Optimal. Leaf size=522 \[ -\frac {2 e (a e+c d x)}{3 d \left (c d^2-a e^2\right ) x^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}+\frac {2 \left (3 c^3 d^6+a c^2 d^4 e^2+11 a^2 c d^2 e^4-7 a^3 e^6+c d e \left (3 c^2 d^4+12 a c d^2 e^2-7 a^2 e^4\right ) x\right )}{3 a d^2 e \left (c d^2-a e^2\right )^3 x^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}-\frac {\left (15 c^3 d^6-9 a c^2 d^4 e^2+61 a^2 c d^2 e^4-35 a^3 e^6\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{6 a^2 d^3 e^2 \left (c d^2-a e^2\right )^3 x^2}+\frac {\left (45 c^4 d^8-30 a c^3 d^6 e^2-36 a^2 c^2 d^4 e^4+190 a^3 c d^2 e^6-105 a^4 e^8\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{12 a^3 d^4 e^3 \left (c d^2-a e^2\right )^3 x}-\frac {5 \left (3 c^2 d^4+6 a c d^2 e^2+7 a^2 e^4\right ) \tanh ^{-1}\left (\frac {2 a d e+\left (c d^2+a e^2\right ) x}{2 \sqrt {a} \sqrt {d} \sqrt {e} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\right )}{8 a^{7/2} d^{9/2} e^{7/2}} \]
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Rubi [A]
time = 0.50, antiderivative size = 522, 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 = {865, 836, 848,
820, 738, 212} \begin {gather*} -\frac {5 \left (7 a^2 e^4+6 a c d^2 e^2+3 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 a^{7/2} d^{9/2} e^{7/2}}+\frac {2 \left (-7 a^3 e^6+c d e x \left (-7 a^2 e^4+12 a c d^2 e^2+3 c^2 d^4\right )+11 a^2 c d^2 e^4+a c^2 d^4 e^2+3 c^3 d^6\right )}{3 a d^2 e x^2 \left (c d^2-a e^2\right )^3 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}-\frac {\left (-35 a^3 e^6+61 a^2 c d^2 e^4-9 a c^2 d^4 e^2+15 c^3 d^6\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{6 a^2 d^3 e^2 x^2 \left (c d^2-a e^2\right )^3}+\frac {\left (-105 a^4 e^8+190 a^3 c d^2 e^6-36 a^2 c^2 d^4 e^4-30 a c^3 d^6 e^2+45 c^4 d^8\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{12 a^3 d^4 e^3 x \left (c d^2-a e^2\right )^3}-\frac {2 e (a e+c d x)}{3 d x^2 \left (c d^2-a e^2\right ) \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 738
Rule 820
Rule 836
Rule 848
Rule 865
Rubi steps
\begin {align*} \int \frac {1}{x^3 (d+e x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx &=\int \frac {a e+c d x}{x^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}} \, dx\\ &=-\frac {2 e (a e+c d x)}{3 d \left (c d^2-a e^2\right ) x^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}-\frac {2 \int \frac {-\frac {1}{2} a e \left (3 c d^2-7 a e^2\right ) \left (c d^2-a e^2\right )+4 a c d e^2 \left (c d^2-a e^2\right ) x}{x^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx}{3 a d e \left (c d^2-a e^2\right )^2}\\ &=-\frac {2 e (a e+c d x)}{3 d \left (c d^2-a e^2\right ) x^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}+\frac {2 \left (3 c^3 d^6+a c^2 d^4 e^2+11 a^2 c d^2 e^4-7 a^3 e^6+c d e \left (3 c^2 d^4+12 a c d^2 e^2-7 a^2 e^4\right ) x\right )}{3 a d^2 e \left (c d^2-a e^2\right )^3 x^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac {4 \int \frac {\frac {1}{4} a e \left (c d^2-a e^2\right ) \left (15 c^3 d^6-9 a c^2 d^4 e^2+61 a^2 c d^2 e^4-35 a^3 e^6\right )+a c d e^2 \left (c d^2-a e^2\right ) \left (3 c^2 d^4+12 a c d^2 e^2-7 a^2 e^4\right ) x}{x^3 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx}{3 a^2 d^2 e^2 \left (c d^2-a e^2\right )^4}\\ &=-\frac {2 e (a e+c d x)}{3 d \left (c d^2-a e^2\right ) x^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}+\frac {2 \left (3 c^3 d^6+a c^2 d^4 e^2+11 a^2 c d^2 e^4-7 a^3 e^6+c d e \left (3 c^2 d^4+12 a c d^2 e^2-7 a^2 e^4\right ) x\right )}{3 a d^2 e \left (c d^2-a e^2\right )^3 x^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}-\frac {\left (15 c^3 d^6-9 a c^2 d^4 e^2+61 a^2 c d^2 e^4-35 a^3 e^6\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{6 a^2 d^3 e^2 \left (c d^2-a e^2\right )^3 