Optimal. Leaf size=345 \[ \frac {1}{24} \left (\frac {a}{c d}-\frac {7 d}{e^2}\right ) x^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}+\frac {x^3 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{4 e}-\frac {\left (105 c^3 d^6-25 a c^2 d^4 e^2-17 a^2 c d^2 e^4-15 a^3 e^6-2 c d e \left (35 c^2 d^4-6 a c d^2 e^2-5 a^2 e^4\right ) x\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{192 c^3 d^3 e^4}+\frac {\left (c d^2-a e^2\right ) \left (35 c^3 d^6+15 a c^2 d^4 e^2+9 a^2 c d^2 e^4+5 a^3 e^6\right ) \tanh ^{-1}\left (\frac {c d^2+a e^2+2 c d e x}{2 \sqrt {c} \sqrt {d} \sqrt {e} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\right )}{128 c^{7/2} d^{7/2} e^{9/2}} \]
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Rubi [A]
time = 0.31, antiderivative size = 345, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 5, integrand size = 40, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {863, 846, 793,
635, 212} \begin {gather*} -\frac {\left (-15 a^3 e^6-2 c d e x \left (-5 a^2 e^4-6 a c d^2 e^2+35 c^2 d^4\right )-17 a^2 c d^2 e^4-25 a c^2 d^4 e^2+105 c^3 d^6\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{192 c^3 d^3 e^4}+\frac {\left (c d^2-a e^2\right ) \left (5 a^3 e^6+9 a^2 c d^2 e^4+15 a c^2 d^4 e^2+35 c^3 d^6\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 )}{128 c^{7/2} d^{7/2} e^{9/2}}+\frac {1}{24} x^2 \left (\frac {a}{c d}-\frac {7 d}{e^2}\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}+\frac {x^3 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{4 e} \end {gather*}
Antiderivative was successfully verified.
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Rule 212
Rule 635
Rule 793
Rule 846
Rule 863
Rubi steps
\begin {align*} \int \frac {x^3 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{d+e x} \, dx &=\int \frac {x^3 (a e+c d x)}{\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx\\ &=\frac {x^3 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{4 e}+\frac {\int \frac {x^2 \left (-3 a c d^2 e-\frac {1}{2} c d \left (7 c d^2-a e^2\right ) x\right )}{\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx}{4 c d e}\\ &=\frac {1}{24} \left (\frac {a}{c d}-\frac {7 d}{e^2}\right ) x^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}+\frac {x^3 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{4 e}+\frac {\int \frac {x \left (a c d^2 e \left (7 c d^2-a e^2\right )+\frac {1}{4} c d \left (35 c^2 d^4-6 a c d^2 e^2-5 a^2 e^4\right ) x\right )}{\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx}{12 c^2 d^2 e^2}\\ &=\frac {1}{24} \left (\frac {a}{c d}-\frac {7 d}{e^2}\right ) x^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}+\frac {x^3 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{4 e}-\frac {\left (105 c^3 d^6-25 a c^2 d^4 e^2-17 a^2 c d^2 e^4-15 a^3 e^6-2 c d e \left (35 c^2 d^4-6 a c d^2 e^2-5 a^2 e^4\right ) x\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{192 c^3 d^3 e^4}+\frac {\left (\left (c d^2-a e^2\right ) \left (35 c^3 d^6+15 a c^2 d^4 e^2+9 a^2 c d^2 e^4+5 a^3 e^6\right )\right ) \int \frac {1}{\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx}{128 c^3 d^3 e^4}\\ &=\frac {1}{24} \left (\frac {a}{c d}-\frac {7 d}{e^2}\right ) x^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}+\frac {x^3 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{4 e}-\frac {\left (105 c^3 d^6-25 a c^2 d^4 e^2-17 a^2 c d^2 e^4-15 a^3 e^6-2 c d e \left (35 c^2 d^4-6 a c d^2 e^2-5 a^2 e^4\right ) x\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{192 c^3 d^3 e^4}+\frac {\left (\left (c d^2-a e^2\right ) \left (35 c^3 d^6+15 a c^2 d^4 e^2+9 a^2 c d^2 e^4+5 a^3 e^6\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 )}{64 c^3 d^3 e^4}\\ &=\frac {1}{24} \left (\frac {a}{c d}-\frac {7 d}{e^2}\right ) x^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}+\frac {x^3 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{4 e}-\frac {\left (105 c^3 d^6-25 a c^2 d^4 e^2-17 a^2 c d^2 e^4-15 a^3 e^6-2 c d e \left (35 c^2 d^4-6 a c d^2 e^2-5 a^2 e^4\right ) x\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{192 c^3 d^3 e^4}+\frac {\left (c d^2-a e^2\right ) \left (35 c^3 d^6+15 a c^2 d^4 e^2+9 a^2 c d^2 e^4+5 a^3 e^6\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 )}{128 c^{7/2} d^{7/2} e^{9/2}}\\ \end {align*}
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Mathematica [A]
time = 1.11, size = 275, normalized size = 0.80 \begin {gather*} \frac {\sqrt {a e+c d x} \sqrt {d+e x} \left (e \sqrt {a e+c d x} \sqrt {d+e x} \left (15 a^3 e^6+a^2 c d e^4 (17 d-10 e x)+a c^2 d^2 e^2 \left (25 d^2-12 d e x+8 e^2 x^2\right )+c^3 d^3 \left (-105 d^3+70 d^2 e x-56 d e^2 x^2+48 e^3 x^3\right )\right )+3 \sqrt {\frac {e}{c d}} \left (-35 c^4 d^8+20 a c^3 d^6 e^2+6 a^2 c^2 d^4 e^4+4 a^3 c d^2 e^6+5 a^4 e^8\right ) \log \left (-\sqrt {\frac {e}{c d}} \sqrt {a e+c d x}+\sqrt {d+e x}\right )\right )}{192 c^3 d^3 e^5 \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. \(954\) vs.
