Optimal. Leaf size=237 \[ -\frac {f^2 \left (4 a e^2-b^2 f^2\right ) \left (2 d e-b f^2\right )^2}{16 e^4 \left (2 e \left (f \sqrt {a+\frac {x \left (b f^2+e^2 x\right )}{f^2}}+e x\right )+b f^2\right )}+\frac {f^2 \left (4 a e^2-b^2 f^2\right ) \left (2 d e-b f^2\right ) \log \left (2 e \left (f \sqrt {a+\frac {x \left (b f^2+e^2 x\right )}{f^2}}+e x\right )+b f^2\right )}{8 e^4}+\frac {f^2 \left (4 a e^2-b^2 f^2\right ) \left (f \sqrt {a+b x+\frac {e^2 x^2}{f^2}}+e x\right )}{8 e^3}+\frac {\left (f \sqrt {a+b x+\frac {e^2 x^2}{f^2}}+d+e x\right )^3}{6 e} \]
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Rubi [A] time = 0.24, antiderivative size = 237, 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} \[ -\frac {f^2 \left (4 a e^2-b^2 f^2\right ) \left (2 d e-b f^2\right )^2}{16 e^4 \left (2 e \left (f \sqrt {a+\frac {x \left (b f^2+e^2 x\right )}{f^2}}+e x\right )+b f^2\right )}+\frac {f^2 \left (4 a e^2-b^2 f^2\right ) \left (2 d e-b f^2\right ) \log \left (2 e \left (f \sqrt {a+\frac {x \left (b f^2+e^2 x\right )}{f^2}}+e x\right )+b f^2\right )}{8 e^4}+\frac {f^2 \left (4 a e^2-b^2 f^2\right ) \left (f \sqrt {a+b x+\frac {e^2 x^2}{f^2}}+e x\right )}{8 e^3}+\frac {\left (f \sqrt {a+b x+\frac {e^2 x^2}{f^2}}+d+e x\right )^3}{6 e} \]
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 )^2 \, dx &=2 \operatorname {Subst}\left (\int \frac {x^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 )\\ &=2 \operatorname {Subst}\left (\int \left (\frac {4 a e^2 f^2-b^2 f^4}{16 e^3}+\frac {x^2}{4 e}+\frac {\left (4 a e^2-b^2 f^2\right ) \left (2 d e f-b f^3\right )^2}{16 e^3 \left (2 d e-b f^2-2 e x\right )^2}-\frac {f^2 \left (2 d e-b f^2\right ) \left (4 a e^2-b^2 f^2\right )}{8 e^3 \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 (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^3}+\frac {\left (d+e x+f \sqrt {a+b x+\frac {e^2 x^2}{f^2}}\right )^3}{6 e}-\frac {f^2 \left (2 d e-b f^2\right )^2 \left (4 a e^2-b^2 f^2\right )}{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 {f^2 \left (2 d e-b f^2\right ) \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 )}{8 e^4}\\ \end {align*}
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Mathematica [A] time = 0.32, size = 213, normalized size = 0.90 \[ \frac {6 f^2 \left (b^2 f^2-4 a e^2\right ) \left (b f^2-2 d e\right ) \log \left (-2 e \left (f \sqrt {a+x \left (b+\frac {e^2 x}{f^2}\right )}+e x\right )-b f^2\right )+\frac {3 \left (b^2 f^2-4 a e^2\right ) \left (b f^3-2 d e f\right )^2}{2 e \left (f \sqrt {a+x \left (b+\frac {e^2 x}{f^2}\right )}+e x\right )+b f^2}+6 e f^2 \left (4 a e^2-b^2 f^2\right ) \left (f \sqrt {a+x \left (b+\frac {e^2 x}{f^2}\right )}+e x\right )+8 e^3 \left (f \sqrt {a+x \left (b+\frac {e^2 x}{f^2}\right )}+d+e x\right )^3}{48 e^4} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.56, size = 219, normalized size = 0.92 \[ \frac {16 \, e^{6} x^{3} + 12 \, {\left (b e^{4} f^{2} + 2 \, d e^{5}\right )} x^{2} + 24 \, {\left (a e^{4} f^{2} + d^{2} e^{4}\right )} x - 3 \, {\left (b^{3} f^{6} + 8 \, a d e^{3} f^{2} - 2 \, {\left (b^{2} d e + 2 \, a b e^{2}\right )} f^{4}\right )} \log \left (-b f^{2} - 2 \, e^{2} x + 2 \, e f \sqrt {\frac {b f^{2} x + e^{2} x^{2} + a f^{2}}{f^{2}}}\right ) - 2 \, {\left (3 \, b^{2} e f^{5} - 8 \, e^{5} f x^{2} - 2 \, {\left (3 \, b d e^{2} + 4 \, a e^{3}\right )} f^{3} - 2 \, {\left (b e^{3} f^{3} + 6 \, d e^{4} f\right )} x\right )} \sqrt {\frac {b f^{2} x + e^{2} x^{2} + a f^{2}}{f^{2}}}}{24 \, e^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.30, size = 224, normalized size = 0.95 \[ \frac {1}{2} \, b f^{2} x^{2} + a f^{2} x + \frac {2}{3} \, x^{3} e^{2} + d x^{2} e + d^{2} x - \frac {1}{8} \, {\left (b^{3} f^{5} {\left | f \right |} - 2 \, b^{2} d f^{3} {\left | f \right |} e - 4 \, a b f^{3} {\left | f \right |} e^{2} + 8 \, a d f {\left | f \right |} e^{3}\right )} e^{\left (-4\right )} \log \left ({\left | -b f^{2} - 2 \, {\left (x e - \sqrt {b f^{2} x + a f^{2} + x^{2} e^{2}}\right )} e \right |}\right ) + \frac {1}{12} \, \sqrt {b f^{2} x + a f^{2} + x^{2} e^{2}} {\left (2 \, {\left (\frac {4 \, x {\left | f \right |} e}{f} + \frac {{\left (b f^{3} {\left | f \right |} e^{3} + 6 \, d f {\left | f \right |} e^{4}\right )} e^{\left (-4\right )}}{f^{2}}\right )} x - \frac {{\left (3 \, b^{2} f^{5} {\left | f \right |} e - 6 \, b d f^{3} {\left | f \right |} e^{2} - 8 \, a f^{3} {\left | f \right |} e^{3}\right )} e^{\left (-4\right )}}{f^{2}}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 409, normalized size = 1.73 \[ \frac {b^{3} f^{5} \ln \left (\frac {\frac {b}{2}+\frac {e^{2} x}{f^{2}}}{\sqrt {\frac {e^{2}}{f^{2}}}}+\sqrt {b x +\frac {e^{2} x^{2}}{f^{2}}+a}\right )}{8 \sqrt {\frac {e^{2}}{f^{2}}}\, e^{3}}-\frac {a b \,f^{3} \ln \left (\frac {\frac {b}{2}+\frac {e^{2} x}{f^{2}}}{\sqrt {\frac {e^{2}}{f^{2}}}}+\sqrt {b x +\frac {e^{2} x^{2}}{f^{2}}+a}\right )}{2 \sqrt {\frac {e^{2}}{f^{2}}}\, e}-\frac {b^{2} d \,f^{3} \ln \left (\frac {\frac {b}{2}+\frac {e^{2} x}{f^{2}}}{\sqrt {\frac {e^{2}}{f^{2}}}}+\sqrt {b x +\frac {e^{2} x^{2}}{f^{2}}+a}\right )}{4 \sqrt {\frac {e^{2}}{f^{2}}}\, e^{2}}+\frac {b \,f^{2} x^{2}}{2}+\frac {2 e^{2} x^{3}}{3}+\frac {a d f \ln \left (\frac {\frac {b}{2}+\frac {e^{2} x}{f^{2}}}{\sqrt {\frac {e^{2}}{f^{2}}}}+\sqrt {b x +\frac {e^{2} x^{2}}{f^{2}}+a}\right )}{\sqrt {\frac {e^{2}}{f^{2}}}}+a \,f^{2} x -\frac {\sqrt {b x +\frac {e^{2} x^{2}}{f^{2}}+a}\, b^{2} f^{5}}{4 e^{3}}-\frac {\sqrt {b x +\frac {e^{2} x^{2}}{f^{2}}+a}\, b \,f^{3} x}{2 e}+d e \,x^{2}+\frac {\sqrt {b x +\frac {e^{2} x^{2}}{f^{2}}+a}\, b d \,f^{3}}{2 e^{2}}+d^{2} x +\sqrt {b x +\frac {e^{2} x^{2}}{f^{2}}+a}\, d f x +\frac {d^{3}}{3 e}+\frac {2 \left (b x +\frac {e^{2} x^{2}}{f^{2}}+a \right )^{\frac {3}{2}} f^{3}}{3 e} \]
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 \[ \int {\left (d+e\,x+f\,\sqrt {a+b\,x+\frac {e^2\,x^2}{f^2}}\right )}^2 \,d x \]
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 \[ \int \left (d + e x + f \sqrt {a + b x + \frac {e^{2} x^{2}}{f^{2}}}\right )^{2}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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