Optimal. Leaf size=370 \[ -\frac {5 f^2 \left (4 a e^2-b^2 f^2\right ) \left (2 d e-b f^2\right )^{3/2} \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {e} \sqrt {f \sqrt {a+b x+\frac {e^2 x^2}{f^2}}+d+e x}}{\sqrt {2 d e-b f^2}}\right )}{16 \sqrt {2} e^{9/2}}+\frac {f^2 \left (4 a e^2-b^2 f^2\right ) \left (2 d e-b f^2\right ) \sqrt {f \sqrt {a+b x+\frac {e^2 x^2}{f^2}}+d+e x}}{4 e^4}-\frac {f^2 \left (4 a e^2-b^2 f^2\right ) \left (2 d e-b f^2\right )^2 \sqrt {f \sqrt {a+b x+\frac {e^2 x^2}{f^2}}+d+e x}}{16 e^4 \left (2 e \left (f \sqrt {a+\frac {x \left (b f^2+e^2 x\right )}{f^2}}+e x\right )+b f^2\right )}+\frac {f^2 \left (4 a e^2-b^2 f^2\right ) \left (f \sqrt {a+b x+\frac {e^2 x^2}{f^2}}+d+e x\right )^{3/2}}{12 e^3}+\frac {\left (f \sqrt {a+b x+\frac {e^2 x^2}{f^2}}+d+e x\right )^{7/2}}{7 e} \]
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Rubi [A] time = 0.60, 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 = {2116, 897, 1257, 1810, 208} \[ \frac {f^2 \left (4 a e^2-b^2 f^2\right ) \left (f \sqrt {a+b x+\frac {e^2 x^2}{f^2}}+d+e x\right )^{3/2}}{12 e^3}+\frac {f^2 \left (4 a e^2-b^2 f^2\right ) \left (2 d e-b f^2\right ) \sqrt {f \sqrt {a+b x+\frac {e^2 x^2}{f^2}}+d+e x}}{4 e^4}-\frac {f^2 \left (4 a e^2-b^2 f^2\right ) \left (2 d e-b f^2\right )^2 \sqrt {f \sqrt {a+b x+\frac {e^2 x^2}{f^2}}+d+e x}}{16 e^4 \left (2 e \left (f \sqrt {a+\frac {x \left (b f^2+e^2 x\right )}{f^2}}+e x\right )+b f^2\right )}-\frac {5 f^2 \left (4 a e^2-b^2 f^2\right ) \left (2 d e-b f^2\right )^{3/2} \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {e} \sqrt {f \sqrt {a+b x+\frac {e^2 x^2}{f^2}}+d+e x}}{\sqrt {2 d e-b f^2}}\right )}{16 \sqrt {2} e^{9/2}}+\frac {\left (f \sqrt {a+b x+\frac {e^2 x^2}{f^2}}+d+e x\right )^{7/2}}{7 e} \]
Antiderivative was successfully verified.
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Rule 208
Rule 897
Rule 1257
Rule 1810
Rule 2116
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 \operatorname {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 \operatorname {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 {\operatorname {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 {\operatorname {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 ) \operatorname {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 [A] time = 1.06, size = 357, normalized size = 0.96 \[ \frac {\frac {4}{3} e^2 f^2 \left (4 a e^2-b^2 f^2\right ) \left (f \sqrt {a+x \left (b+\frac {e^2 x}{f^2}\right )}+d+e x\right )^{3/2}+4 e f^2 \left (4 a e^2-b^2 f^2\right ) \left (2 d e-b f^2\right ) \sqrt {f \sqrt {a+x \left (b+\frac {e^2 x}{f^2}\right )}+d+e x}-\frac {5 \sqrt {e} f^2 \left (4 a e^2-b^2 f^2\right ) \left (2 d e-b f^2\right )^{3/2} \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {e} \sqrt {f \sqrt {a+x \left (b+\frac {e^2 x}{f^2}\right )}+d+e x}}{\sqrt {2 d e-b f^2}}\right )}{\sqrt {2}}-\frac {\left (4 a e^3 f^2-b^2 e f^4\right ) \left (b f^2-2 d e\right )^2 \sqrt {f \sqrt {a+x \left (b+\frac {e^2 x}{f^2}\right )}+d+e x}}{2 e \left (f \sqrt {a+x \left (b+\frac {e^2 x}{f^2}\right )}+e x\right )+b f^2}+\frac {16}{7} e^4 \left (f \sqrt {a+x \left (b+\frac {e^2 x}{f^2}\right )}+d+e x\right )^{7/2}}{16 e^5} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.70, size = 923, normalized size = 2.49 \[ \left [\frac {105 \, \sqrt {\frac {1}{2}} {\left (b^{3} f^{6} + 8 \, a d e^{3} f^{2} - 2 \, {\left (b^{2} d e + 2 \, a