Optimal. Leaf size=302 \[ -\frac {2 (d+e x)^4}{c d \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac {35 e \left (c d^2-a e^2\right )^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{8 c^4 d^4}+\frac {35 e \left (c d^2-a e^2\right ) (d+e x) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{12 c^3 d^3}+\frac {7 e (d+e x)^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{3 c^2 d^2}+\frac {35 \sqrt {e} \left (c d^2-a e^2\right )^3 \tanh ^{-1}\left (\frac {c d^2+a e^2+2 c d e x}{2 \sqrt {c} \sqrt {d} \sqrt {e} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\right )}{16 c^{9/2} d^{9/2}} \]
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Rubi [A]
time = 0.17, antiderivative size = 302, 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 = {682, 684, 654,
635, 212} \begin {gather*} \frac {35 \sqrt {e} \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^{9/2} d^{9/2}}+\frac {35 e \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^4 d^4}+\frac {35 e (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^3 d^3}+\frac {7 e (d+e x)^2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{3 c^2 d^2}-\frac {2 (d+e x)^4}{c d \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 212
Rule 635
Rule 654
Rule 682
Rule 684
Rubi steps
\begin {align*} \int \frac {(d+e x)^5}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx &=-\frac {2 (d+e x)^4}{c d \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac {(7 e) \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}{c d}\\ &=-\frac {2 (d+e x)^4}{c d \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac {7 e (d+e x)^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{3 c^2 d^2}+\frac {\left (35 e \left (c d^2-a e^2\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 c^2 d^2}\\ &=-\frac {2 (d+e x)^4}{c d \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac {35 e \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^3 d^3}+\frac {7 e (d+e x)^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{3 c^2 d^2}+\frac {\left (35 e \left (c d^2-a e^2\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 c^3 d^3}\\ &=-\frac {2 (d+e x)^4}{c d \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac {35 e \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^4 d^4}+\frac {35 e \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^3 d^3}+\frac {7 e (d+e x)^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{3 c^2 d^2}+\frac {\left (35 e \left (c d^2-a e^2\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 c^4 d^4}\\ &=-\frac {2 (d+e x)^4}{c d \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac {35 e \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^4 d^4}+\frac {35 e \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^3 d^3}+\frac {7 e (d+e x)^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{3 c^2 d^2}+\frac {\left (35 e \left (c d^2-a e^2\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 c^4 d^4}\\ &=-\frac {2 (d+e x)^4}{c d \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac {35 e \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^4 d^4}+\frac {35 e \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^3 d^3}+\frac {7 e (d+e x)^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{3 c^2 d^2}+\frac {35 \sqrt {e} \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^{9/2} d^{9/2}}\\ \end {align*}
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Mathematica [A]
time = 0.54, size = 235, normalized size = 0.78 \begin {gather*} \frac {\left (c d^2-a e^2\right )^3 \left (-\frac {\sqrt {c} \sqrt {d} (a e+c d x)^4 (d+e x)^2 \left (-105 e^3+\frac {280 c d e^2 (d+e x)}{a e+c d x}-\frac {231 c^2 d^2 e (d+e x)^2}{(a e+c d x)^2}+\frac {48 c^3 d^3 (d+e x)^3}{(a e+c d x)^3}\right )}{\left (c d^2-a e^2\right )^3}+105 \sqrt {e} (a e+c d x)^{3/2} (d+e x)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {d} \sqrt {d+e x}}{\sqrt {e} \sqrt {a e+c d x}}\right )\right )}{24 c^{9/2} d^{9/2} ((a e+c d x) (d+e x))^{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. \(3116\) vs.
