Optimal. Leaf size=294 \[ -\frac {2 (d+e x)^5}{3 c d \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}-\frac {14 e (d+e x)^3}{3 c^2 d^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac {35 e^2 \left (c d^2-a e^2\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{4 c^4 d^4}+\frac {35 e^2 (d+e x) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{6 c^3 d^3}+\frac {35 e^{3/2} \left (c d^2-a e^2\right )^2 \tanh ^{-1}\left (\frac {c d^2+a e^2+2 c d e x}{2 \sqrt {c} \sqrt {d} \sqrt {e} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\right )}{8 c^{9/2} d^{9/2}} \]
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Rubi [A]
time = 0.15, antiderivative size = 294, 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 e^{3/2} \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 c^{9/2} d^{9/2}}+\frac {35 e^2 \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}}{4 c^4 d^4}+\frac {35 e^2 (d+e x) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{6 c^3 d^3}-\frac {14 e (d+e x)^3}{3 c^2 d^2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}-\frac {2 (d+e x)^5}{3 c d \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}} \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)^6}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}} \, dx &=-\frac {2 (d+e x)^5}{3 c d \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}+\frac {(7 e) \int \frac {(d+e x)^4}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx}{3 c d}\\ &=-\frac {2 (d+e x)^5}{3 c d \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}-\frac {14 e (d+e x)^3}{3 c^2 d^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac {\left (35 e^2\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}{3 c^2 d^2}\\ &=-\frac {2 (d+e x)^5}{3 c d \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}-\frac {14 e (d+e x)^3}{3 c^2 d^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac {35 e^2 (d+e x) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{6 c^3 d^3}+\frac {\left (35 e^2 \left (c d^2-a e^2\right )\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}{4 c^3 d^3}\\ &=-\frac {2 (d+e x)^5}{3 c d \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}-\frac {14 e (d+e x)^3}{3 c^2 d^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac {35 e^2 \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 c^4 d^4}+\frac {35 e^2 (d+e x) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{6 c^3 d^3}+\frac {\left (35 e^2 \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 c^4 d^4}\\ &=-\frac {2 (d+e x)^5}{3 c d \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}-\frac {14 e (d+e x)^3}{3 c^2 d^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac {35 e^2 \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 c^4 d^4}+\frac {35 e^2 (d+e x) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{6 c^3 d^3}+\frac {\left (35 e^2 \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 c^4 d^4}\\ &=-\frac {2 (d+e x)^5}{3 c d \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}-\frac {14 e (d+e x)^3}{3 c^2 d^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac {35 e^2 \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 c^4 d^4}+\frac {35 e^2 (d+e x) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{6 c^3 d^3}+\frac {35 e^{3/2} \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 c^{9/2} d^{9/2}}\\ \end {align*}
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Mathematica [A]
time = 0.42, size = 212, normalized size = 0.72 \begin {gather*} \frac {\sqrt {(a e+c d x) (d+e x)} \left (-\sqrt {c} \sqrt {d} (a e+c d x) \left (105 e^3-\frac {175 c d e^2 (d+e x)}{a e+c d x}+\frac {56 c^2 d^2 e (d+e x)^2}{(a e+c d x)^2}+\frac {8 c^3 d^3 (d+e x)^3}{(a e+c d x)^3}\right )+\frac {105 e^{3/2} \left (c d^2-a e^2\right )^2 \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {d} \sqrt {d+e x}}{\sqrt {e} \sqrt {a e+c d x}}\right )}{\sqrt {a e+c d x} \sqrt {d+e x}}\right )}{12 c^{9/2} d^{9/2}} \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. \(7419\) vs.
