Optimal. Leaf size=328 \[ \frac {7 \left (c d^2-a e^2\right )^3 \left (c d^2+a e^2+2 c d e x\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{128 c^4 d^4 e}+\frac {7 \left (c d^2-a e^2\right )^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{48 c^3 d^3}+\frac {7 \left (c d^2-a e^2\right ) (d+e x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{40 c^2 d^2}+\frac {(d+e x)^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{5 c d}-\frac {7 \left (c d^2-a e^2\right )^5 \tanh ^{-1}\left (\frac {c d^2+a e^2+2 c d e x}{2 \sqrt {c} \sqrt {d} \sqrt {e} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\right )}{256 c^{9/2} d^{9/2} e^{3/2}} \]
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Rubi [A]
time = 0.18, antiderivative size = 328, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 5, integrand size = 37, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.135, Rules used = {684, 654, 626,
635, 212} \begin {gather*} -\frac {7 \left (c d^2-a e^2\right )^5 \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 )}{256 c^{9/2} d^{9/2} e^{3/2}}+\frac {7 \left (c d^2-a e^2\right )^3 \left (a e^2+c d^2+2 c d e x\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{128 c^4 d^4 e}+\frac {7 \left (c d^2-a e^2\right )^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{48 c^3 d^3}+\frac {7 (d+e x) \left (c d^2-a e^2\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{40 c^2 d^2}+\frac {(d+e x)^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{5 c d} \end {gather*}
Antiderivative was successfully verified.
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Rule 212
Rule 626
Rule 635
Rule 654
Rule 684
Rubi steps
\begin {align*} \int (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 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{5 c d}+\frac {\left (7 \left (d^2-\frac {a e^2}{c}\right )\right ) \int (d+e x)^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2} \, dx}{10 d}\\ &=\frac {7 \left (c d^2-a e^2\right ) (d+e x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{40 c^2 d^2}+\frac {(d+e x)^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{5 c d}+\frac {\left (7 \left (d^2-\frac {a e^2}{c}\right )^2\right ) \int (d+e x) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2} \, dx}{16 d^2}\\ &=\frac {7 \left (c d^2-a e^2\right )^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{48 c^3 d^3}+\frac {7 \left (c d^2-a e^2\right ) (d+e x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{40 c^2 d^2}+\frac {(d+e x)^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{5 c d}+\frac {\left (7 \left (d^2-\frac {a e^2}{c}\right )^3\right ) \int \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2} \, dx}{32 d^3}\\ &=\frac {7 \left (c d^2-a e^2\right )^3 \left (c d^2+a e^2+2 c d e x\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{128 c^4 d^4 e}+\frac {7 \left (c d^2-a e^2\right )^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{48 c^3 d^3}+\frac {7 \left (c d^2-a e^2\right ) (d+e x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{40 c^2 d^2}+\frac {(d+e x)^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{5 c d}-\frac {\left (7 \left (c d^2-a e^2\right )^5\right ) \int \frac {1}{\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx}{256 c^4 d^4 e}\\ &=\frac {7 \left (c d^2-a e^2\right )^3 \left (c d^2+a e^2+2 c d e x\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{128 c^4 d^4 e}+\frac {7 \left (c d^2-a e^2\right )^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{48 c^3 d^3}+\frac {7 \left (c d^2-a e^2\right ) (d+e x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{40 c^2 d^2}+\frac {(d+e x)^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{5 c d}-\frac {\left (7 \left (c d^2-a e^2\right )^5\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 )}{128 c^4 d^4 e}\\ &=\frac {7 \left (c d^2-a e^2\right )^3 \left (c d^2+a e^2+2 c d e x\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{128 c^4 d^4 e}+\frac {7 \left (c d^2-a e^2\right )^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{48 c^3 d^3}+\frac {7 \left (c d^2-a e^2\right ) (d+e x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{40 c^2 d^2}+\frac {(d+e x)^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{5 c d}-\frac {7 \left (c d^2-a e^2\right )^5 \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 )}{256 c^{9/2} d^{9/2} e^{3/2}}\\ \end {align*}
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Mathematica [A]
time = 0.49, size = 257, normalized size = 0.78 \begin {gather*} \frac {\left (c d^2-a e^2\right )^5 \sqrt {(a e+c d x) (d+e x)} \left (\frac {\sqrt {c} \sqrt {d} \sqrt {e} (d+e x)^4 \left (105 c^4 d^4-\frac {105 e^4 (a e+c d x)^4}{(d+e x)^4}+\frac {490 c d e^3 (a e+c d x)^3}{(d+e x)^3}-\frac {896 c^2 d^2 e^2 (a e+c d x)^2}{(d+e x)^2}+\frac {790 c^3 d^3 e (a e+c d x)}{d+e x}\right )}{\left (c d^2-a e^2\right )^5}-\frac {105 \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {a e+c d x}}{\sqrt {c} \sqrt {d} \sqrt {d+e x}}\right )}{\sqrt {a e+c d x} \sqrt {d+e x}}\right )}{1920 c^{9/2} d^{9/2} e^{3/2}} \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. \(1542\) vs.
