Optimal. Leaf size=356 \[ \frac {5 \left (c d^2-a e^2\right )^5 \left (c d^2+a e^2+2 c d e x\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{1024 c^4 d^4 e^3}-\frac {5 \left (c d^2-a e^2\right )^3 \left (c d^2+a e^2+2 c d e x\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{384 c^3 d^3 e^2}+\frac {\left (c d^2-a e^2\right ) \left (c d^2+a e^2+2 c d e x\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{24 c^2 d^2 e}+\frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{7 c d}-\frac {5 \left (c d^2-a e^2\right )^7 \tanh ^{-1}\left (\frac {c d^2+a e^2+2 c d e x}{2 \sqrt {c} \sqrt {d} \sqrt {e} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\right )}{2048 c^{9/2} d^{9/2} e^{7/2}} \]
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Rubi [A]
time = 0.15, antiderivative size = 356, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 4, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.114, Rules used = {654, 626, 635,
212} \begin {gather*} -\frac {5 \left (c d^2-a e^2\right )^7 \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 )}{2048 c^{9/2} d^{9/2} e^{7/2}}+\frac {5 \left (c d^2-a e^2\right )^5 \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}}{1024 c^4 d^4 e^3}-\frac {5 \left (c d^2-a e^2\right )^3 \left (a e^2+c d^2+2 c d e x\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{384 c^3 d^3 e^2}+\frac {\left (c d^2-a e^2\right ) \left (a e^2+c d^2+2 c d e x\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{24 c^2 d^2 e}+\frac {\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{7 c d} \end {gather*}
Antiderivative was successfully verified.
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Rule 212
Rule 626
Rule 635
Rule 654
Rubi steps
\begin {align*} \int (d+e x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2} \, dx &=\frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{7 c d}+\frac {\left (d^2-\frac {a e^2}{c}\right ) \int \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2} \, dx}{2 d}\\ &=\frac {\left (c d^2-a e^2\right ) \left (c d^2+a e^2+2 c d e x\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{24 c^2 d^2 e}+\frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{7 c d}-\frac {\left (5 \left (c d^2-a e^2\right )^3\right ) \int \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2} \, dx}{48 c^2 d^2 e}\\ &=-\frac {5 \left (c d^2-a e^2\right )^3 \left (c d^2+a e^2+2 c d e x\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{384 c^3 d^3 e^2}+\frac {\left (c d^2-a e^2\right ) \left (c d^2+a e^2+2 c d e x\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{24 c^2 d^2 e}+\frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{7 c d}+\frac {\left (5 \left (c d^2-a e^2\right )^5\right ) \int \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2} \, dx}{256 c^3 d^3 e^2}\\ &=\frac {5 \left (c d^2-a e^2\right )^5 \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}}{1024 c^4 d^4 e^3}-\frac {5 \left (c d^2-a e^2\right )^3 \left (c d^2+a e^2+2 c d e x\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{384 c^3 d^3 e^2}+\frac {\left (c d^2-a e^2\right ) \left (c d^2+a e^2+2 c d e