Optimal. Leaf size=339 \[ -\frac {3 \left (a e \left (3 c d^2+a e^2\right )-c d \left (c d^2+3 a e^2\right ) x\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{4 x}-\frac {(a e-c d x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{2 x^2}+\frac {3 \sqrt {c} \sqrt {d} \left (c^2 d^4+10 a c d^2 e^2+5 a^2 e^4\right ) \tanh ^{-1}\left (\frac {c d^2+a e^2+2 c d e x}{2 \sqrt {c} \sqrt {d} \sqrt {e} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\right )}{8 \sqrt {e}}-\frac {3 \sqrt {a} \sqrt {e} \left (5 c^2 d^4+10 a c d^2 e^2+a^2 e^4\right ) \tanh ^{-1}\left (\frac {2 a d e+\left (c d^2+a e^2\right ) x}{2 \sqrt {a} \sqrt {d} \sqrt {e} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\right )}{8 \sqrt {d}} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.25, antiderivative size = 339, normalized size of antiderivative = 1.00, number of steps
used = 8, number of rules used = 6, integrand size = 40, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {863, 826, 857,
635, 212, 738} \begin {gather*} \frac {3 \sqrt {c} \sqrt {d} \left (5 a^2 e^4+10 a c d^2 e^2+c^2 d^4\right ) \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 \sqrt {e}}-\frac {3 \sqrt {a} \sqrt {e} \left (a^2 e^4+10 a c d^2 e^2+5 c^2 d^4\right ) \tanh ^{-1}\left (\frac {x \left (a e^2+c d^2\right )+2 a d e}{2 \sqrt {a} \sqrt {d} \sqrt {e} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{8 \sqrt {d}}-\frac {(a e-c d x) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{2 x^2}-\frac {3 \left (a e \left (a e^2+3 c d^2\right )-c d x \left (3 a e^2+c d^2\right )\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{4 x} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 212
Rule 635
Rule 738
Rule 826
Rule 857
Rule 863
Rubi steps
\begin {align*} \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{x^3 (d+e x)} \, dx &=\int \frac {(a e+c d x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{x^3} \, dx\\ &=-\frac {(a e-c d x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{2 x^2}-\frac {3}{8} \int \frac {\left (-2 a e \left (3 c d^2+a e^2\right )-2 c d \left (c d^2+3 a e^2\right ) x\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{x^2} \, dx\\ &=-\frac {3 \left (a e \left (3 c d^2+a e^2\right )-c d \left (c d^2+3 a e^2\right ) x\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{4 x}-\frac {(a e-c d x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{2 x^2}+\frac {3}{16} \int \frac {2 a e \left (5 c^2 d^4+10 a c d^2 e^2+a^2 e^4\right )+2 c d \left (c^2 d^4+10 a c d^2 e^2+5 a^2 e^4\right ) x}{x \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx\\ &=-\frac {3 \left (a e \left (3 c d^2+a e^2\right )-c d \left (c d^2+3 a e^2\right ) x\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{4 x}-\frac {(a e-c d x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{2 x^2}+\frac {1}{8} \left (3 a e \left (5 c^2 d^4+10 a c d^2 e^2+a^2 e^4\right )\right ) \int \frac {1}{x \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx+\frac {1}{8} \left (3 c d \left (c^2 d^4+10 a c d^2 e^2+5 a^2 e^4\right )\right ) \int \frac {1}{\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx\\ &=-\frac {3 \left (a e \left (3 c d^2+a e^2\right )-c d \left (c d^2+3 a e^2\right ) x\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{4 x}-\frac {(a e-c d x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{2 x^2}-\frac {1}{4} \left (3 a e \left (5 c^2 d^4+10 a c d^2 e^2+a^2 e^4\right )\right ) \text {Subst}\left (\int \frac {1}{4 a d e-x^2} \, dx,x,\frac {2 a d e-\left (-c d^2-a e^2\right ) x}{\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\right )+\frac {1}{4} \left (3 c d \left (c^2 d^4+10 a c d^2 e^2+5 a^2 e^4\right )\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 )\\ &=-\frac {3 \left (a e \left (3 c d^2+a e^2\right )-c d \left (c d^2+3 a e^2\right ) x\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{4 x}-\frac {(a e-c d x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{2 x^2}+\frac {3 \sqrt {c} \sqrt {d} \left (c^2 d^4+10 a c d^2 e^2+5 a^2 e^4\right ) \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 \sqrt {e}}-\frac {3 \sqrt {a} \sqrt {e} \left (5 c^2 d^4+10 a c d^2 e^2+a^2 e^4\right ) \tanh ^{-1}\left (\frac {2 a d e+\left (c d^2+a e^2\right ) x}{2 \sqrt {a} \sqrt {d} \sqrt {e} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\right )}{8 \sqrt {d}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.93, size = 287, normalized size = 0.85 \begin {gather*} \frac {\sqrt {a e+c d x} \sqrt {d+e x} \left (\sqrt {d} \sqrt {e} \sqrt {a e+c d x} \sqrt {d+e x} \left (-9 a c d e x (d-e x)+c^2 d^2 x^2 (5 d+2 e x)-a^2 e^2 (2 d+5 e x)\right )-3 \sqrt {a} e \left (5 c^2 d^4+10 a c d^2 e^2+a^2 e^4\right ) x^2 \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a e+c d x}}{\sqrt {a} \sqrt {e} \sqrt {d+e x}}\right )+3 \sqrt {c} d \left (c^2 d^4+10 a c d^2 e^2+5 a^2 e^4\right ) x^2 \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {a e+c d x}}{\sqrt {c} \sqrt {d} \sqrt {d+e x}}\right )\right )}{4 \sqrt {d} \sqrt {e} x^2 \sqrt {(a e+c d x) (d+e x)}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(3892\) vs.
