Optimal. Leaf size=255 \[ \frac {5 \left (c d^2-a e^2\right )^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{8 c^3 d^3}+\frac {5 \left (c d^2-a e^2\right ) (d+e x) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{12 c^2 d^2}+\frac {(d+e x)^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{3 c d}+\frac {5 \left (c d^2-a e^2\right )^3 \tanh ^{-1}\left (\frac {c d^2+a e^2+2 c d e x}{2 \sqrt {c} \sqrt {d} \sqrt {e} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\right )}{16 c^{7/2} d^{7/2} \sqrt {e}} \]
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Rubi [A]
time = 0.14, antiderivative size = 255, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 4, integrand size = 37, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.108, Rules used = {684, 654, 635,
212} \begin {gather*} \frac {5 \left (c d^2-a e^2\right )^3 \tanh ^{-1}\left (\frac {a e^2+c d^2+2 c d e x}{2 \sqrt {c} \sqrt {d} \sqrt {e} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{16 c^{7/2} d^{7/2} \sqrt {e}}+\frac {5 \left (c d^2-a e^2\right )^2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{8 c^3 d^3}+\frac {5 (d+e x) \left (c d^2-a e^2\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{12 c^2 d^2}+\frac {(d+e x)^2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{3 c d} \end {gather*}
Antiderivative was successfully verified.
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Rule 212
Rule 635
Rule 654
Rule 684
Rubi steps
\begin {align*} \int \frac {(d+e x)^3}{\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx &=\frac {(d+e x)^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{3 c d}+\frac {\left (5 \left (d^2-\frac {a e^2}{c}\right )\right ) \int \frac {(d+e x)^2}{\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx}{6 d}\\ &=\frac {5 \left (c d^2-a e^2\right ) (d+e x) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{12 c^2 d^2}+\frac {(d+e x)^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{3 c d}+\frac {\left (5 \left (d^2-\frac {a e^2}{c}\right )^2\right ) \int \frac {d+e x}{\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx}{8 d^2}\\ &=\frac {5 \left (c d^2-a e^2\right )^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{8 c^3 d^3}+\frac {5 \left (c d^2-a e^2\right ) (d+e x) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{12 c^2 d^2}+\frac {(d+e x)^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{3 c d}+\frac {\left (5 \left (d^2-\frac {a e^2}{c}\right )^3\right ) \int \frac {1}{\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx}{16 d^3}\\ &=\frac {5 \left (c d^2-a e^2\right )^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{8 c^3 d^3}+\frac {5 \left (c d^2-a e^2\right ) (d+e x) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{12 c^2 d^2}+\frac {(d+e x)^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{3 c d}+\frac {\left (5 \left (d^2-\frac {a e^2}{c}\right )^3\right ) \text {Subst}\left (\int \frac {1}{4 c d e-x^2} \, dx,x,\frac {c d^2+a e^2+2 c d e x}{\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\right )}{8 d^3}\\ &=\frac {5 \left (c d^2-a e^2\right )^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{8 c^3 d^3}+\frac {5 \left (c d^2-a e^2\right ) (d+e x) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{12 c^2 d^2}+\frac {(d+e x)^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{3 c d}+\frac {5 \left (c d^2-a e^2\right )^3 \tanh ^{-1}\left (\frac {c d^2+a e^2+2 c d e x}{2 \sqrt {c} \sqrt {d} \sqrt {e} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\right )}{16 c^{7/2} d^{7/2} \sqrt {e}}\\ \end {align*}
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Mathematica [A]
time = 0.30, size = 192, normalized size = 0.75 \begin {gather*} \frac {\sqrt {c} \sqrt {d} \sqrt {e} (a e+c d x) (d+e x) \left (15 a^2 e^4-10 a c d e^2 (4 d+e x)+c^2 d^2 \left (33 d^2+26 d e x+8 e^2 x^2\right )\right )+15 \left (c d^2-a e^2\right )^3 \sqrt {a e+c d x} \sqrt {d+e x} \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {a e+c d x}}{\sqrt {c} \sqrt {d} \sqrt {d+e x}}\right )}{24 c^{7/2} d^{7/2} \sqrt {e} \sqrt {(a e+c d x) (d+e x)}} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(887\) vs.
