Optimal. Leaf size=388 \[ \frac {21 \left (c d^2-a e^2\right )^4 \left (c d^2+a e^2+2 c d e x\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{512 c^5 d^5 e}+\frac {7 \left (c d^2-a e^2\right )^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{64 c^4 d^4}+\frac {21 \left (c d^2-a e^2\right )^2 (d+e x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{160 c^3 d^3}+\frac {3 \left (c d^2-a e^2\right ) (d+e x)^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{20 c^2 d^2}+\frac {(d+e x)^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{6 c d}-\frac {21 \left (c d^2-a e^2\right )^6 \tanh ^{-1}\left (\frac {c d^2+a e^2+2 c d e x}{2 \sqrt {c} \sqrt {d} \sqrt {e} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\right )}{1024 c^{11/2} d^{11/2} e^{3/2}} \]
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Rubi [A]
time = 0.30, antiderivative size = 388, normalized size of antiderivative = 1.00, number of steps
used = 7, 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 {21 \left (c d^2-a e^2\right )^6 \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 )}{1024 c^{11/2} d^{11/2} e^{3/2}}+\frac {21 \left (c d^2-a e^2\right )^4 \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}}{512 c^5 d^5 e}+\frac {7 \left (c d^2-a e^2\right )^3 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{64 c^4 d^4}+\frac {21 (d+e x) \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}}{160 c^3 d^3}+\frac {3 (d+e x)^2 \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}}{20 c^2 d^2}+\frac {(d+e x)^3 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{6 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)^4 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2} \, dx &=\frac {(d+e x)^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{6 c d}+\frac {\left (3 \left (d^2-\frac {a e^2}{c}\right )\right ) \int (d+e x)^3 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2} \, dx}{4 d}\\ &=\frac {3 \left (c d^2-a e^2\right ) (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}}{20 c^2 d^2}+\frac {(d+e x)^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{6 c d}+\frac {\left (21 \left (d^2-\frac {a e^2}{c}\right )^2\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}{40 d^2}\\ &=\frac {21 \left (c d^2-a e^2\right )^2 (d+e x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{160 c^3 d^3}+\frac {3 \left (c d^2-a e^2\right ) (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}}{20 c^2 d^2}+\frac {(d+e x)^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{6 c d}+\frac {\left (21 \left (d^2-\frac {a e^2}{c}\right )^3\right ) \int (d+e x) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2} \, dx}{64 d^3}\\ &=\frac {7 \left (c d^2-a e^2\right )^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{64 c^4 d^4}+\frac {21 \left (c d^2-a e^2\right )^2 (d+e x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{160 c^3 d^3}+\frac {3 \left (c d^2-a e^2\right ) (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}}{20 c^2 d^2}+\frac {(d+e x)^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{6 c d}+\frac {\left (21 \left (d^2-\frac {a e^2}{c}\right )^4\right ) \int \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2} \, dx}{128 d^4}\\ &=\frac {21 \left (c d^2-a e^2\right )^4 \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}}{512 c^5 d^5 e}+\frac {7 \left (c d^2-a e^2\right )^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{64 c^4 d^4}+\frac {21 \left (c d^2-a e^2\right )^2 (d+e x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{160 c^3 d^3}+\frac {3 \left (c d^2-a e^2\right ) (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}}{20 c^2 d^2}+\frac {(d+e x)^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{6 c d}-\frac {\left (21 \left (c d^2-a e^2\right )^6\right ) \int \frac {1}{\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx}{1024 c^5 d^5 e}\\ &=\frac {21 \left (c d^2-a e^2\right )^4 \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}}{512 c^5 d^5 e}+\frac {7 \left (c d^2-a e^2\right )^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{64 c^4 d^4}+\frac {21 \left (c d^2-a e^2\right )^2 (d+e x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{160 c^3 d^3}+\frac {3 \left (c d^2-a e^2\right ) (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}}{20 c^2 d^2}+\frac {(d+e x)^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{6 c d}-\frac {\left (21 \left (c d^2-a e^2\right )^6\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 )}{512 c^5 d^5 e}\\ &=\frac {21 \left (c d^2-a e^2\right )^4 \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}}{512 c^5 d^5 e}+\frac {7 \left (c d^2-a e^2\right )^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{64 c^4 d^4}+\frac {21 \left (c d^2-a e^2\right )^2 (d+e x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{160 c^3 d^3}+\frac {3 \left (c d^2-a e^2\right ) (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}}{20 c^2 d^2}+\frac {(d+e x)^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{6 c d}-\frac {21 \left (c d^2-a e^2\right )^6 \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 )}{1024 c^{11/2} d^{11/2} e^{3/2}}\\ \end {align*}
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Mathematica [A]
time = 0.63, size = 285, normalized size = 0.73 \begin {gather*} \frac {\left (c d^2-a e^2\right )^6 \sqrt {(a e+c d x) (d+e x)} \left (\frac {\sqrt {c} \sqrt {d} \sqrt {e} (d+e x)^5 \left (315 c^5 d^5+\frac {315 e^5 (a e+c d x)^5}{(d+e x)^5}-\frac {1785 c d e^4 (a e+c d x)^4}{(d+e x)^4}+\frac {4158 c^2 d^2 e^3 (a e+c d x)^3}{(d+e x)^3}-\frac {5058 c^3 d^3 e^2 (a e+c d x)^2}{(d+e x)^2}+\frac {3335 c^4 d^4 e (a e+c d x)}{d+e x}\right )}{\left (c d^2-a e^2\right )^6}-\frac {315 \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 )}{7680 c^{11/2} d^{11/2} 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. \(2774\) vs.
\(2(350)=700\).
time = 0.70, size = 2775, normalized size = 7.15
method | result | size |
default | \(\text {Expression too large to display}\) | \(2775\) |
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 = 2.23, size = 1009, normalized size = 2.60 \begin {gather*} \left [\frac {{\left (315 \, {\left (c^{6} d^{12} - 6 \, a c^{5} d^{10} e^{2} + 15 \, a^{2} c^{4} d^{8} e^{4} - 20 \, a^{3} c^{3} d^{6} e^{6} + 15 \, a^{4} c^{2} d^{4} e^{8} - 6 \, a^{5} c d^{2} e^{10} + a^{6} e^{12}\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 (4910 \, c^{6} d^{10} x e^{2} + 315 \, c^{6} d^{11} e - 210 \, a^{4} c^{2} d^{2} x e^{10} + 315 \, a^{5} c d e^{11} + 21 \, {\left (8 \, a^{3} c^{3} d^{3} x^{2} - 85 \, a^{4} c^{2} d^{3}\right )} e^{9} - 24 \, {\left (6 \, a^{2} c^{4} d^{4} x^{3} - 49 \, a^{3} c^{3} d^{4} x\right )} e^{8} + 2 \, {\left (64 \, a c^{5} d^{5} x^{4} - 