\(2(315)=630\).
time = 0.10, size = 955, normalized size = 2.77
method | result | size |
default | \(\frac {\frac {x \left (a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d e \,x^{2}\right )^{\frac {3}{2}}}{4 c d e}-\frac {5 \left (a \,e^{2}+c \,d^{2}\right ) \left (\frac {\left (a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d e \,x^{2}\right )^{\frac {3}{2}}}{3 c d e}-\frac {\left (a \,e^{2}+c \,d^{2}\right ) \left (\frac {\left (2 c d e x +a \,e^{2}+c \,d^{2}\right ) \sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d e \,x^{2}}}{4 c d e}+\frac {\left (4 a c \,d^{2} e^{2}-\left (a \,e^{2}+c \,d^{2}\right )^{2}\right ) \ln \left (\frac {\frac {1}{2} a \,e^{2}+\frac {1}{2} c \,d^{2}+c d e x}{\sqrt {c d e}}+\sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d e \,x^{2}}\right )}{8 c d e \sqrt {c d e}}\right )}{2 c d e}\right )}{8 c d e}-\frac {a \left (\frac {\left (2 c d e x +a \,e^{2}+c \,d^{2}\right ) \sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d e \,x^{2}}}{4 c d e}+\frac {\left (4 a c \,d^{2} e^{2}-\left (a \,e^{2}+c \,d^{2}\right )^{2}\right ) \ln \left (\frac {\frac {1}{2} a \,e^{2}+\frac {1}{2} c \,d^{2}+c d e x}{\sqrt {c d e}}+\sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d e \,x^{2}}\right )}{8 c d e \sqrt {c d e}}\right )}{4 c}}{e}-\frac {d \left (\frac {\left (a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d e \,x^{2}\right )^{\frac {3}{2}}}{3 c d e}-\frac {\left (a \,e^{2}+c \,d^{2}\right ) \left (\frac {\left (2 c d e x +a \,e^{2}+c \,d^{2}\right ) \sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d e \,x^{2}}}{4 c d e}+\frac {\left (4 a c \,d^{2} e^{2}-\left (a \,e^{2}+c \,d^{2}\right )^{2}\right ) \ln \left (\frac {\frac {1}{2} a \,e^{2}+\frac {1}{2} c \,d^{2}+c d e x}{\sqrt {c d e}}+\sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d e \,x^{2}}\right )}{8 c d e \sqrt {c d e}}\right )}{2 c d e}\right )}{e^{2}}+\frac {d^{2} \left (\frac {\left (2 c d e x +a \,e^{2}+c \,d^{2}\right ) \sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d e \,x^{2}}}{4 c d e}+\frac {\left (4 a c \,d^{2} e^{2}-\left (a \,e^{2}+c \,d^{2}\right )^{2}\right ) \ln \left (\frac {\frac {1}{2} a \,e^{2}+\frac {1}{2} c \,d^{2}+c d e x}{\sqrt {c d e}}+\sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d e \,x^{2}}\right )}{8 c d e \sqrt {c d e}}\right )}{e^{3}}-\frac {d^{3} \left (\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 {\left (a \,e^{2}-c \,d^{2}\right ) \ln \left (\frac {\frac {a \,e^{2}}{2}-\frac {c \,d^{2}}{2}+c d e \left (x +\frac {d}{e}\right )}{\sqrt {c d e}}+\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 )}{2 \sqrt {c d e}}\right )}{e^{4}}\) | \(955\) |