x^2}-\frac {2 \int \frac {\frac {1}{8} a e \left (c d^2-a e^2\right ) \left (45 c^4 d^8-30 a c^3 d^6 e^2-36 a^2 c^2 d^4 e^4+190 a^3 c d^2 e^6-105 a^4 e^8\right )+\frac {1}{4} a c d e^2 \left (c d^2-a e^2\right ) \left (15 c^3 d^6-9 a c^2 d^4 e^2+61 a^2 c d^2 e^4-35 a^3 e^6\right ) x}{x^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx}{3 a^3 d^3 e^3 \left (c d^2-a e^2\right )^4}\\ &=-\frac {2 e (a e+c d x)}{3 d \left (c d^2-a e^2\right ) x^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}+\frac {2 \left (3 c^3 d^6+a c^2 d^4 e^2+11 a^2 c d^2 e^4-7 a^3 e^6+c d e \left (3 c^2 d^4+12 a c d^2 e^2-7 a^2 e^4\right ) x\right )}{3 a d^2 e \left (c d^2-a e^2\right )^3 x^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}-\frac {\left (15 c^3 d^6-9 a c^2 d^4 e^2+61 a^2 c d^2 e^4-35 a^3 e^6\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{6 a^2 d^3 e^2 \left (c d^2-a e^2\right )^3 x^2}+\frac {\left (45 c^4 d^8-30 a c^3 d^6 e^2-36 a^2 c^2 d^4 e^4+190 a^3 c d^2 e^6-105 a^4 e^8\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{12 a^3 d^4 e^3 \left (c d^2-a e^2\right )^3 x}+\frac {\left (5 \left (3 c^2 d^4+6 a c d^2 e^2+7 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}{8 a^3 d^4 e^3}\\ &=-\frac {2 e (a e+c d x)}{3 d \left (c d^2-a e^2\right ) x^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}+\frac {2 \left (3 c^3 d^6+a c^2 d^4 e^2+11 a^2 c d^2 e^4-7 a^3 e^6+c d e \left (3 c^2 d^4+12 a c d^2 e^2-7 a^2 e^4\right ) x\right )}{3 a d^2 e \left (c d^2-a e^2\right )^3 x^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}-\frac {\left (15 c^3 d^6-9 a c^2 d^4 e^2+61 a^2 c d^2 e^4-35 a^3 e^6\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{6 a^2 d^3 e^2 \left (c d^2-a e^2\right )^3 x^2}+\frac {\left (45 c^4 d^8-30 a c^3 d^6 e^2-36 a^2 c^2 d^4 e^4+190 a^3 c d^2 e^6-105 a^4 e^8\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{12 a^3 d^4 e^3 \left (c d^2-a e^2\right )^3 x}-\frac {\left (5 \left (3 c^2 d^4+6 a c d^2 e^2+7 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 )}{4 a^3 d^4 e^3}\\ &=-\frac {2 e (a e+c d x)}{3 d \left (c d^2-a e^2\right ) x^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}+\frac {2 \left (3 c^3 d^6+a c^2 d^4 e^2+11 a^2 c d^2 e^4-7 a^3 e^6+c d e \left (3 c^2 d^4+12 a c d^2 e^2-7 a^2 e^4\right ) x\right )}{3 a d^2 e \left (c d^2-a e^2\right )^3 x^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}-\frac {\left (15 c^3 d^6-9 a c^2 d^4 e^2+61 a^2 c d^2 e^4-35 a^3 e^6\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{6 a^2 d^3 e^2 \left (c d^2-a e^2\right )^3 x^2}+\frac {\left (45 c^4 d^8-30 a c^3 d^6 e^2-36 a^2 c^2 d^4 e^4+190 a^3 c d^2 e^6-105 a^4 e^8\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{12 a^3 d^4 e^3 \left (c d^2-a e^2\right )^3 x}-\frac {5 \left (3 c^2 d^4+6 a c d^2 e^2+7 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 a^{7/2} d^{9/2} e^{7/2}}\\ \end {align*}
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Mathematica [A]
time = 0.80, size = 390, normalized size = 0.75 \begin {gather*} \frac {\frac {\sqrt {a} \sqrt {d} \sqrt {e} (a e+c d x) \left (-45 c^5 d^9 x^2 (d+e x)^2-15 a c^4 d^7 e x (d-2 e x) (d+e x)^2+6 a^2 c^3 d^5 e^2 (d+e x)^2 \left (d^2+2 d e x+6 e^2 x^2\right )+a^5 e^8 \left (-6 d^3+21 d^2 e x+140 d e^2 x^2+105 e^3 x^3\right )-2 a^3 c^2 d^3 e^4 \left (9 d^4-9 d^3 e x-6 d^2 e^2 x^2+111 d e^3 x^3+95 e^4 x^4\right )+a^4 c d e^6 \left (18 d^4-48 d^3 e x-237 d^2 e^2 x^2-50 d e^3 x^3+105 e^4 x^4\right )\right )}{\left (-c d^2+a e^2\right )^3 x^2}-15 \left (3 c^2 d^4+6 a c d^2 e^2+7 a^2 e^4\right ) (a e+c d x)^{3/2} (d+e x)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {a} \sqrt {e} \sqrt {d+e x}}{\sqrt {d} \sqrt {a e+c d x}}\right )}{12 a^{7/2} d^{9/2} e^{7/2} ((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. \(1353\) vs.