\(2(294)=588\).
time = 0.71, size = 1543, normalized size = 4.70
method | result | size |
default | \(\text {Expression too large to display}\) | \(1543\) |
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 = 2.80, size = 819, normalized size = 2.50 \begin {gather*} \left [\frac {{\left (105 \, {\left (c^{5} d^{10} - 5 \, a c^{4} d^{8} e^{2} + 10 \, a^{2} c^{3} d^{6} e^{4} - 10 \, a^{3} c^{2} d^{4} e^{6} + 5 \, a^{4} c d^{2} e^{8} - a^{5} e^{10}\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 (1210 \, c^{5} d^{8} x e^{2} + 105 \, c^{5} d^{9} e + 70 \, a^{3} c^{2} d^{2} x e^{8} - 105 \, a^{4} c d e^{9} - 14 \, {\left (4 \, a^{2} c^{3} d^{3} x^{2} - 35 \, a^{3} c^{2} d^{3}\right )} e^{7} + 2 \, {\left (24 \, a c^{4} d^{4} x^{3} - 161 \, a^{2} c^{3} d^{4} x\right )} e^{6} + 128 \, {\left (3 \, c^{5} d^{5} x^{4} + 2 \, a c^{4} d^{5} x^{2} - 7 \, a^{2} c^{3} d^{5}\right )} e^{5} + 2 \, {\left (744 \, c^{5} d^{6} x^{3} + 289 \, a c^{4} d^{6} x\right )} e^{4} + 2 \, {\left (1052 \, c^{5} d^{7} x^{2} + 395 \, a c^{4} d^{7}\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 (-2\right )}}{7680 \, c^{5} d^{5}}, \frac {{\left (105 \, {\left (c^{5} d^{10} - 5 \, a c^{4} d^{8} e^{2} + 10 \, a^{2} c^{3} d^{6} e^{4} - 10 \, a^{3} c^{2} d^{4} e^{6} + 5 \, a^{4} c d^{2} e^{8} - a^{5} e^{10}\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 (1210 \, c^{5} d^{8} x e^{2} + 105 \, c^{5} d^{9} e + 70 \, a^{3} c^{2} d^{2} x e^{8} - 105 \, a^{4} c d e^{9} - 14 \, {\left (4 \, a^{2} c^{3} d^{3} x^{2} - 35 \, a^{3} c^{2} d^{3}\right )} e^{7} + 2 \, {\left (24 \, a c^{4} d^{4} x^{3} - 161 \, a^{2} c^{3} d^{4} x\right )} e^{6} + 128 \, {\left (3 \, c^{5} d^{5} x^{4} + 2 \, a c^{4} d^{5} x^{2} - 7 \, a^{2} c^{3} d^{5}\right )} e^{5} + 2 \, {\left (744 \, c^{5} d^{6} x^{3} + 289 \, a c^{4} d^{6} x\right )} e^{4} + 2 \, {\left (1052 \, c^{5} d^{7} x^{2} + 395 \, a c^{4} d^{7}\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 (-2\right )}}{3840 \, c^{5} d^{5}}\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 \sqrt {\left (d + e x\right ) \left (a e + c d x\right )} \left (d + e x\right )^{3}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.73, size = 377, normalized size = 1.15 \begin {gather*} \frac {1}{1920} \, \sqrt {c d x^{2} e + c d^{2} x + a x e^{2} + a d e} {\left (2 \, {\left (4 \, {\left (6 \, {\left (8 \, x e^{3} + \frac {{\left (31 \, c^{4} d^{5} e^{6} + a c^{3} d^{3} e^{8}\right )} e^{\left (-4\right )}}{c^{4} d^{4}}\right )} x + \frac {{\left (263 \, c^{4} d^{6} e^{5} + 32 \, a c^{3} d^{4} e^{7} - 7 \, a^{2} c^{2} d^{2} e^{9}\right )} e^{\left (-4\right )}}{c^{4} d^{4}}\right )} x + \frac {{\left (605 \, c^{4} d^{7} e^{4} + 289 \, a c^{3} d^{5} e^{6} - 161 \, a^{2} c^{2} d^{3} e^{8} + 35 \, a^{3} c d e^{10}\right )} e^{\left (-4\right )}}{c^{4} d^{4}}\right )} x + \frac {{\left (105 \, c^{4} d^{8} e^{3} + 790 \, a c^{3} d^{6} e^{5} - 896 \, a^{2} c^{2} d^{4} e^{7} + 490 \, a^{3} c d^{2} e^{9} - 105 \, a^{4} e^{11}\right )} e^{\left (-4\right )}}{c^{4} d^{4}}\right )} + \frac {7 \, {\left (c^{5} d^{10} - 5 \, a c^{4} d^{8} e^{2} + 10 \, a^{2} c^{3} d^{6} e^{4} - 10 \, a^{3} c^{2} d^{4} e^{6} + 5 \, a^{4} c d^{2} e^{8} - a^{5} e^{10}\right )} e^{\left (-\frac {3}{2}\right )} \log \left ({\left | -c d^{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 )} \sqrt {c d} e^{\frac {1}{2}} - a e^{2} \right |}\right )}{256 \, \sqrt {c d} 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 {\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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