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 = 3.75, size = 653, normalized size = 1.89 \begin {gather*} \left [-\frac {{\left (3 \, {\left (35 \, c^{4} d^{8} - 20 \, a c^{3} d^{6} e^{2} - 6 \, a^{2} c^{2} d^{4} e^{4} - 4 \, a^{3} c d^{2} e^{6} - 5 \, a^{4} e^{8}\right )} \sqrt {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 \, \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^{\frac {1}{2}} + 2 \, {\left (4 \, c^{2} d^{2} x^{2} + 3 \, a c d^{2}\right )} e^{2}\right ) - 4 \, {\left (70 \, c^{4} d^{6} x e^{2} - 105 \, c^{4} d^{7} e - 10 \, a^{2} c^{2} d^{2} x e^{6} + 15 \, a^{3} c d e^{7} + {\left (8 \, a c^{3} d^{3} x^{2} + 17 \, a^{2} c^{2} d^{3}\right )} e^{5} + 12 \, {\left (4 \, c^{4} d^{4} x^{3} - a c^{3} d^{4} x\right )} e^{4} - {\left (56 \, c^{4} d^{5} x^{2} - 25 \, a c^{3} d^{5}\right )} e^{3}\right )} \sqrt {c d^{2} x + a x e^{2} + {\left (c d x^{2} + a d\right )} e}\right )} e^{\left (-5\right )}}{768 \, c^{4} d^{4}}, -\frac {{\left (3 \, {\left (35 \, c^{4} d^{8} - 20 \, a c^{3} d^{6} e^{2} - 6 \, a^{2} c^{2} d^{4} e^{4} - 4 \, a^{3} c d^{2} e^{6} - 5 \, a^{4} e^{8}\right )} \sqrt {-c d e} \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}}{2 \, {\left (c^{2} d^{3} x e + a c d x e^{3} + {\left (c^{2} d^{2} x^{2} + a c d^{2}\right )} e^{2}\right )}}\right ) - 2 \, {\left (70 \, c^{4} d^{6} x e^{2} - 105 \, c^{4} d^{7} e - 10 \, a^{2} c^{2} d^{2} x e^{6} + 15 \, a^{3} c d e^{7} + {\left (8 \, a c^{3} d^{3} x^{2} + 17 \, a^{2} c^{2} d^{3}\right )} e^{5} + 12 \, {\left (4 \, c^{4} d^{4} x^{3} - a c^{3} d^{4} x\right )} e^{4} - {\left (56 \, c^{4} d^{5} x^{2} - 25 \, a c^{3} d^{5}\right )} e^{3}\right )} \sqrt {c d^{2} x + a x e^{2} + {\left (c d x^{2} + a d\right )} e}\right )} e^{\left (-5\right )}}{384 \, c^{4} d^{4}}\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 {x^{3} \sqrt {\left (d + e x\right ) \left (a e + c d x\right )}}{d + e x}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 2.48, size = 292, normalized size = 0.85 \begin {gather*} \frac {1}{192} \, \sqrt {c d x^{2} e + c d^{2} x + a x e^{2} + a d e} {\left (2 \, {\left (4 \, {\left (6 \, x e^{\left (-1\right )} - \frac {{\left (7 \, c^{3} d^{4} e^{2} - a c^{2} d^{2} e^{4}\right )} e^{\left (-4\right )}}{c^{3} d^{3}}\right )} x + \frac {{\left (35 \, c^{3} d^{5} e - 6 \, a c^{2} d^{3} e^{3} - 5 \, a^{2} c d e^{5}\right )} e^{\left (-4\right )}}{c^{3} d^{3}}\right )} x - \frac {{\left (105 \, c^{3} d^{6} - 25 \, a c^{2} d^{4} e^{2} - 17 \, a^{2} c d^{2} e^{4} - 15 \, a^{3} e^{6}\right )} e^{\left (-4\right )}}{c^{3} d^{3}}\right )} - \frac {{\left (35 \, c^{4} d^{8} - 20 \, a c^{3} d^{6} e^{2} - 6 \, a^{2} c^{2} d^{4} e^{4} - 4 \, a^{3} c d^{2} e^{6} - 5 \, a^{4} e^{8}\right )} e^{\left (-\frac {9}{2}\right )} \log \left ({\left | -c d^{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 )} \sqrt {c d} e^{\frac {1}{2}} - a e^{2} \right |}\right )}{128 \, \sqrt {c d} 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 {x^3\,\sqrt {c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e}}{d+e\,x} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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