\(2(270)=540\).
time = 0.72, size = 3117, normalized size = 10.32
method | result | size |
default | \(\text {Expression too large to display}\) | \(3117\) |
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 = 3.40, size = 740, normalized size = 2.45 \begin {gather*} \left [\frac {105 \, {\left (c^{4} d^{7} x - 3 \, a c^{3} d^{5} x e^{2} + a c^{3} d^{6} e + 3 \, a^{2} c^{2} d^{3} x e^{4} - 3 \, a^{2} c^{2} d^{4} e^{3} - a^{3} c d x e^{6} + 3 \, a^{3} c d^{2} e^{5} - a^{4} e^{7}\right )} \sqrt {\frac {1}{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 \, {\left (2 \, c^{2} d^{2} x e + c^{2} d^{3} + a c d e^{2}\right )} \sqrt {c d^{2} x + a x e^{2} + {\left (c d x^{2} + a d\right )} e} \sqrt {\frac {1}{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 (87 \, c^{3} d^{5} x e - 48 \, c^{3} d^{6} + 35 \, a^{2} c d x e^{5} + 105 \, a^{3} e^{6} - 14 \, {\left (a c^{2} d^{2} x^{2} + 20 \, a^{2} c d^{2}\right )} e^{4} + 2 \, {\left (4 \, c^{3} d^{3} x^{3} - 49 \, a c^{2} d^{3} x\right )} e^{3} + {\left (38 \, c^{3} d^{4} x^{2} + 231 \, a c^{2} d^{4}\right )} e^{2}\right )} \sqrt {c d^{2} x + a x e^{2} + {\left (c d x^{2} + a d\right )} e}}{96 \, {\left (c^{5} d^{5} x + a c^{4} d^{4} e\right )}}, -\frac {105 \, {\left (c^{4} d^{7} x - 3 \, a c^{3} d^{5} x e^{2} + a c^{3} d^{6} e + 3 \, a^{2} c^{2} d^{3} x e^{4} - 3 \, a^{2} c^{2} d^{4} e^{3} - a^{3} c d x e^{6} + 3 \, a^{3} c d^{2} e^{5} - a^{4} e^{7}\right )} \sqrt {-\frac {e}{c d}} \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 {-\frac {e}{c d}}}{2 \, {\left (c d^{2} x e + a x e^{3} + {\left (c d x^{2} + a d\right )} e^{2}\right )}}\right ) - 2 \, {\left (87 \, c^{3} d^{5} x e - 48 \, c^{3} d^{6} + 35 \, a^{2} c d x e^{5} + 105 \, a^{3} e^{6} - 14 \, {\left (a c^{2} d^{2} x^{2} + 20 \, a^{2} c d^{2}\right )} e^{4} + 2 \, {\left (4 \, c^{3} d^{3} x^{3} - 49 \, a c^{2} d^{3} x\right )} e^{3} + {\left (38 \, c^{3} d^{4} x^{2} + 231 \, a c^{2} d^{4}\right )} e^{2}\right )} \sqrt {c d^{2} x + a x e^{2} + {\left (c d x^{2} + a d\right )} e}}{48 \, {\left (c^{5} d^{5} x + a c^{4} d^{4} e\right )}}\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 )^{5}}{\left (\left (d + e x\right ) \left (a e + c d x\right )\right )^{\frac {3}{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.40, size = 372, normalized size = 1.23 \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^{3}}{c^{2} d^{2}} + \frac {{\left (19 \, c^{11} d^{12} e^{4} - 11 \, a c^{10} d^{10} e^{6}\right )} e^{\left (-2\right )}}{c^{13} d^{13}}\right )} + \frac {{\left (87 \, c^{11} d^{13} e^{3} - 136 \, a c^{10} d^{11} e^{5} + 57 \, a^{2} c^{9} d^{9} e^{7}\right )} e^{\left (-2\right )}}{c^{13} d^{13}}\right )} - \frac {35 \, {\left (\sqrt {c d} c^{3} d^{6} e^{\frac {3}{2}} - 3 \, \sqrt {c d} a c^{2} d^{4} e^{\frac {7}{2}} + 3 \, \sqrt {c d} a^{2} c d^{2} e^{\frac {11}{2}} - \sqrt {c d} a^{3} e^{\frac {15}{2}}\right )} e^{\left (-1\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^{5} d^{5}} + \frac {2 \, {\left (c^{4} d^{8} - 4 \, a c^{3} d^{6} e^{2} + 6 \, a^{2} c^{2} d^{4} e^{4} - 4 \, a^{3} c d^{2} e^{6} + a^{4} e^{8}\right )}}{{\left ({\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 + \sqrt {c d} a e^{\frac {3}{2}}\right )} 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 )}^5}{{\left (c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e\right )}^{3/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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