\(2(225)=450\).
time = 0.88, size = 888, normalized size = 3.48
method | result | size |
default | \(e^{3} \left (\frac {x^{2} \sqrt {a d e +\left (e^{2} a +c \,d^{2}\right ) x +c d e \,x^{2}}}{3 c d e}-\frac {5 \left (e^{2} a +c \,d^{2}\right ) \left (\frac {x \sqrt {a d e +\left (e^{2} a +c \,d^{2}\right ) x +c d e \,x^{2}}}{2 c d e}-\frac {3 \left (e^{2} a +c \,d^{2}\right ) \left (\frac {\sqrt {a d e +\left (e^{2} a +c \,d^{2}\right ) x +c d e \,x^{2}}}{c d e}-\frac {\left (e^{2} a +c \,d^{2}\right ) \ln \left (\frac {\frac {1}{2} e^{2} a +\frac {1}{2} c \,d^{2}+c d e x}{\sqrt {c d e}}+\sqrt {a d e +\left (e^{2} a +c \,d^{2}\right ) x +c d e \,x^{2}}\right )}{2 c d e \sqrt {c d e}}\right )}{4 c d e}-\frac {a \ln \left (\frac {\frac {1}{2} e^{2} a +\frac {1}{2} c \,d^{2}+c d e x}{\sqrt {c d e}}+\sqrt {a d e +\left (e^{2} a +c \,d^{2}\right ) x +c d e \,x^{2}}\right )}{2 c \sqrt {c d e}}\right )}{6 c d e}-\frac {2 a \left (\frac {\sqrt {a d e +\left (e^{2} a +c \,d^{2}\right ) x +c d e \,x^{2}}}{c d e}-\frac {\left (e^{2} a +c \,d^{2}\right ) \ln \left (\frac {\frac {1}{2} e^{2} a +\frac {1}{2} c \,d^{2}+c d e x}{\sqrt {c d e}}+\sqrt {a d e +\left (e^{2} a +c \,d^{2}\right ) x +c d e \,x^{2}}\right )}{2 c d e \sqrt {c d e}}\right )}{3 c}\right )+3 d \,e^{2} \left (\frac {x \sqrt {a d e +\left (e^{2} a +c \,d^{2}\right ) x +c d e \,x^{2}}}{2 c d e}-\frac {3 \left (e^{2} a +c \,d^{2}\right ) \left (\frac {\sqrt {a d e +\left (e^{2} a +c \,d^{2}\right ) x +c d e \,x^{2}}}{c d e}-\frac {\left (e^{2} a +c \,d^{2}\right ) \ln \left (\frac {\frac {1}{2} e^{2} a +\frac {1}{2} c \,d^{2}+c d e x}{\sqrt {c d e}}+\sqrt {a d e +\left (e^{2} a +c \,d^{2}\right ) x +c d e \,x^{2}}\right )}{2 c d e \sqrt {c d e}}\right )}{4 c d e}-\frac {a \ln \left (\frac {\frac {1}{2} e^{2} a +\frac {1}{2} c \,d^{2}+c d e x}{\sqrt {c d e}}+\sqrt {a d e +\left (e^{2} a +c \,d^{2}\right ) x +c d e \,x^{2}}\right )}{2 c \sqrt {c d e}}\right )+3 d^{2} e \left (\frac {\sqrt {a d e +\left (e^{2} a +c \,d^{2}\right ) x +c d e \,x^{2}}}{c d e}-\frac {\left (e^{2} a +c \,d^{2}\right ) \ln \left (\frac {\frac {1}{2} e^{2} a +\frac {1}{2} c \,d^{2}+c d e x}{\sqrt {c d e}}+\sqrt {a d e +\left (e^{2} a +c \,d^{2}\right ) x +c d e \,x^{2}}\right )}{2 c d e \sqrt {c d e}}\right )+\frac {d^{3} \ln \left (\frac {\frac {1}{2} e^{2} a +\frac {1}{2} c \,d^{2}+c d e x}{\sqrt {c d e}}+\sqrt {a d e +\left (e^{2} a +c \,d^{2}\right ) x +c d e \,x^{2}}\right )}{\sqrt {c d e}}\) | \(888\) |
