Optimal. Leaf size=235 \[ \frac {15 c d \left (a-\frac {c d^2}{e^2}\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{4 e}+\frac {5 c d \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{2 e^2 (d+e x)}-\frac {2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{e (d+e x)^3}+\frac {15 \sqrt {c} \sqrt {d} \left (c d^2-a e^2\right )^2 \tanh ^{-1}\left (\frac {c d^2+a e^2+2 c d e x}{2 \sqrt {c} \sqrt {d} \sqrt {e} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\right )}{8 e^{7/2}} \]
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Rubi [A]
time = 0.12, antiderivative size = 235, 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 = {676, 678, 635,
212} \begin {gather*} -\frac {2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{e (d+e x)^3}+\frac {5 c d \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{2 e^2 (d+e x)}+\frac {15 c d \left (a-\frac {c d^2}{e^2}\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{4 e}+\frac {15 \sqrt {c} \sqrt {d} \left (c d^2-a e^2\right )^2 \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 )}{8 e^{7/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 212
Rule 635
Rule 676
Rule 678
Rubi steps
\begin {align*} \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{(d+e x)^4} \, dx &=-\frac {2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{e (d+e x)^3}+\frac {(5 c d) \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{(d+e x)^2} \, dx}{e}\\ &=\frac {5 c d \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{2 e^2 (d+e x)}-\frac {2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{e (d+e x)^3}-\frac {\left (15 c d \left (c d^2-a e^2\right )\right ) \int \frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{d+e x} \, dx}{4 e^2}\\ &=-\frac {15 c d \left (c d^2-a e^2\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{4 e^3}+\frac {5 c d \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{2 e^2 (d+e x)}-\frac {2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{e (d+e x)^3}+\frac {\left (15 c d \left (c d^2-a e^2\right )^2\right ) \int \frac {1}{\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx}{8 e^3}\\ &=-\frac {15 c d \left (c d^2-a e^2\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{4 e^3}+\frac {5 c d \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{2 e^2 (d+e x)}-\frac {2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{e (d+e x)^3}+\frac {\left (15 c d \left (c d^2-a e^2\right )^2\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 )}{4 e^3}\\ &=-\frac {15 c d \left (c d^2-a e^2\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{4 e^3}+\frac {5 c d \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{2 e^2 (d+e x)}-\frac {2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{e (d+e x)^3}+\frac {15 \sqrt {c} \sqrt {d} \left (c d^2-a e^2\right )^2 \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 )}{8 e^{7/2}}\\ \end {align*}
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Mathematica [A]
time = 0.42, size = 186, normalized size = 0.79 \begin {gather*} \frac {((a e+c d x) (d+e x))^{5/2} \left (\frac {\sqrt {e} \left (-8 a^2 e^4+a c d e^2 (25 d+9 e x)+c^2 d^2 \left (-15 d^2-5 d e x+2 e^2 x^2\right )\right )}{(a e+c d x)^2 (d+e x)^3}+\frac {15 \sqrt {c} \sqrt {d} \left (c d^2-a e^2\right )^2 \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {a e+c d x}}{\sqrt {c} \sqrt {d} \sqrt {d+e x}}\right )}{(a e+c d x)^{5/2} (d+e x)^{5/2}}\right )}{4 e^{7/2}} \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. \(569\) vs.
\(2(207)=414\).
time = 0.80, size = 570, normalized size = 2.43
method | result | size |
