Optimal. Leaf size=240 \[ -\frac {(a e-c d x) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{x}+\frac {\sqrt {c} \sqrt {d} \left (c d^2+3 a e^2\right ) \tanh ^{-1}\left (\frac {c d^2+a e^2+2 c d e x}{2 \sqrt {c} \sqrt {d} \sqrt {e} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\right )}{2 \sqrt {e}}-\frac {\sqrt {a} \sqrt {e} \left (3 c d^2+a e^2\right ) \tanh ^{-1}\left (\frac {2 a d e+\left (c d^2+a e^2\right ) x}{2 \sqrt {a} \sqrt {d} \sqrt {e} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\right )}{2 \sqrt {d}} \]
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Rubi [A]
time = 0.18, antiderivative size = 240, normalized size of antiderivative = 1.00, number of steps
used = 7, 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 {\sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2} (a e-c d x)}{x}+\frac {\sqrt {c} \sqrt {d} \left (3 a e^2+c d^2\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 )}{2 \sqrt {e}}-\frac {\sqrt {a} \sqrt {e} \left (a e^2+3 c d^2\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 )}{2 \sqrt {d}} \end {gather*}
Antiderivative was successfully verified.
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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 )^{3/2}}{x^2 (d+e x)} \, dx &=\int \frac {(a e+c d x) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{x^2} \, dx\\ &=-\frac {(a e-c d x) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{x}-\frac {1}{2} \int \frac {-a e \left (3 c d^2+a e^2\right )-c d \left (c d^2+3 a e^2\right ) x}{x \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx\\ &=-\frac {(a e-c d x) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{x}+\frac {1}{2} \left (a e \left (3 c d^2+a e^2\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}{2} \left (c d \left (c d^2+3 a e^2\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 {(a e-c d x) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{x}-\left (a e \left (3 c d^2+a e^2\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 )+\left (c d \left (c d^2+3 a e^2\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 {(a e-c d x) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{x}+\frac {\sqrt {c} \sqrt {d} \left (c d^2+3 a e^2\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 )}{2 \sqrt {e}}-\frac {\sqrt {a} \sqrt {e} \left (3 c d^2+a e^2\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 )}{2 \sqrt {d}}\\ \end {align*}
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Mathematica [A]
time = 0.43, size = 213, normalized size = 0.89 \begin {gather*} -\frac {\sqrt {a e+c d x} \sqrt {d+e x} \left (\sqrt {d} \sqrt {e} (a e-c d x) \sqrt {a e+c d x} \sqrt {d+e x}-\sqrt {c} d \left (c d^2+3 a e^2\right ) x \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {d} \sqrt {d+e x}}{\sqrt {e} \sqrt {a e+c d x}}\right )+\sqrt {a} e \left (3 c d^2+a e^2\right ) x \tanh ^{-1}\left (\frac {\sqrt {a} \sqrt {e} \sqrt {d+e x}}{\sqrt {d} \sqrt {a e+c d x}}\right )\right )}{\sqrt {d} \sqrt {e} x \sqrt {(a e+c d x) (d+e x)}} \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. \(1299\) vs.
\(2(202)=404\).
time = 0.08, size = 1300, normalized size = 5.42
method | result | size |
default | \(\text {Expression too large to display}\) | \(1300\) |
Verification of antiderivative is not currently implemented for this CAS.
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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.
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Fricas [A]
time = 3.68, size = 1212, normalized size = 5.05 \begin {gather*} \left [\frac {{\left (c d^{2} x + 3 \, a x e^{2}\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 ) + {\left (3 \, c d^{2} x + a x e^{2}\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 \, \sqrt {c d^{2} x + a x e^{2} + {\left (c d x^{2} + a d\right )} e} {\left (c d x - a e\right )}}{4 \, x}, \frac {{\left (3 \, c d^{2} x + a x e^{2}\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 ) - 2 \, {\left (c d^{2} x + 3 \, a x e^{2}\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 \, \sqrt {c d^{2} x + a x e^{2} + {\left (c d x^{2} + a d\right )} e} {\left (c d x - a e\right )}}{4 \, x}, \frac {{\left (c d^{2} x + 3 \, a x e^{2}\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 ) + 2 \, {\left (3 \, c d^{2} x + a x e^{2}\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 \, \sqrt {c d^{2} x + a x e^{2} + {\left (c d x^{2} + a d\right )} e} {\left (c d x - a e\right )}}{4 \, x}, -\frac {{\left (c d^{2} x + 3 \, a x e^{2}\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 ) - {\left (3 \, c d^{2} x + a x e^{2}\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 \, \sqrt {c d^{2} x + a x e^{2} + {\left (c d x^{2} + a d\right )} e} {\left (c d x - a e\right )}}{2 \, x}\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 (\left (d + e x\right ) \left (a e + c d x\right )\right )^{\frac {3}{2}}}{x^{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 = 1.37, size = 373, normalized size = 1.55 \begin {gather*} \sqrt {c d x^{2} e + c d^{2} x + a x e^{2} + a d e} c d + \frac {{\left (3 \, a c d^{2} e + a^{2} e^{3}\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 )}{\sqrt {-a d e}} - \frac {{\left (\sqrt {c d} c^{2} d^{3} e^{\frac {1}{2}} + 3 \, \sqrt {c d} a c d e^{\frac {5}{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 )}{2 \, c d} - \frac {{\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 c d^{2} e + 2 \, \sqrt {c d} a^{2} d e^{\frac {5}{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 )} a^{2} e^{3}}{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}} \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 (c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e\right )}^{3/2}}{x^2\,\left (d+e\,x\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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