Optimal. Leaf size=303 \[ -\frac {f^2 \left (4 a e^2-b^2 f^2\right ) \left (2 d e-b f^2\right )^3}{32 e^5 \left (2 e \left (f \sqrt {a+\frac {x \left (b f^2+e^2 x\right )}{f^2}}+e x\right )+b f^2\right )}+\frac {3 f^2 \left (4 a e^2-b^2 f^2\right ) \left (2 d e-b f^2\right )^2 \log \left (2 e \left (f \sqrt {a+\frac {x \left (b f^2+e^2 x\right )}{f^2}}+e x\right )+b f^2\right )}{32 e^5}+\frac {f^2 \left (4 a e^2-b^2 f^2\right ) \left (2 d e-b f^2\right ) \left (f \sqrt {a+b x+\frac {e^2 x^2}{f^2}}+e x\right )}{8 e^4}+\frac {f^2 \left (4 a e^2-b^2 f^2\right ) \left (f \sqrt {a+b x+\frac {e^2 x^2}{f^2}}+d+e x\right )^2}{16 e^3}+\frac {\left (f \sqrt {a+b x+\frac {e^2 x^2}{f^2}}+d+e x\right )^4}{8 e} \]
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Rubi [A] time = 0.38, antiderivative size = 303, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {2116, 893} \[ \frac {f^2 \left (4 a e^2-b^2 f^2\right ) \left (f \sqrt {a+b x+\frac {e^2 x^2}{f^2}}+d+e x\right )^2}{16 e^3}+\frac {f^2 \left (4 a e^2-b^2 f^2\right ) \left (2 d e-b f^2\right ) \left (f \sqrt {a+b x+\frac {e^2 x^2}{f^2}}+e x\right )}{8 e^4}-\frac {f^2 \left (4 a e^2-b^2 f^2\right ) \left (2 d e-b f^2\right )^3}{32 e^5 \left (2 e \left (f \sqrt {a+\frac {x \left (b f^2+e^2 x\right )}{f^2}}+e x\right )+b f^2\right )}+\frac {3 f^2 \left (4 a e^2-b^2 f^2\right ) \left (2 d e-b f^2\right )^2 \log \left (2 e \left (f \sqrt {a+\frac {x \left (b f^2+e^2 x\right )}{f^2}}+e x\right )+b f^2\right )}{32 e^5}+\frac {\left (f \sqrt {a+b x+\frac {e^2 x^2}{f^2}}+d+e x\right )^4}{8 e} \]
Antiderivative was successfully verified.
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Rule 893
Rule 2116
Rubi steps
\begin {align*} \int \left (d+e x+f \sqrt {a+b x+\frac {e^2 x^2}{f^2}}\right )^3 \, dx &=2 \operatorname {Subst}\left (\int \frac {x^3 \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 )\\ &=2 \operatorname {Subst}\left (\int \left (\frac {f^2 \left (2 d e-b f^2\right ) \left (4 a e^2-b^2 f^2\right )}{16 e^4}+\frac {f^2 \left (4 a e^2-b^2 f^2\right ) x}{16 e^3}+\frac {x^3}{4 e}+\frac {f^2 \left (2 d e-b f^2\right )^3 \left (4 a e^2-b^2 f^2\right )}{32 e^4 \left (2 d e-b f^2-2 e x\right )^2}-\frac {3 \left (4 a e^2-b^2 f^2\right ) \left (2 d e f-b f^3\right )^2}{32 e^4 \left (2 d e-b f^2-2 e x\right )}\right ) \, dx,x,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 ) \left (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 ) \left (d+e x+f \sqrt {a+b x+\frac {e^2 x^2}{f^2}}\right )^2}{16 e^3}+\frac {\left (d+e x+f \sqrt {a+b x+\frac {e^2 x^2}{f^2}}\right )^4}{8 e}-\frac {f^2 \left (2 d e-b f^2\right )^3 \left (4 a e^2-b^2 f^2\right )}{32 e^5 \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 \left (2 d e-b f^2\right )^2 \left (4 a e^2-b^2 f^2\right ) \log \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 )}{32 e^5}\\ \end {align*}