\(2(295)=590\).
time = 0.09, size = 3893, normalized size = 11.48
method | result | size |
default | \(\text {Expression too large to display}\) | \(3893\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 8.84, size = 1581, normalized size = 4.66 \begin {gather*} \left [\frac {3 \, {\left (c^{2} d^{4} x^{2} + 10 \, a c d^{2} x^{2} e^{2} + 5 \, a^{2} x^{2} e^{4}\right )} \sqrt {c d} e^{\left (-\frac {1}{2}\right )} \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^{2} + c d^{2} e + a e^{3}\right )} \sqrt {c d} e^{\left (-\frac {1}{2}\right )} + 2 \, {\left (4 \, c^{2} d^{2} x^{2} + 3 \, a c d^{2}\right )} e^{2}\right ) + 3 \, {\left (5 \, c^{2} d^{4} x^{2} + 10 \, a c d^{2} x^{2} e^{2} + a^{2} x^{2} e^{4}\right )} \sqrt {\frac {a}{d}} e^{\frac {1}{2}} \log \left (\frac {c^{2} d^{4} x^{2} + 8 \, a c d^{3} x e + a^{2} x^{2} e^{4} + 8 \, a^{2} d x e^{3} - 4 \, {\left (c d^{3} x + a d x e^{2} + 2 \, a d^{2} e\right )} \sqrt {c d^{2} x + a x e^{2} + {\left (c d x^{2} + a d\right )} e} \sqrt {\frac {a}{d}} e^{\frac {1}{2}} + 2 \, {\left (3 \, a c d^{2} x^{2} + 4 \, a^{2} d^{2}\right )} e^{2}}{x^{2}}\right ) + 4 \, {\left (5 \, c^{2} d^{3} x^{2} - 5 \, a^{2} x e^{3} + {\left (9 \, a c d x^{2} - 2 \, a^{2} d\right )} e^{2} + {\left (2 \, c^{2} d^{2} x^{3} - 9 \, a c d^{2} x\right )} e\right )} \sqrt {c d^{2} x + a x e^{2} + {\left (c d x^{2} + a d\right )} e}}{16 \, x^{2}}, \frac {3 \, {\left (5 \, c^{2} d^{4} x^{2} + 10 \, a c d^{2} x^{2} e^{2} + a^{2} x^{2} e^{4}\right )} \sqrt {\frac {a}{d}} e^{\frac {1}{2}} \log \left (\frac {c^{2} d^{4} x^{2} + 8 \, a c d^{3} x e + a^{2} x^{2} e^{4} + 8 \, a^{2} d x e^{3} - 4 \, {\left (c d^{3} x + a d x e^{2} + 2 \, a d^{2} e\right )} \sqrt {c d^{2} x + a x e^{2} + {\left (c d x^{2} + a d\right )} e} \sqrt {\frac {a}{d}} e^{\frac {1}{2}} + 2 \, {\left (3 \, a c d^{2} x^{2} + 4 \, a^{2} d^{2}\right )} e^{2}}{x^{2}}\right ) - 6 \, {\left (c^{2} d^{4} x^{2} + 10 \, a c d^{2} x^{2} e^{2} + 5 \, a^{2} x^{2} e^{4}\right )} \sqrt {-c d e^{\left (-1\right )}} \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^{\left (-1\right )}}}{2 \, {\left (c^{2} d^{3} x + a c d x e^{2} + {\left (c^{2} d^{2} x^{2} + a c d^{2}\right )} e\right )}}\right ) + 4 \, {\left (5 \, c^{2} d^{3} x^{2} - 5 \, a^{2} x e^{3} + {\left (9 \, a c d x^{2} - 2 \, a^{2} d\right )} e^{2} + {\left (2 \, c^{2} d^{2} x^{3} - 9 \, a c d^{2} x\right )} e\right )} \sqrt {c d^{2} x + a x e^{2} + {\left (c d x^{2} + a d\right )} e}}{16 \, x^{2}}, \frac {3 \, {\left (c^{2} d^{4} x^{2} + 10 \, a c d^{2} x^{2} e^{2} + 5 \, a^{2} x^{2} e^{4}\right )} \sqrt {c d} e^{\left (-\frac {1}{2}\right )} \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^{2} + c d^{2} e + a e^{3}\right )} \sqrt {c d} e^{\left (-\frac {1}{2}\right )} + 2 \, {\left (4 \, c^{2} d^{2} x^{2} + 3 \, a c d^{2}\right )} e^{2}\right ) + 6 \, {\left (5 \, c^{2} d^{4} x^{2} + 10 \, a c d^{2} x^{2} e^{2} + a^{2} x^{2} e^{4}\right )} \sqrt {-\frac {a e}{d}} \arctan \left (\frac {{\left (c d^{2} x + a x e^{2} + 2 \, a d e\right )} \sqrt {c d^{2} x + a x e^{2} + {\left (c d x^{2} + a d\right )} e} \sqrt {-\frac {a e}{d}}}{2 \, {\left (a c d^{2} x e + a^{2} x e^{3} + {\left (a c d x^{2} + a^{2} d\right )} e^{2}\right )}}\right ) + 4 \, {\left (5 \, c^{2} d^{3} x^{2} - 5 \, a^{2} x e^{3} + {\left (9 \, a c d x^{2} - 2 \, a^{2} d\right )} e^{2} + {\left (2 \, c^{2} d^{2} x^{3} - 9 \, a c d^{2} x\right )} e\right )} \sqrt {c d^{2} x + a x e^{2} + {\left (c d x^{2} + a d\right )} e}}{16 \, x^{2}}, -\frac {3 \, {\left (c^{2} d^{4} x^{2} + 10 \, a c d^{2} x^{2} e^{2} + 5 \, a^{2} x^{2} e^{4}\right )} \sqrt {-c d e^{\left (-1\right )}} \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^{\left (-1\right )}}}{2 \, {\left (c^{2} d^{3} x + a c d x e^{2} + {\left (c^{2} d^{2} x^{2} + a c d^{2}\right )} e\right )}}\right ) - 3 \, {\left (5 \, c^{2} d^{4} x^{2} + 10 \, a c d^{2} x^{2} e^{2} + a^{2} x^{2} e^{4}\right )} \sqrt {-\frac {a e}{d}} \arctan \left (\frac {{\left (c d^{2} x + a x e^{2} + 2 \, a d e\right )} \sqrt {c d^{2} x + a x e^{2} + {\left (c d x^{2} + a d\right )} e} \sqrt {-\frac {a e}{d}}}{2 \, {\left (a c d^{2} x e + a^{2} x e^{3} + {\left (a c d x^{2} + a^{2} d\right )} e^{2}\right )}}\right ) - 2 \, {\left (5 \, c^{2} d^{3} x^{2} - 5 \, a^{2} x e^{3} + {\left (9 \, a c d x^{2} - 2 \, a^{2} d\right )} e^{2} + {\left (2 \, c^{2} d^{2} x^{3} - 9 \, a c d^{2} x\right )} e\right )} \sqrt {c d^{2} x + a x e^{2} + {\left (c d x^{2} + a d\right )} e}}{8 \, x^{2}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 785 vs.
\(2 (291) = 582\).
time = 1.78, size = 785, normalized size = 2.32 \begin {gather*} \frac {1}{4} \, {\left (2 \, c^{2} d^{2} x e + \frac {{\left (5 \, c^{3} d^{4} e + 9 \, a c^{2} d^{2} e^{3}\right )} e^{\left (-1\right )}}{c d}\right )} \sqrt {c d x^{2} e + c d^{2} x + a x e^{2} + a d e} + \frac {3 \, {\left (5 \, a c^{2} d^{4} e + 10 \, a^{2} c d^{2} e^{3} + a^{3} e^{5}\right )} \arctan \left (-\frac {\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}}{\sqrt {-a d e}}\right )}{4 \, \sqrt {-a d e}} - \frac {3 \, {\left (\sqrt {c d} c^{3} d^{5} e^{\frac {1}{2}} + 10 \, \sqrt {c d} a c^{2} d^{3} e^{\frac {5}{2}} + 5 \, \sqrt {c d} a^{2} c d e^{\frac {9}{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 d} - \frac {7 \, {\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 )} a^{2} c^{2} d^{5} e^{2} - 9 \, {\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 )}^{3} a c^{2} d^{4} e + 16 \, \sqrt {c d} a^{3} c d^{4} e^{\frac {7}{2}} - 24 \, {\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} \sqrt {c d} a^{2} c d^{3} e^{\frac {5}{2}} + 6 \, {\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 )} a^{3} c d^{3} e^{4} - 18 \, {\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 )}^{3} a^{2} c d^{2} e^{3} + 8 \, \sqrt {c d} a^{4} d^{2} e^{\frac {11}{2}} - 16 \, {\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} \sqrt {c d} a^{3} d e^{\frac {9}{2}} + 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 )} a^{4} d e^{6} - 5 \, {\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 )}^{3} a^{3} e^{5}}{4 \, {\left (a d e - {\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}\right )}^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\left (c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e\right )}^{5/2}}{x^3\,\left (d+e\,x\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________