468 \, a^{2} c^{4} d^{5} x^{2} + 2079 \, a^{3} c^{3} d^{5}\right )} e^{7} + 20 \, {\left (64 \, c^{6} d^{6} x^{5} + 40 \, a c^{5} d^{6} x^{3} - 135 \, a^{2} c^{4} d^{6} x\right )} e^{6} + 2 \, {\left (3136 \, c^{6} d^{7} x^{4} + 1068 \, a c^{5} d^{7} x^{2} - 2529 \, a^{2} c^{4} d^{7}\right )} e^{5} + 8 \, {\left (1518 \, c^{6} d^{8} x^{3} + 403 \, a c^{5} d^{8} x\right )} e^{4} + {\left (11432 \, c^{6} d^{9} x^{2} + 3335 \, a c^{5} d^{9}\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 )}}{30720 \, c^{6} d^{6}}, \frac {{\left (315 \, {\left (c^{6} d^{12} - 6 \, a c^{5} d^{10} e^{2} + 15 \, a^{2} c^{4} d^{8} e^{4} - 20 \, a^{3} c^{3} d^{6} e^{6} + 15 \, a^{4} c^{2} d^{4} e^{8} - 6 \, a^{5} c d^{2} e^{10} + a^{6} e^{12}\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 (4910 \, c^{6} d^{10} x e^{2} + 315 \, c^{6} d^{11} e - 210 \, a^{4} c^{2} d^{2} x e^{10} + 315 \, a^{5} c d e^{11} + 21 \, {\left (8 \, a^{3} c^{3} d^{3} x^{2} - 85 \, a^{4} c^{2} d^{3}\right )} e^{9} - 24 \, {\left (6 \, a^{2} c^{4} d^{4} x^{3} - 49 \, a^{3} c^{3} d^{4} x\right )} e^{8} + 2 \, {\left (64 \, a c^{5} d^{5} x^{4} - 468 \, a^{2} c^{4} d^{5} x^{2} + 2079 \, a^{3} c^{3} d^{5}\right )} e^{7} + 20 \, {\left (64 \, c^{6} d^{6} x^{5} + 40 \, a c^{5} d^{6} x^{3} - 135 \, a^{2} c^{4} d^{6} x\right )} e^{6} + 2 \, {\left (3136 \, c^{6} d^{7} x^{4} + 1068 \, a c^{5} d^{7} x^{2} - 2529 \, a^{2} c^{4} d^{7}\right )} e^{5} + 8 \, {\left (1518 \, c^{6} d^{8} x^{3} + 403 \, a c^{5} d^{8} x\right )} e^{4} + {\left (11432 \, c^{6} d^{9} x^{2} + 3335 \, a c^{5} d^{9}\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 )}}{15360 \, c^{6} d^{6}}\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 )^{4}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.67, size = 476, normalized size = 1.23 \begin {gather*} \frac {1}{7680} \, \sqrt {c d x^{2} e + c d^{2} x + a x e^{2} + a d e} {\left (2 \, {\left (4 \, {\left (2 \, {\left (8 \, {\left (10 \, x e^{4} + \frac {{\left (49 \, c^{5} d^{6} e^{8} + a c^{4} d^{4} e^{10}\right )} e^{\left (-5\right )}}{c^{5} d^{5}}\right )} x + \frac {{\left (759 \, c^{5} d^{7} e^{7} + 50 \, a c^{4} d^{5} e^{9} - 9 \, a^{2} c^{3} d^{3} e^{11}\right )} e^{\left (-5\right )}}{c^{5} d^{5}}\right )} x + \frac {{\left (1429 \, c^{5} d^{8} e^{6} + 267 \, a c^{4} d^{6} e^{8} - 117 \, a^{2} c^{3} d^{4} e^{10} + 21 \, a^{3} c^{2} d^{2} e^{12}\right )} e^{\left (-5\right )}}{c^{5} d^{5}}\right )} x + \frac {{\left (2455 \, c^{5} d^{9} e^{5} + 1612 \, a c^{4} d^{7} e^{7} - 1350 \, a^{2} c^{3} d^{5} e^{9} + 588 \, a^{3} c^{2} d^{3} e^{11} - 105 \, a^{4} c d e^{13}\right )} e^{\left (-5\right )}}{c^{5} d^{5}}\right )} x + \frac {{\left (315 \, c^{5} d^{10} e^{4} + 3335 \, a c^{4} d^{8} e^{6} - 5058 \, a^{2} c^{3} d^{6} e^{8} + 4158 \, a^{3} c^{2} d^{4} e^{10} - 1785 \, a^{4} c d^{2} e^{12} + 315 \, a^{5} e^{14}\right )} e^{\left (-5\right )}}{c^{5} d^{5}}\right )} + \frac {21 \, {\left (c^{6} d^{12} - 6 \, a c^{5} d^{10} e^{2} + 15 \, a^{2} c^{4} d^{8} e^{4} - 20 \, a^{3} c^{3} d^{6} e^{6} + 15 \, a^{4} c^{2} d^{4} e^{8} - 6 \, a^{5} c d^{2} e^{10} + a^{6} e^{12}\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 )}{1024 \, \sqrt {c d} c^{5} d^{5}} \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 )}^4\,\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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