Rubi [A]
time = 0.42, antiderivative size = 370, 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 {5 f^2 \left (4 a e^2-b^2 f^2\right ) \left (2 d e-b f^2\right )^{3/2} \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 )}{16 \sqrt {2} e^{9/2}}+\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}}{4 e^4}-\frac {f^2 \left (4 a e^2-b^2 f^2\right ) \left (2 d e-b f^2\right )^2 \sqrt {f \sqrt {a+b x+\frac {e^2 x^2}{f^2}}+d+e x}}{16 e^4 \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 {f^2 \left (4 a e^2-b^2 f^2\right ) \left (f \sqrt {a+b x+\frac {e^2 x^2}{f^2}}+d+e x\right )^{3/2}}{12 e^3}+\frac {\left (f \sqrt {a+b x+\frac {e^2 x^2}{f^2}}+d+e x\right )^{7/2}}{7 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 )^{5/2} \, dx &=2 \text {Subst}\left (\int \frac {x^{5/2} \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^6 \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 )^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}}}}{16 e^4 \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 )^2 \left (4 a e^2-b^2 f^2\right )-4 e^2 f^2 \left (2 d e-b f^2\right ) \left (4 a e^2-b^2 f^2\right ) x^2-8 e^3 f^2 \left (4 a e^2-b^2 f^2\right ) x^4+16 e^4 \left (2 d e-b f^2\right ) x^6-32 e^5 x^8}{-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 )}{16 e^5}\\ &=-\frac {f^2 \left (2 d e-b f^2\right )^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}}}}{16 e^4 \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 (-4 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^2-16 e^4 x^6-\frac {5 \left (16 a d^2 e^5 f^2-4 b^2 d^2 e^3 f^4-16 a b d e^4 f^4+4 b^3 d e^2 f^6+4 a b^2 e^3 f^6-b^4 e f^8\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 )}{16 e^5}\\ &=\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}}}}{4 e^4}+\frac {f^2 \left (4 a e^2-b^2 f^2\right ) \left (d+e x+f \sqrt {a+b x+\frac {e^2 x^2}{f^2}}\right )^{3/2}}{12 e^3}+\frac {\left (d+e x+f \sqrt {a+b x+\frac {e^2 x^2}{f^2}}\right )^{7/2}}{7 e}-\frac {f^2 \left (2 d e-b f^2\right )^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}}}}{16 e^4 \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 (5 f^2 \left (2 d e-b f^2\right )^2 \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 )}{16 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}}}}{4 e^4}+\frac {f^2 \left (4 a e^2-b^2 f^2\right ) \left (d+e x+f \sqrt {a+b x+\frac {e^2 x^2}{f^2}}\right )^{3/2}}{12 e^3}+\frac {\left (d+e x+f \sqrt {a+b x+\frac {e^2 x^2}{f^2}}\right )^{7/2}}{7 e}-\frac {f^2 \left (2 d e-b f^2\right )^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}}}}{16 e^4 \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 {5 f^2 \left (2 d e-b f^2\right )^{3/2} \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 )}{16 \sqrt {2} e^{9/2}}\\ \end {align*}
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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(985\) vs. \(2(370)=740\).