\(2(260)=520\).
time = 0.72, size = 7420, normalized size = 25.24
method | result | size |
default | \(\text {Expression too large to display}\) | \(7420\) |
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 = 10.82, size = 822, normalized size = 2.80 \begin {gather*} \left [\frac {105 \, {\left (c^{4} d^{6} x^{2} e + 2 \, a c^{3} d^{5} x e^{2} - 4 \, a^{2} c^{2} d^{3} x e^{4} + 2 \, a^{3} c d x e^{6} + a^{4} e^{7} + {\left (a^{2} c^{2} d^{2} x^{2} - 2 \, a^{3} c d^{2}\right )} e^{5} - {\left (2 \, a c^{3} d^{4} x^{2} - a^{2} c^{2} d^{4}\right )} e^{3}\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 (80 \, c^{3} d^{5} x e + 8 \, c^{3} d^{6} + 140 \, a^{2} c d x e^{5} + 105 \, a^{3} e^{6} + 7 \, {\left (3 \, a c^{2} d^{2} x^{2} - 25 \, a^{2} c d^{2}\right )} e^{4} - 2 \, {\left (3 \, c^{3} d^{3} x^{3} + 119 \, a c^{2} d^{3} x\right )} e^{3} - {\left (39 \, c^{3} d^{4} x^{2} - 56 \, 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^{6} d^{6} x^{2} + 2 \, a c^{5} d^{5} x e + a^{2} c^{4} d^{4} e^{2}\right )}}, -\frac {105 \, {\left (c^{4} d^{6} x^{2} e + 2 \, a c^{3} d^{5} x e^{2} - 4 \, a^{2} c^{2} d^{3} x e^{4} + 2 \, a^{3} c d x e^{6} + a^{4} e^{7} + {\left (a^{2} c^{2} d^{2} x^{2} - 2 \, a^{3} c d^{2}\right )} e^{5} - {\left (2 \, a c^{3} d^{4} x^{2} - a^{2} c^{2} d^{4}\right )} e^{3}\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 (80 \, c^{3} d^{5} x e + 8 \, c^{3} d^{6} + 140 \, a^{2} c d x e^{5} + 105 \, a^{3} e^{6} + 7 \, {\left (3 \, a c^{2} d^{2} x^{2} - 25 \, a^{2} c d^{2}\right )} e^{4} - 2 \, {\left (3 \, c^{3} d^{3} x^{3} + 119 \, a c^{2} d^{3} x\right )} e^{3} - {\left (39 \, c^{3} d^{4} x^{2} - 56 \, 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}}{24 \, {\left (c^{6} d^{6} x^{2} + 2 \, a c^{5} d^{5} x e + a^{2} c^{4} d^{4} e^{2}\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 )^{6}}{\left (\left (d + e x\right ) \left (a e + c d x\right )\right )^{\frac {5}{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 798 vs.
\(2 (263) = 526\).
time = 1.51, size = 798, normalized size = 2.71 \begin {gather*} \frac {1}{4} \, \sqrt {c d x^{2} e + c d^{2} x + a x e^{2} + a d e} {\left (\frac {2 \, x e^{3}}{c^{3} d^{3}} + \frac {{\left (13 \, c^{8} d^{9} e^{3} - 11 \, a c^{7} d^{7} e^{5}\right )} e^{\left (-1\right )}}{c^{11} d^{11}}\right )} - \frac {35 \, {\left (\sqrt {c d} c^{2} d^{4} e^{\frac {5}{2}} - 2 \, \sqrt {c d} a c d^{2} e^{\frac {9}{2}} + \sqrt {c d} a^{2} e^{\frac {13}{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 )}{8 \, c^{5} d^{5}} + \frac {2 \, {\left (c^{5} d^{10} + 3 \, {\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} c^{4} d^{8} e^{\frac {1}{2}} - 2 \, a c^{4} d^{8} e^{2} + 12 \, {\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 )}^{2} c^{4} d^{7} e + 12 \, {\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} a c^{3} d^{6} e^{\frac {5}{2}} + 10 \, a^{2} c^{3} d^{6} e^{4} - 36 \, {\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 )}^{2} a c^{3} d^{5} e^{3} - 54 \, {\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} a^{2} c^{2} d^{4} e^{\frac {9}{2}} - 28 \, a^{3} c^{2} d^{4} e^{6} + 36 \, {\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 )}^{2} a^{2} c^{2} d^{3} e^{5} + 60 \, {\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} a^{3} c d^{2} e^{\frac {13}{2}} + 29 \, a^{4} c d^{2} e^{8} - 12 \, {\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 )}^{2} a^{3} c d e^{7} - 21 \, {\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} a^{4} e^{\frac {17}{2}} - 10 \, a^{5} e^{10}\right )}}{3 \, {\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 )}^{3} c^{3} d^{3}} \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 )}^6}{{\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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