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 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 2.83, size = 513, normalized size = 2.01 \begin {gather*} \left [\frac {{\left (15 \, {\left (c^{3} d^{6} - 3 \, a c^{2} d^{4} e^{2} + 3 \, a^{2} c d^{2} e^{4} - a^{3} e^{6}\right )} \sqrt {c d} e^{\frac {1}{2}} \log \left (8 \, c^{2} d^{3} x e + c^{2} d^{4} + 8 \, a c d x e^{3} + a^{2} e^{4} + 4 \, \sqrt {c d^{2} x + a x e^{2} + {\left (c d x^{2} + a d\right )} e} {\left (2 \, c d x e + c d^{2} + a e^{2}\right )} \sqrt {c d} e^{\frac {1}{2}} + 2 \, {\left (4 \, c^{2} d^{2} x^{2} + 3 \, a c d^{2}\right )} e^{2}\right ) + 4 \, {\left (26 \, c^{3} d^{4} x e^{2} + 33 \, c^{3} d^{5} e - 10 \, a c^{2} d^{2} x e^{4} + 15 \, a^{2} c d e^{5} + 8 \, {\left (c^{3} d^{3} x^{2} - 5 \, a c^{2} d^{3}\right )} e^{3}\right )} \sqrt {c d^{2} x + a x e^{2} + {\left (c d x^{2} + a d\right )} e}\right )} e^{\left (-1\right )}}{96 \, c^{4} d^{4}}, -\frac {{\left (15 \, {\left (c^{3} d^{6} - 3 \, a c^{2} d^{4} e^{2} + 3 \, a^{2} c d^{2} e^{4} - a^{3} e^{6}\right )} \sqrt {-c d e} \arctan \left (\frac {\sqrt {c d^{2} x + a x e^{2} + {\left (c d x^{2} + a d\right )} e} {\left (2 \, c d x e + c d^{2} + a e^{2}\right )} \sqrt {-c d e}}{2 \, {\left (c^{2} d^{3} x e + a c d x e^{3} + {\left (c^{2} d^{2} x^{2} + a c d^{2}\right )} e^{2}\right )}}\right ) - 2 \, {\left (26 \, c^{3} d^{4} x e^{2} + 33 \, c^{3} d^{5} e - 10 \, a c^{2} d^{2} x e^{4} + 15 \, a^{2} c d e^{5} + 8 \, {\left (c^{3} d^{3} x^{2} - 5 \, a c^{2} d^{3}\right )} e^{3}\right )} \sqrt {c d^{2} x + a x e^{2} + {\left (c d x^{2} + a d\right )} e}\right )} e^{\left (-1\right )}}{48 \, c^{4} d^{4}}\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 \frac {\left (d + e x\right )^{3}}{\sqrt {\left (d + e x\right ) \left (a e + c d x\right )}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.83, size = 232, normalized size = 0.91 \begin {gather*} \frac {1}{24} \, \sqrt {c d x^{2} e + c d^{2} x + a x e^{2} + a d e} {\left (2 \, x {\left (\frac {4 \, x e^{2}}{c d} + \frac {{\left (13 \, c^{2} d^{3} e^{3} - 5 \, a c d e^{5}\right )} e^{\left (-2\right )}}{c^{3} d^{3}}\right )} + \frac {{\left (33 \, c^{2} d^{4} e^{2} - 40 \, a c d^{2} e^{4} + 15 \, a^{2} e^{6}\right )} e^{\left (-2\right )}}{c^{3} d^{3}}\right )} - \frac {5 \, {\left (c^{3} d^{6} - 3 \, a c^{2} d^{4} e^{2} + 3 \, a^{2} c d^{2} e^{4} - a^{3} e^{6}\right )} \sqrt {c d} e^{\left (-\frac {1}{2}\right )} \log \left ({\left | -\sqrt {c d} c d^{2} e^{\frac {1}{2}} - 2 \, {\left (\sqrt {c d} x e^{\frac {1}{2}} - \sqrt {c d x^{2} e + c d^{2} x + a x e^{2} + a d e}\right )} c d e - \sqrt {c d} a e^{\frac {5}{2}} \right |}\right )}{16 \, c^{4} d^{4}} \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 \frac {{\left (d+e\,x\right )}^3}{\sqrt {c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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