Optimal. Leaf size=335 \[ -\frac {4 f^2 \left (4 a e^2-b^2 f^2\right )}{\left (2 d e-b f^2\right )^3 \sqrt {f \sqrt {a+b x+\frac {e^2 x^2}{f^2}}+d+e x}}-\frac {2 e f^2 \left (4 a e^2-b^2 f^2\right ) \sqrt {f \sqrt {a+b x+\frac {e^2 x^2}{f^2}}+d+e x}}{\left (2 d e-b f^2\right )^3 \left (2 e \left (f \sqrt {a+\frac {x \left (b f^2+e^2 x\right )}{f^2}}+e x\right )+b f^2\right )}+\frac {5 \sqrt {2} \sqrt {e} f^2 \left (4 a e^2-b^2 f^2\right ) \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {e} \sqrt {f \sqrt {a+b x+\frac {e^2 x^2}{f^2}}+d+e x}}{\sqrt {2 d e-b f^2}}\right )}{\left (2 d e-b f^2\right )^{7/2}}-\frac {4 \left (a e f^2-b d f^2+d^2 e\right )}{3 \left (2 d e-b f^2\right )^2 \left (f \sqrt {a+b x+\frac {e^2 x^2}{f^2}}+d+e x\right )^{3/2}} \]
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Rubi [A] time = 0.50, antiderivative size = 335, 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, 1259, 1261, 208} \[ -\frac {4 f^2 \left (4 a e^2-b^2 f^2\right )}{\left (2 d e-b f^2\right )^3 \sqrt {f \sqrt {a+b x+\frac {e^2 x^2}{f^2}}+d+e x}}-\frac {2 e f^2 \left (4 a e^2-b^2 f^2\right ) \sqrt {f \sqrt {a+b x+\frac {e^2 x^2}{f^2}}+d+e x}}{\left (2 d e-b f^2\right )^3 \left (2 e \left (f \sqrt {a+\frac {x \left (b f^2+e^2 x\right )}{f^2}}+e x\right )+b f^2\right )}+\frac {5 \sqrt {2} \sqrt {e} f^2 \left (4 a e^2-b^2 f^2\right ) \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {e} \sqrt {f \sqrt {a+b x+\frac {e^2 x^2}{f^2}}+d+e x}}{\sqrt {2 d e-b f^2}}\right )}{\left (2 d e-b f^2\right )^{7/2}}-\frac {4 \left (a e f^2-b d f^2+d^2 e\right )}{3 \left (2 d e-b f^2\right )^2 \left (f \sqrt {a+b x+\frac {e^2 x^2}{f^2}}+d+e x\right )^{3/2}} \]
Antiderivative was successfully verified.
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Rule 208
Rule 897
Rule 1259
Rule 1261
Rule 2116
Rubi steps
\begin {align*} \int \frac {1}{\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 {d^2 e-(b d-a e) f^2-\left (2 d e-b f^2\right ) x+e x^2}{x^{5/2} \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 {d^2 e-(b d-a e) f^2+\left (-2 d e+b f^2\right ) x^2+e x^4}{x^4 \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 {2 e 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}}}}{\left (2 d e-b f^2\right )^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 {\operatorname {Subst}\left (\int \frac {8 e^2 \left (2 d e-b f^2\right )^2 \left (d^2 e-b d f^2+a e f^2\right )-8 e^2 \left (2 d e-b f^2\right ) \left (2 d^2 e^2-2 b d e f^2-2 a e^2 f^2+b^2 f^4\right ) x^2+4 e^3 f^2 \left (4 a e^2-b^2 f^2\right ) x^4}{x^4 \left (-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 )}{2 e^2 \left (2 d e-b f^2\right )^3}\\ &=-\frac {2 e 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}}}}{\left (2 d e-b f^2\right )^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 {\operatorname {Subst}\left (\int \left (-\frac {8 e^2 \left (2 d e-b f^2\right ) \left (d^2 e-b d f^2+a e f^2\right )}{x^4}-\frac {8 \left (4 a e^4 f^2-b^2 e^2 f^4\right )}{x^2}-\frac {20 \left (4 a e^5 f^2-b^2 e^3 f^4\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 )}{2 e^2 \left (2 d e-b f^2\right )^3}\\ &=-\frac {4 \left (d^2 e-b d f^2+a e f^2\right )}{3 \left (2 d e-b f^2\right )^2 \left (d+e x+f \sqrt {a+b x+\frac {e^2 x^2}{f^2}}\right )^{3/2}}-\frac {4 f^2 \left (4 a e^2-b^2 f^2\right )}{\left (2 d e-b f^2\right )^3 \sqrt {d+e x+f \sqrt {a+b x+\frac {e^2 x^2}{f^2}}}}-\frac {2 e 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}}}}{\left (2 d e-b f^2\right )^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 (10 e f^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 )}{\left (2 d e-b f^2\right )^3}\\ &=-\frac {4 \left (d^2 e-b d f^2+a e f^2\right )}{3 \left (2 d e-b f^2\right )^2 \left (d+e x+f \sqrt {a+b x+\frac {e^2 x^2}{f^2}}\right )^{3/2}}-\frac {4 f^2 \left (4 a e^2-b^2 f^2\right )}{\left (2 d e-b f^2\right )^3 \sqrt {d+e x+f \sqrt {a+b x+\frac {e^2 x^2}{f^2}}}}-\frac {2 e 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}}}}{\left (2 d e-b f^2\right )^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 {5 \sqrt {2} \sqrt {e} 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 )}{\left (2 d e-b f^2\right )^{7/2}}\\ \end {align*}
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Mathematica [A] time = 0.86, size = 315, normalized size = 0.94 \[ \frac {\frac {8 f^2 \left (b^2 f^2-4 a e^2\right )}{\sqrt {f \sqrt {a+x \left (b+\frac {e^2 x}{f^2}\right )}+d+e x}}+\frac {10 \sqrt {2} \sqrt {e} f^2 \left (4 a e^2-b^2 f^2\right ) \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {e} \sqrt {f \sqrt {a+x \left (b+\frac {e^2 x}{f^2}\right )}+d+e x}}{\sqrt {2 d e-b f^2}}\right )}{\sqrt {2 d e-b f^2}}-\frac {4 \left (4 a e^3 f^2-b^2 e f^4\right ) \sqrt {f \sqrt {a+x \left (b+\frac {e^2 x}{f^2}\right )}+d+e x}}{2 e \left (f \sqrt {a+x \left (b+\frac {e^2 x}{f^2}\right )}+e x\right )+b f^2}-\frac {8 \left (2 d e-b f^2\right ) \left (a e f^2-b d f^2+d^2 e\right )}{3 \left (f \sqrt {a+x \left (b+\frac {e^2 x}{f^2}\right )}+d+e x\right )^{3/2}}}{2 \left (2 d e-b f^2\right )^3} \]
Antiderivative was successfully verified.
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fricas [B] time = 1.38, size = 2514, normalized size = 7.50 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
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 \[ \int \frac {1}{\left (e x +d +\sqrt {b x +\frac {e^{2} x^{2}}{f^{2}}+a}\, f \right )^{\frac {5}{2}}}\, dx \]
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 \[ \int \frac {1}{{\left (e x + \sqrt {b x + \frac {e^{2} x^{2}}{f^{2}} + a} f + d\right )}^{\frac {5}{2}}}\,{d x} \]
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 \[ \int \frac {1}{{\left (d+e\,x+f\,\sqrt {a+b\,x+\frac {e^2\,x^2}{f^2}}\right )}^{5/2}} \,d x \]
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 \[ \int \frac {1}{\left (d + e x + f \sqrt {a + b x + \frac {e^{2} x^{2}}{f^{2}}}\right )^{\frac {5}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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