default | \(\frac {-\frac {2 \left (c d e \left (x +\frac {d}{e}\right )^{2}+\left (e^{2} a -c \,d^{2}\right ) \left (x +\frac {d}{e}\right )\right )^{\frac {7}{2}}}{\left (e^{2} a -c \,d^{2}\right ) \left (x +\frac {d}{e}\right )^{4}}+\frac {6 c d e \left (\frac {2 \left (c d e \left (x +\frac {d}{e}\right )^{2}+\left (e^{2} a -c \,d^{2}\right ) \left (x +\frac {d}{e}\right )\right )^{\frac {7}{2}}}{\left (e^{2} a -c \,d^{2}\right ) \left (x +\frac {d}{e}\right )^{3}}-\frac {8 c d e \left (\frac {2 \left (c d e \left (x +\frac {d}{e}\right )^{2}+\left (e^{2} a -c \,d^{2}\right ) \left (x +\frac {d}{e}\right )\right )^{\frac {7}{2}}}{3 \left (e^{2} a -c \,d^{2}\right ) \left (x +\frac {d}{e}\right )^{2}}-\frac {10 c d e \left (\frac {\left (c d e \left (x +\frac {d}{e}\right )^{2}+\left (e^{2} a -c \,d^{2}\right ) \left (x +\frac {d}{e}\right )\right )^{\frac {5}{2}}}{5}+\frac {\left (e^{2} a -c \,d^{2}\right ) \left (\frac {\left (2 c d e \left (x +\frac {d}{e}\right )+e^{2} a -c \,d^{2}\right ) \left (c d e \left (x +\frac {d}{e}\right )^{2}+\left (e^{2} a -c \,d^{2}\right ) \left (x +\frac {d}{e}\right )\right )^{\frac {3}{2}}}{8 c d e}-\frac {3 \left (e^{2} a -c \,d^{2}\right )^{2} \left (\frac {\left (2 c d e \left (x +\frac {d}{e}\right )+e^{2} a -c \,d^{2}\right ) \sqrt {c d e \left (x +\frac {d}{e}\right )^{2}+\left (e^{2} a -c \,d^{2}\right ) \left (x +\frac {d}{e}\right )}}{4 c d e}-\frac {\left (e^{2} a -c \,d^{2}\right )^{2} \ln \left (\frac {\frac {e^{2} a}{2}-\frac {c \,d^{2}}{2}+c d e \left (x +\frac {d}{e}\right )}{\sqrt {c d e}}+\sqrt {c d e \left (x +\frac {d}{e}\right )^{2}+\left (e^{2} a -c \,d^{2}\right ) \left (x +\frac {d}{e}\right )}\right )}{8 c d e \sqrt {c d e}}\right )}{16 c d e}\right )}{2}\right )}{3 \left (e^{2} a -c \,d^{2}\right )}\right )}{e^{2} a -c \,d^{2}}\right )}{e^{2} a -c \,d^{2}}}{e^{4}}\) | \(570\) |
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 = 4.44, size = 529, normalized size = 2.25 \begin {gather*} \left [\frac {15 \, {\left (c^{2} d^{4} x e + c^{2} d^{5} - 2 \, a c d^{2} x e^{3} - 2 \, a c d^{3} e^{2} + a^{2} x e^{5} + a^{2} d e^{4}\right )} \sqrt {c d} e^{\left (-\frac {1}{2}\right )} \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^{2} + c d^{2} e + a e^{3}\right )} \sqrt {c d} e^{\left (-\frac {1}{2}\right )} + 2 \, {\left (4 \, c^{2} d^{2} x^{2} + 3 \, a c d^{2}\right )} e^{2}\right ) - 4 \, {\left (5 \, c^{2} d^{3} x e + 15 \, c^{2} d^{4} - 9 \, a c d x e^{3} + 8 \, a^{2} e^{4} - {\left (2 \, c^{2} d^{2} x^{2} + 25 \, a c d^{2}\right )} e^{2}\right )} \sqrt {c d^{2} x + a x e^{2} + {\left (c d x^{2} + a d\right )} e}}{16 \, {\left (x e^{4} + d e^{3}\right )}}, -\frac {15 \, {\left (c^{2} d^{4} x e + c^{2} d^{5} - 2 \, a c d^{2} x e^{3} - 2 \, a c d^{3} e^{2} + a^{2} x e^{5} + a^{2} d e^{4}\right )} \sqrt {-c d e^{\left (-1\right )}} \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^{\left (-1\right )}}}{2 \, {\left (c^{2} d^{3} x + a c d x e^{2} + {\left (c^{2} d^{2} x^{2} + a c d^{2}\right )} e\right )}}\right ) + 2 \, {\left (5 \, c^{2} d^{3} x e + 15 \, c^{2} d^{4} - 9 \, a c d x e^{3} + 8 \, a^{2} e^{4} - {\left (2 \, c^{2} d^{2} x^{2} + 25 \, a c d^{2}\right )} e^{2}\right )} \sqrt {c d^{2} x + a x e^{2} + {\left (c d x^{2} + a d\right )} e}}{8 \, {\left (x e^{4} + d e^{3}\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.96, size = 294, normalized size = 1.25 \begin {gather*} \frac {1}{4} \, {\left (2 \, c^{2} d^{2} x e^{\left (-2\right )} - \frac {{\left (7 \, c^{3} d^{4} e^{5} - 9 \, a c^{2} d^{2} e^{7}\right )} e^{\left (-8\right )}}{c d}\right )} \sqrt {c d x^{2} e + c d^{2} x + a x e^{2} + a d e} - \frac {2 \, {\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 )} e^{\left (-3\right )}}{\sqrt {c d} d e^{\frac {1}{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 )} e} - \frac {15 \, {\left (\sqrt {c d} c^{3} d^{5} e^{\frac {1}{2}} - 2 \, \sqrt {c d} a c^{2} d^{3} e^{\frac {5}{2}} + \sqrt {c d} a^{2} c d e^{\frac {9}{2}}\right )} e^{\left (-4\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 )}{8 \, c d} \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 (c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e\right )}^{5/2}}{{\left (d+e\,x\right )}^4} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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