b e^{2}\right )} f^{4}\right )} \sqrt {-\frac {b f^{2} - 2 \, d e}{e}} \log \left (-b^{2} f^{4} + 4 \, {\left (b d e - a e^{2}\right )} f^{2} - 4 \, {\left (b e^{2} f^{2} - 2 \, d e^{3}\right )} x + 4 \, {\left (2 \, \sqrt {\frac {1}{2}} e^{2} f \sqrt {-\frac {b f^{2} - 2 \, d e}{e}} \sqrt {\frac {b f^{2} x + e^{2} x^{2} + a f^{2}}{f^{2}}} - \sqrt {\frac {1}{2}} {\left (b e f^{2} + 2 \, e^{3} x\right )} \sqrt {-\frac {b f^{2} - 2 \, d e}{e}}\right )} \sqrt {e x + f \sqrt {\frac {b f^{2} x + e^{2} x^{2} + a f^{2}}{f^{2}}} + d} + 4 \, {\left (b e f^{3} - 2 \, d e^{2} f\right )} \sqrt {\frac {b f^{2} x + e^{2} x^{2} + a f^{2}}{f^{2}}}\right ) + 2 \, {\left (105 \, b^{3} f^{6} + 192 \, e^{6} x^{3} + 48 \, d^{3} e^{3} - 56 \, {\left (5 \, b^{2} d e + 6 \, a b e^{2}\right )} f^{4} + 4 \, {\left (21 \, b d^{2} e^{2} + 232 \, a d e^{3}\right )} f^{2} + 144 \, {\left (b e^{4} f^{2} + 2 \, d e^{5}\right )} x^{2} + 2 \, {\left (7 \, b^{2} e^{2} f^{4} + 156 \, d^{2} e^{4} - 4 \, {\left (3 \, b d e^{3} - 32 \, a e^{4}\right )} f^{2}\right )} x - 2 \, {\left (35 \, b^{2} e f^{5} - 96 \, e^{5} f x^{2} + 12 \, d^{2} e^{3} f - 4 \, {\left (21 \, b d e^{2} + 20 \, a e^{3}\right )} f^{3} - 24 \, {\left (b e^{3} f^{3} + 6 \, d e^{4} f\right )} x\right )} \sqrt {\frac {b f^{2} x + e^{2} x^{2} + a f^{2}}{f^{2}}}\right )} \sqrt {e x + f \sqrt {\frac {b f^{2} x + e^{2} x^{2} + a f^{2}}{f^{2}}} + d}}{672 \, e^{4}}, -\frac {105 \, \sqrt {\frac {1}{2}} {\left (b^{3} f^{6} + 8 \, a d e^{3} f^{2} - 2 \, {\left (b^{2} d e + 2 \, a b e^{2}\right )} f^{4}\right )} \sqrt {\frac {b f^{2} - 2 \, d e}{e}} \arctan \left (\frac {2 \, \sqrt {\frac {1}{2}} \sqrt {e x + f \sqrt {\frac {b f^{2} x + e^{2} x^{2} + a f^{2}}{f^{2}}} + d} e \sqrt {\frac {b f^{2} - 2 \, d e}{e}}}{b f^{2} - 2 \, d e}\right ) - {\left (105 \, b^{3} f^{6} + 192 \, e^{6} x^{3} + 48 \, d^{3} e^{3} - 56 \, {\left (5 \, b^{2} d e + 6 \, a b e^{2}\right )} f^{4} + 4 \, {\left (21 \, b d^{2} e^{2} + 232 \, a d e^{3}\right )} f^{2} + 144 \, {\left (b e^{4} f^{2} + 2 \, d e^{5}\right )} x^{2} + 2 \, {\left (7 \, b^{2} e^{2} f^{4} + 156 \, d^{2} e^{4} - 4 \, {\left (3 \, b d e^{3} - 32 \, a e^{4}\right )} f^{2}\right )} x - 2 \, {\left (35 \, b^{2} e f^{5} - 96 \, e^{5} f x^{2} + 12 \, d^{2} e^{3} f - 4 \, {\left (21 \, b d e^{2} + 20 \, a e^{3}\right )} f^{3} - 24 \, {\left (b e^{3} f^{3} + 6 \, d e^{4} f\right )} x\right )} \sqrt {\frac {b f^{2} x + e^{2} x^{2} + a f^{2}}{f^{2}}}\right )} \sqrt {e x + f \sqrt {\frac {b f^{2} x + e^{2} x^{2} + a f^{2}}{f^{2}}} + d}}{336 \, e^{4}}\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 \[ \int {\left (e x + \sqrt {b x + \frac {e^{2} x^{2}}{f^{2}} + a} f + d\right )}^{\frac {5}{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.05, size = 0, normalized size = 0.00 \[ \int \left (e x +d +\sqrt {b x +\frac {e^{2} x^{2}}{f^{2}}+a}\, f \right )^{\frac {5}{2}}\, dx \]
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 \[ \int {\left (e x + \sqrt {b x + \frac {e^{2} x^{2}}{f^{2}} + a} f + d\right )}^{\frac {5}{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int {\left (d+e\,x+f\,\sqrt {a+b\,x+\frac {e^2\,x^2}{f^2}}\right )}^{5/2} \,d x \]
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 \[ \int \left (d + e x + f \sqrt {a + b x + \frac {e^{2} x^{2}}{f^{2}}}\right )^{\frac {5}{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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