\(2(488)=976\).
time = 0.08, size = 1354, normalized size = 2.59
method | result | size |
default | \(-\frac {e^{2} \left (-\frac {2}{3 \left (a \,e^{2}-c \,d^{2}\right ) \left (x +\frac {d}{e}\right ) \sqrt {c d e \left (x +\frac {d}{e}\right )^{2}+\left (a \,e^{2}-c \,d^{2}\right ) \left (x +\frac {d}{e}\right )}}+\frac {8 c d e \left (2 c d e \left (x +\frac {d}{e}\right )+a \,e^{2}-c \,d^{2}\right )}{3 \left (a \,e^{2}-c \,d^{2}\right )^{3} \sqrt {c d e \left (x +\frac {d}{e}\right )^{2}+\left (a \,e^{2}-c \,d^{2}\right ) \left (x +\frac {d}{e}\right )}}\right )}{d^{3}}+\frac {-\frac {1}{2 a d e \,x^{2} \sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d e \,x^{2}}}-\frac {5 \left (a \,e^{2}+c \,d^{2}\right ) \left (-\frac {1}{a d e x \sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d e \,x^{2}}}-\frac {3 \left (a \,e^{2}+c \,d^{2}\right ) \left (\frac {1}{a d e \sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d e \,x^{2}}}-\frac {\left (a \,e^{2}+c \,d^{2}\right ) \left (2 c d e x +a \,e^{2}+c \,d^{2}\right )}{a d e \left (4 a c \,d^{2} e^{2}-\left (a \,e^{2}+c \,d^{2}\right )^{2}\right ) \sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d e \,x^{2}}}-\frac {\ln \left (\frac {2 a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +2 \sqrt {a d e}\, \sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d e \,x^{2}}}{x}\right )}{a d e \sqrt {a d e}}\right )}{2 a d e}-\frac {4 c \left (2 c d e x +a \,e^{2}+c \,d^{2}\right )}{a \left (4 a c \,d^{2} e^{2}-\left (a \,e^{2}+c \,d^{2}\right )^{2}\right ) \sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d e \,x^{2}}}\right )}{4 a d e}-\frac {3 c \left (\frac {1}{a d e \sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d e \,x^{2}}}-\frac {\left (a \,e^{2}+c \,d^{2}\right ) \left (2 c d e x +a \,e^{2}+c \,d^{2}\right )}{a d e \left (4 a c \,d^{2} e^{2}-\left (a \,e^{2}+c \,d^{2}\right )^{2}\right ) \sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d e \,x^{2}}}-\frac {\ln \left (\frac {2 a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +2 \sqrt {a d e}\, \sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d e \,x^{2}}}{x}\right )}{a d e \sqrt {a d e}}\right )}{2 a}}{d}-\frac {e \left (-\frac {1}{a d e x \sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d e \,x^{2}}}-\frac {3 \left (a \,e^{2}+c \,d^{2}\right ) \left (\frac {1}{a d e \sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d e \,x^{2}}}-\frac {\left (a \,e^{2}+c \,d^{2}\right ) \left (2 c d e x +a \,e^{2}+c \,d^{2}\right )}{a d e \left (4 a c \,d^{2} e^{2}-\left (a \,e^{2}+c \,d^{2}\right )^{2}\right ) \sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d e \,x^{2}}}-\frac {\ln \left (\frac {2 a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +2 \sqrt {a d e}\, \sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d e \,x^{2}}}{x}\right )}{a d e \sqrt {a d e}}\right )}{2 a d e}-\frac {4 c \left (2 c d e x +a \,e^{2}+c \,d^{2}\right )}{a \left (4 a c \,d^{2} e^{2}-\left (a \,e^{2}+c \,d^{2}\right )^{2}\right ) \sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d e \,x^{2}}}\right )}{d^{2}}+\frac {e^{2} \left (\frac {1}{a d e \sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d e \,x^{2}}}-\frac {\left (a \,e^{2}+c \,d^{2}\right ) \left (2 c d e x +a \,e^{2}+c \,d^{2}\right )}{a d e \left (4 a c \,d^{2} e^{2}-\left (a \,e^{2}+c \,d^{2}\right )^{2}\right ) \sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d e \,x^{2}}}-\frac {\ln \left (\frac {2 a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +2 \sqrt {a d e}\, \sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d e \,x^{2}}}{x}\right )}{a d e \sqrt {a d e}}\right )}{d^{3}}\) | \(1354\) |
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 [B] Leaf count of result is larger than twice the leaf count of optimal. 1133 vs.
\(2 (473) = 946\).
time = 73.55, size = 2291, normalized size = 4.39 \begin {gather*} \text {Too large to display} \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 {1}{x^{3} \left (\left (d + e x\right ) \left (a e + c d x\right )\right )^{\frac {3}{2}} \left (d + e x\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \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 {1}{x^3\,\left (d+e\,x\right )\,{\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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