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Mathematica [A] time = 0.55, size = 276, normalized size = 0.91 \[ \frac {2 e^2 f^2 \left (4 a e^2-b^2 f^2\right ) \left (f \sqrt {a+x \left (b+\frac {e^2 x}{f^2}\right )}+d+e x\right )^2+4 e f^2 \left (4 a e^2-b^2 f^2\right ) \left (2 d e-b f^2\right ) \left (f \sqrt {a+x \left (b+\frac {e^2 x}{f^2}\right )}+e x\right )-\frac {f^2 \left (b^2 f^2-4 a e^2\right ) \left (b f^2-2 d e\right )^3}{2 e \left (f \sqrt {a+x \left (b+\frac {e^2 x}{f^2}\right )}+e x\right )+b f^2}-3 \left (b^2 f^2-4 a e^2\right ) \left (b f^3-2 d e f\right )^2 \log \left (-2 e \left (f \sqrt {a+x \left (b+\frac {e^2 x}{f^2}\right )}+e x\right )-b f^2\right )+4 e^4 \left (f \sqrt {a+x \left (b+\frac {e^2 x}{f^2}\right )}+d+e x\right )^4}{32 e^5} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.48, size = 345, normalized size = 1.14 \[ \frac {32 \, e^{8} x^{4} + 32 \, {\left (b e^{6} f^{2} + 2 \, d e^{7}\right )} x^{3} + 48 \, {\left (d^{2} e^{6} + {\left (b d e^{5} + a e^{6}\right )} f^{2}\right )} x^{2} + 32 \, {\left (3 \, a d e^{5} f^{2} + d^{3} e^{5}\right )} x + 3 \, {\left (b^{4} f^{8} - 16 \, a d^{2} e^{4} f^{2} - 4 \, {\left (b^{3} d e + a b^{2} e^{2}\right )} f^{6} + 4 \, {\left (b^{2} d^{2} e^{2} + 4 \, a b d e^{3}\right )} f^{4}\right )} \log \left (-b f^{2} - 2 \, e^{2} x + 2 \, e f \sqrt {\frac {b f^{2} x + e^{2} x^{2} + a f^{2}}{f^{2}}}\right ) + 2 \, {\left (3 \, b^{3} e f^{7} + 16 \, e^{7} f x^{3} - 4 \, {\left (3 \, b^{2} d e^{2} + 2 \, a b e^{3}\right )} f^{5} + 4 \, {\left (3 \, b d^{2} e^{3} + 8 \, a d e^{4}\right )} f^{3} + 8 \, {\left (b e^{5} f^{3} + 4 \, d e^{6} f\right )} x^{2} - 2 \, {\left (b^{2} e^{3} f^{5} - 12 \, d^{2} e^{5} f - 4 \, {\left (b d e^{4} + 2 \, a e^{5}\right )} f^{3}\right )} x\right )} \sqrt {\frac {b f^{2} x + e^{2} x^{2} + a f^{2}}{f^{2}}}}{32 \, e^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.37, size = 373, normalized size = 1.23 \[ b f^{2} x^{3} e + \frac {3}{2} \, b d f^{2} x^{2} + \frac {3}{2} \, a f^{2} x^{2} e + 3 \, a d f^{2} x + x^{4} e^{3} + 2 \, d x^{3} e^{2} + \frac {3}{2} \, d^{2} x^{2} e + d^{3} x + \frac {3}{32} \, {\left (b^{4} f^{7} {\left | f \right |} - 4 \, b^{3} d f^{5} {\left | f \right |} e - 4 \, a b^{2} f^{5} {\left | f \right |} e^{2} + 4 \, b^{2} d^{2} f^{3} {\left | f \right |} e^{2} + 16 \, a b d f^{3} {\left | f \right |} e^{3} - 16 \, a d^{2} f {\left | f \right |} e^{4}\right )} e^{\left (-5\right )} \log \left ({\left | -b f^{2} - 2 \, {\left (x e - \sqrt {b f^{2} x + a f^{2} + x^{2} e^{2}}\right )} e \right |}\right ) + \frac {1}{16} \, \sqrt {b f^{2} x + a f^{2} + x^{2} e^{2}} {\left (2 \, {\left (4 \, {\left (\frac {2 \, x {\left | f \right |} e^{2}}{f} + \frac {{\left (b f^{4} {\left | f \right |} e^{6} + 4 \, d f^{2} {\left | f \right |} e^{7}\right )} e^{\left (-6\right )}}{f^{3}}\right )} x - \frac {{\left (b^{2} f^{6} {\left | f \right |} e^{4} - 4 \, b d f^{4} {\left | f \right |} e^{5} - 8 \, a f^{4} {\left | f \right |} e^{6} - 12 \, d^{2} f^{2} {\left | f \right |} e^{6}\right )} e^{\left (-6\right )}}{f^{3}}\right )} x + \frac {{\left (3 \, b^{3} f^{8} {\left | f \right |} e^{2} - 12 \, b^{2} d f^{6} {\left | f \right |} e^{3} - 8 \, a b f^{6} {\left | f \right |} e^{4} + 12 \, b d^{2} f^{4} {\left | f \right |} e^{4} + 32 \, a d f^{4} {\left | f \right |} e^{5}\right )} e^{\left (-6\right )}}{f^{3}}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.02, size = 685, normalized size = 2.26 \[ -\frac {3 b^{4} f^{7} \ln \left (\frac {\frac {b}{2}+\frac {e^{2} x}{f^{2}}}{\sqrt {\frac {e^{2}}{f^{2}}}}+\sqrt {b x +\frac {e^{2} x^{2}}{f^{2}}+a}\right )}{32 \sqrt {\frac {e^{2}}{f^{2}}}\, e^{4}}+\frac {3 a \,b^{2} f^{5} \ln \left (\frac {\frac {b}{2}+\frac {e^{2} x}{f^{2}}}{\sqrt {\frac {e^{2}}{f^{2}}}}+\sqrt {b x +\frac {e^{2} x^{2}}{f^{2}}+a}\right )}{8 \sqrt {\frac {e^{2}}{f^{2}}}\, e^{2}}+\frac {3 b^{3} d \,f^{5} \ln \left (\frac {\frac {b}{2}+\frac {e^{2} x}{f^{2}}}{\sqrt {\frac {e^{2}}{f^{2}}}}+\sqrt {b x +\frac {e^{2} x^{2}}{f^{2}}+a}\right )}{8 \sqrt {\frac {e^{2}}{f^{2}}}\, e^{3}}+b e \,f^{2} x^{3}+e^{3} x^{4}-\frac {3 a b d \,f^{3} \ln \left (\frac {\frac {b}{2}+\frac {e^{2} x}{f^{2}}}{\sqrt {\frac {e^{2}}{f^{2}}}}+\sqrt {b x +\frac {e^{2} x^{2}}{f^{2}}+a}\right )}{2 \sqrt {\frac {e^{2}}{f^{2}}}\, e}+\frac {3 a e \,f^{2} x^{2}}{2}+\frac {3 \sqrt {b x +\frac {e^{2} x^{2}}{f^{2}}+a}\, b^{3} f^{7}}{16 e^{4}}-\frac {3 b^{2} d^{2} f^{3} \ln \left (\frac {\frac {b}{2}+\frac {e^{2} x}{f^{2}}}{\sqrt {\frac {e^{2}}{f^{2}}}}+\sqrt {b x +\frac {e^{2} x^{2}}{f^{2}}+a}\right )}{8 \sqrt {\frac {e^{2}}{f^{2}}}\, e^{2}}+\frac {3 \sqrt {b x +\frac {e^{2} x^{2}}{f^{2}}+a}\, b^{2} f^{5} x}{8 e^{2}}+\frac {3 b d \,f^{2} x^{2}}{2}+2 d \,e^{2} x^{3}+\frac {3 a \,d^{2} f \ln \left (\frac {\frac {b}{2}+\frac {e^{2} x}{f^{2}}}{\sqrt {\frac {e^{2}}{f^{2}}}}+\sqrt {b x +\frac {e^{2} x^{2}}{f^{2}}+a}\right )}{2 \sqrt {\frac {e^{2}}{f^{2}}}}+3 a d \,f^{2} x -\frac {3 \sqrt {b x +\frac {e^{2} x^{2}}{f^{2}}+a}\, b^{2} d \,f^{5}}{4 e^{3}}-\frac {3 \sqrt {b x +\frac {e^{2} x^{2}}{f^{2}}+a}\, b d \,f^{3} x}{2 e}+\frac {3 d^{2} e \,x^{2}}{2}+\frac {3 \sqrt {b x +\frac {e^{2} x^{2}}{f^{2}}+a}\, b \,d^{2} f^{3}}{4 e^{2}}-\frac {\left (b x +\frac {e^{2} x^{2}}{f^{2}}+a \right )^{\frac {3}{2}} b \,f^{5}}{2 e^{2}}+d^{3} x +\frac {3 \sqrt {b x +\frac {e^{2} x^{2}}{f^{2}}+a}\, d^{2} f x}{2}+\left (b x +\frac {e^{2} x^{2}}{f^{2}}+a \right )^{\frac {3}{2}} f^{3} x +\frac {d^{4}}{4 e}+\frac {2 \left (b x +\frac {e^{2} x^{2}}{f^{2}}+a \right )^{\frac {3}{2}} d \,f^{3}}{e} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: ValueError} \]
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 \[ \int {\left (d+e\,x+f\,\sqrt {a+b\,x+\frac {e^2\,x^2}{f^2}}\right )}^3 \,d x \]
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 \[ \int \left (d + e x + f \sqrt {a + b x + \frac {e^{2} x^{2}}{f^{2}}}\right )^{3}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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