x\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{24 c^2 d^2 e}+\frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{7 c d}-\frac {\left (5 \left (c d^2-a e^2\right )^7\right ) \int \frac {1}{\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx}{2048 c^4 d^4 e^3}\\ &=\frac {5 \left (c d^2-a e^2\right )^5 \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}}{1024 c^4 d^4 e^3}-\frac {5 \left (c d^2-a e^2\right )^3 \left (c d^2+a e^2+2 c d e x\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{384 c^3 d^3 e^2}+\frac {\left (c d^2-a e^2\right ) \left (c d^2+a e^2+2 c d e x\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{24 c^2 d^2 e}+\frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{7 c d}-\frac {\left (5 \left (c d^2-a e^2\right )^7\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 )}{1024 c^4 d^4 e^3}\\ &=\frac {5 \left (c d^2-a e^2\right )^5 \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}}{1024 c^4 d^4 e^3}-\frac {5 \left (c d^2-a e^2\right )^3 \left (c d^2+a e^2+2 c d e x\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{384 c^3 d^3 e^2}+\frac {\left (c d^2-a e^2\right ) \left (c d^2+a e^2+2 c d e x\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{24 c^2 d^2 e}+\frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{7 c d}-\frac {5 \left (c d^2-a e^2\right )^7 \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 )}{2048 c^{9/2} d^{9/2} e^{7/2}}\\ \end {align*}
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Mathematica [A]
time = 1.21, size = 323, normalized size = 0.91 \begin {gather*} \frac {\left (c d^2-a e^2\right )^7 ((a e+c d x) (d+e x))^{5/2} \left (\frac {\sqrt {c} \sqrt {d} \sqrt {e} (a e+c d x)^4 \left (-105 e^6+\frac {700 c d e^5 (d+e x)}{a e+c d x}-\frac {1981 c^2 d^2 e^4 (d+e x)^2}{(a e+c d x)^2}+\frac {3072 c^3 d^3 e^3 (d+e x)^3}{(a e+c d x)^3}+\frac {1981 c^4 d^4 e^2 (d+e x)^4}{(a e+c d x)^4}-\frac {700 c^5 d^5 e (d+e x)^5}{(a e+c d x)^5}+\frac {105 c^6 d^6 (d+e x)^6}{(a e+c d x)^6}\right )}{\left (c d^2-a e^2\right )^7 (d+e x)^2}-\frac {105 \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {d} \sqrt {d+e x}}{\sqrt {e} \sqrt {a e+c d x}}\right )}{(a e+c d x)^{5/2} (d+e x)^{5/2}}\right )}{21504 c^{9/2} d^{9/2} 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. \(744\) vs.
\(2(322)=644\).
time = 0.66, size = 745, normalized size = 2.09
method | result | size |
default | \(e \left (\frac {\left (a d e +\left (e^{2} a +c \,d^{2}\right ) x +c d e \,x^{2}\right )^{\frac {7}{2}}}{7 c d e}-\frac {\left (e^{2} a +c \,d^{2}\right ) \left (\frac {\left (2 c d e x +e^{2} a +c \,d^{2}\right ) \left (a d e +\left (e^{2} a +c \,d^{2}\right ) x +c d e \,x^{2}\right )^{\frac {5}{2}}}{12 c d e}+\frac {5 \left (4 a c \,d^{2} e^{2}-\left (e^{2} a +c \,d^{2}\right )^{2}\right ) \left (\frac {\left (2 c d e x +e^{2} a +c \,d^{2}\right ) \left (a d e +\left (e^{2} a +c \,d^{2}\right ) x +c d e \,x^{2}\right )^{\frac {3}{2}}}{8 c d e}+\frac {3 \left (4 a c \,d^{2} e^{2}-\left (e^{2} a +c \,d^{2}\right )^{2}\right ) \left (\frac {\left (2 c d e x +e^{2} a +c \,d^{2}\right ) \sqrt {a d e +\left (e^{2} a +c \,d^{2}\right ) x +c d e \,x^{2}}}{4 c d e}+\frac {\left (4 a c \,d^{2} e^{2}-\left (e^{2} a +c \,d^{2}\right )^{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 )}{8 c d e \sqrt {c d e}}\right )}{16 c d e}\right )}{24 c d e}\right )}{2 c d e}\right )+d \left (\frac {\left (2 c d e x +e^{2} a +c \,d^{2}\right ) \left (a d e +\left (e^{2} a +c \,d^{2}\right ) x +c d e \,x^{2}\right )^{\frac {5}{2}}}{12 c d e}+\frac {5 \left (4 a c \,d^{2} e^{2}-\left (e^{2} a +c \,d^{2}\right )^{2}\right ) \left (\frac {\left (2 c d e x +e^{2} a +c \,d^{2}\right ) \left (a d e +\left (e^{2} a +c \,d^{2}\right ) x +c d e \,x^{2}\right )^{\frac {3}{2}}}{8 c d e}+\frac {3 \left (4 a c \,d^{2} e^{2}-\left (e^{2} a +c \,d^{2}\right )^{2}\right ) \left (\frac {\left (2 c d e x +e^{2} a +c \,d^{2}\right ) \sqrt {a d e +\left (e^{2} a +c \,d^{2}\right ) x +c d e \,x^{2}}}{4 c d e}+\frac {\left (4 a c \,d^{2} e^{2}-\left (e^{2} a +c \,d^{2}\right )^{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 )}{8 c d e \sqrt {c d e}}\right )}{16 c d e}\right )}{24 c d e}\right )\) | \(745\) |
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 = 3.21, size = 1231, normalized size = 3.46 \begin {gather*} \left [\frac {{\left (105 \, {\left (c^{7} d^{14} - 7 \, a c^{6} d^{12} e^{2} + 21 \, a^{2} c^{5} d^{10} e^{4} - 35 \, a^{3} c^{4} d^{8} e^{6} + 35 \, a^{4} c^{3} d^{6} e^{8} - 21 \, a^{5} c^{2} d^{4} e^{10} + 7 \, a^{6} c d^{2} e^{12} - a^{7} e^{14}\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 (70 \, c^{7} d^{12} x e^{2} - 105 \, c^{7} d^{13} e - 70 \, a^{5} c^{2} d^{2} x e^{12} + 105 \, a^{6} c d e^{13} + 28 \, {\left (2 \, a^{4} c^{3} d^{3} x^{2} - 25 \, a^{5} c^{2} d^{3}\right )} e^{11} - 6 \, {\left (8 \, a^{3} c^{4} d^{4} x^{3} - 77 \, a^{4} c^{3} d^{4} x\right )} e^{10} - {\left (4736 \, a^{2} c^{5} d^{5} x^{4} + 368 \, a^{3} c^{4} d^{5} x^{2} - 1981 \, a^{4} c^{3} d^{5}\right )} e^{9} - 4 \, {\left (1856 \, a c^{6} d^{6} x^{5} + 4700 \, a^{2} c^{5} d^{6} x^{3} + 323 \, a^{3} c^{4} d^{6} x\right )} e^{8} - 3072 \, {\left (c^{7} d^{7} x^{6} + 9 \, a c^{6} d^{7} x^{4} + 9 \, a^{2} c^{5} d^{7} x^{2} + a^{3} c^{4} d^{7}\right )} e^{7} - 4 \, {\left (2752 \, c^{7} d^{8} x^{5} + 9124 \, a c^{6} d^{8} x^{3} + 4285 \, a^{2} c^{5} d^{8} x\right )} e^{6} - {\left (13696 \, c^{7} d^{9} x^{4} + 18064 \, a c^{6} d^{9} x^{2} + 1981 \, a^{2} c^{5} d^{9}\right )} e^{5} - 6 \, {\left (1016 \, c^{7} d^{10} x^{3} + 77 \, a c^{6} d^{10} x\right )} e^{4} - 28 \, {\left (2 \, c^{7} d^{11} x^{2} - 25 \, a c^{6} d^{11}\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 (-4\right )}}{86016 \, c^{5} d^{5}}, \frac {{\left (105 \, {\left (c^{7} d^{14} - 7 \, a c^{6} d^{12} e^{2} + 21 \, a^{2} c^{5} d^{10} e^{4} - 35 \, a^{3} c^{4} d^{8} e^{6} + 35 \, a^{4} c^{3} d^{6} e^{8} - 21 \, a^{5} c^{2} d^{4} e^{10} + 7 \, a^{6} c d^{2} e^{12} - a^{7} e^{14}\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 (70 \, c^{7} d^{12} x e^{2} - 105 \, c^{7} d^{13} e - 70 \, a^{5} c^{2} d^{2} x e^{12} + 105 \, a^{6} c d e^{13} + 28 \, {\left (2 \, a^{4} c^{3} d^{3} x^{2} - 25 \, a^{5} c^{2} d^{3}\right )} e^{11} - 6 \, {\left (8 \, a^{3} c^{4} d^{4} x^{3} - 77 \, a^{4} c^{3} d^{4} x\right )} e^{10} - {\left (4736 \, a^{2} c^{5} d^{5} x^{4} + 368 \, a^{3} c^{4} d^{5} x^{2} - 1981 \, a^{4} c^{3} d^{5}\right )} e^{9} - 4 \, {\left (1856 \, a c^{6} d^{6} x^{5} + 4700 \, a^{2} c^{5} d^{6} x^{3} + 323 \, a^{3} c^{4} d^{6} x\right )} e^{8} - 3072 \, {\left (c^{7} d^{7} x^{6} + 9 \, a c^{6} d^{7} x^{4} + 9 \, a^{2} c^{5} d^{7} x^{2} + a^{3} c^{4} d^{7}\right )} e^{7} - 4 \, {\left (2752 \, c^{7} d^{8} x^{5} + 9124 \, a c^{6} d^{8} x^{3} + 4285 \, a^{2} c^{5} d^{8} x\right )} e^{6} - {\left (13696 \, c^{7} d^{9} x^{4} + 18064 \, a c^{6} d^{9} x^{2} + 1981 \, a^{2} c^{5} d^{9}\right )} e^{5} - 6 \, {\left (1016 \, c^{7} d^{10} x^{3} + 77 \, a c^{6} d^{10} x\right )} e^{4} - 28 \, {\left (2 \, c^{7} d^{11} x^{2} - 25 \, a c^{6} d^{11}\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 (-4\right )}}{43008 \, 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 \left (\left (d + e x\right ) \left (a e + c d x\right )\right )^{\frac {5}{2}} \left (d + e x\right )\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.93, size = 610, normalized size = 1.71 \begin {gather*} \frac {1}{21504} \, \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 (2 \, {\left (12 \, c^{2} d^{2} x e^{3} + \frac {{\left (43 \, c^{8} d^{9} e^{8} + 29 \, a c^{7} d^{7} e^{10}\right )} e^{\left (-6\right )}}{c^{6} d^{6}}\right )} x + \frac {{\left (107 \, c^{8} d^{10} e^{7} + 216 \, a c^{7} d^{8} e^{9} + 37 \, a^{2} c^{6} d^{6} e^{11}\right )} e^{\left (-6\right )}}{c^{6} d^{6}}\right )} x + \frac {{\left (381 \, c^{8} d^{11} e^{6} + 2281 \, a c^{7} d^{9} e^{8} + 1175 \, a^{2} c^{6} d^{7} e^{10} + 3 \, a^{3} c^{5} d^{5} e^{12}\right )} e^{\left (-6\right )}}{c^{6} d^{6}}\right )} x + \frac {{\left (7 \, c^{8} d^{12} e^{5} + 2258 \, a c^{7} d^{10} e^{7} + 3456 \, a^{2} c^{6} d^{8} e^{9} + 46 \, a^{3} c^{5} d^{6} e^{11} - 7 \, a^{4} c^{4} d^{4} e^{13}\right )} e^{\left (-6\right )}}{c^{6} d^{6}}\right )} x - \frac {{\left (35 \, c^{8} d^{13} e^{4} - 231 \, a c^{7} d^{11} e^{6} - 8570 \, a^{2} c^{6} d^{9} e^{8} - 646 \, a^{3} c^{5} d^{7} e^{10} + 231 \, a^{4} c^{4} d^{5} e^{12} - 35 \, a^{5} c^{3} d^{3} e^{14}\right )} e^{\left (-6\right )}}{c^{6} d^{6}}\right )} x + \frac {{\left (105 \, c^{8} d^{14} e^{3} - 700 \, a c^{7} d^{12} e^{5} + 1981 \, a^{2} c^{6} d^{10} e^{7} + 3072 \, a^{3} c^{5} d^{8} e^{9} - 1981 \, a^{4} c^{4} d^{6} e^{11} + 700 \, a^{5} c^{3} d^{4} e^{13} - 105 \, a^{6} c^{2} d^{2} e^{15}\right )} e^{\left (-6\right )}}{c^{6} d^{6}}\right )} + \frac {5 \, {\left (c^{7} d^{14} - 7 \, a c^{6} d^{12} e^{2} + 21 \, a^{2} c^{5} d^{10} e^{4} - 35 \, a^{3} c^{4} d^{8} e^{6} + 35 \, a^{4} c^{3} d^{6} e^{8} - 21 \, a^{5} c^{2} d^{4} e^{10} + 7 \, a^{6} c d^{2} e^{12} - a^{7} e^{14}\right )} e^{\left (-\frac {7}{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 )}{2048 \, \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 )\,{\left (c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e\right )}^{5/2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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