time = 5.00, size = 985, normalized size = 2.66 \begin {gather*} \frac {1}{32} \left (\frac {2 \sqrt {d+e x+f \sqrt {a+x \left (b+\frac {e^2 x}{f^2}\right )}} \left (105 b^4 f^8+28 b^3 e f^6 \left (-10 d+3 e x+5 f \sqrt {a+x \left (b+\frac {e^2 x}{f^2}\right )}\right )+4 b^2 e^2 f^4 \left (21 d^2-119 a f^2+16 e x \left (2 e x-f \sqrt {a+x \left (b+\frac {e^2 x}{f^2}\right )}\right )-2 d \left (31 e x+49 f \sqrt {a+x \left (b+\frac {e^2 x}{f^2}\right )}\right )\right )+16 b e^3 f^2 \left (3 d^3+79 a d f^2+36 d e x \left (2 e x+f \sqrt {a+x \left (b+\frac {e^2 x}{f^2}\right )}\right )+9 d^2 \left (3 e x+f \sqrt {a+x \left (b+\frac {e^2 x}{f^2}\right )}\right )+4 \left (15 e^3 x^3-8 a f^3 \sqrt {a+x \left (b+\frac {e^2 x}{f^2}\right )}+9 e^2 f x^2 \sqrt {a+x \left (b+\frac {e^2 x}{f^2}\right )}\right )\right )+16 e^4 \left (20 a^2 f^4+6 (d+2 e x)^3 \left (e x+f \sqrt {a+x \left (b+\frac {e^2 x}{f^2}\right )}\right )+a f^2 \left (-3 d^2+4 e x \left (19 e x+13 f \sqrt {a+x \left (b+\frac {e^2 x}{f^2}\right )}\right )+4 d \left (38 e x+29 f \sqrt {a+x \left (b+\frac {e^2 x}{f^2}\right )}\right )\right )\right )\right )}{21 e^4 \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 {20 b^2 d^2 f^4 \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 )}{e^{5/2} \sqrt {-d e+\frac {b f^2}{2}}}-\frac {80 a b d f^4 \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 )}{e^{3/2} \sqrt {-d e+\frac {b f^2}{2}}}+\frac {20 b^3 d f^6 \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 )}{e^{7/2} \sqrt {-d e+\frac {b f^2}{2}}}+\frac {20 a b^2 f^6 \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 )}{e^{5/2} \sqrt {-d e+\frac {b f^2}{2}}}-\frac {5 b^4 f^8 \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 )}{e^{9/2} \sqrt {-d e+\frac {b f^2}{2}}}+\frac {80 \sqrt {2} a d^2 f^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 )}{\sqrt {e} \sqrt {-2 d e+b f^2}}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.02, size = 0, normalized size = 0.00 \[\int \left (d +e x +f \sqrt {a +b x +\frac {e^{2} x^{2}}{f^{2}}}\right )^{\frac {5}{2}}\, dx\]
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 = 0.55, size = 858, normalized size = 2.32 \begin {gather*} \left [\frac {1}{672} \, {\left (105 \, \sqrt {\frac {1}{2}} {\left (b^{3} f^{6} - 2 \, b^{2} d f^{4} e - 4 \, a b f^{4} e^{2} + 8 \, a d f^{2} e^{3}\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 (105 \, b^{3} f^{6} - 280 \, b^{2} d f^{4} e + 192 \, x^{3} e^{6} + 288 \, d x^{2} e^{5} + 8 \, {\left (18 \, b f^{2} x^{2} + {\left (32 \, a f^{2} + 39 \, d^{2}\right )} x\right )} e^{4} - 8 \, {\left (3 \, b d f^{2} x - 116 \, a d f^{2} - 6 \, d^{3}\right )} e^{3} + 14 \, {\left (b^{2} f^{4} x - 24 \, a b f^{4} + 6 \, b d^{2} f^{2}\right )} e^{2} - 2 \, {\left (35 \, b^{2} f^{5} e - 84 \, b d f^{3} e^{2} - 96 \, f x^{2} e^{5} - 144 \, d f x e^{4} - 4 \, {\left (6 \, b f^{3} x + 20 \, a f^{3} - 3 \, d^{2} f\right )} e^{3}\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 (-4\right )}, \frac {1}{336} \, {\left (105 \, \sqrt {\frac {1}{2}} {\left (b^{3} f^{6} - 2 \, b^{2} d f^{4} e - 4 \, a b f^{4} e^{2} + 8 \, a d f^{2} e^{3}\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 (105 \, b^{3} f^{6} - 280 \, b^{2} d f^{4} e + 192 \, x^{3} e^{6} + 288 \, d x^{2} e^{5} + 8 \, {\left (18 \, b f^{2} x^{2} + {\left (32 \, a f^{2} + 39 \, d^{2}\right )} x\right )} e^{4} - 8 \, {\left (3 \, b d f^{2} x - 116 \, a d f^{2} - 6 \, d^{3}\right )} e^{3} + 14 \, {\left (b^{2} f^{4} x - 24 \, a b f^{4} + 6 \, b d^{2} f^{2}\right )} e^{2} - 2 \, {\left (35 \, b^{2} f^{5} e - 84 \, b d f^{3} e^{2} - 96 \, f x^{2} e^{5} - 144 \, d f x e^{4} - 4 \, {\left (6 \, b f^{3} x + 20 \, a f^{3} - 3 \, d^{2} f\right )} e^{3}\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 (-4\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 {5}{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 )}^{5/2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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