Optimal. Leaf size=171 \[ -\frac {16 \left (c d^2-a e^2\right )^2 \sqrt {d+e x}}{3 c^3 d^3 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac {8 \left (c d^2-a e^2\right ) (d+e x)^{3/2}}{3 c^2 d^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac {2 (d+e x)^{5/2}}{3 c d \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \]
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Rubi [A]
time = 0.07, antiderivative size = 171, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 39, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.051, Rules used = {670, 662}
\begin {gather*} -\frac {16 \sqrt {d+e x} \left (c d^2-a e^2\right )^2}{3 c^3 d^3 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}+\frac {8 (d+e x)^{3/2} \left (c d^2-a e^2\right )}{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)^{5/2}}{3 c d \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 662
Rule 670
Rubi steps
\begin {align*} \int \frac {(d+e x)^{7/2}}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx &=\frac {2 (d+e x)^{5/2}}{3 c d \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac {\left (4 \left (d^2-\frac {a e^2}{c}\right )\right ) \int \frac {(d+e x)^{5/2}}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx}{3 d}\\ &=\frac {8 \left (c d^2-a e^2\right ) (d+e x)^{3/2}}{3 c^2 d^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac {2 (d+e x)^{5/2}}{3 c d \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac {\left (8 \left (c d^2-a e^2\right )^2\right ) \int \frac {(d+e x)^{3/2}}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx}{3 c^2 d^2}\\ &=-\frac {16 \left (c d^2-a e^2\right )^2 \sqrt {d+e x}}{3 c^3 d^3 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac {8 \left (c d^2-a e^2\right ) (d+e x)^{3/2}}{3 c^2 d^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac {2 (d+e x)^{5/2}}{3 c d \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\\ \end {align*}
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Mathematica [A]
time = 0.06, size = 87, normalized size = 0.51 \begin {gather*} -\frac {2 \sqrt {d+e x} \left (8 a^2 e^4+4 a c d e^2 (-3 d+e x)+c^2 d^2 \left (3 d^2-6 d e x-e^2 x^2\right )\right )}{3 c^3 d^3 \sqrt {(a e+c d x) (d+e x)}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.86, size = 102, normalized size = 0.60
method | result | size |
default | \(-\frac {2 \sqrt {\left (c d x +a e \right ) \left (e x +d \right )}\, \left (-e^{2} x^{2} c^{2} d^{2}+4 a c d \,e^{3} x -6 c^{2} d^{3} e x +8 a^{2} e^{4}-12 a c \,d^{2} e^{2}+3 c^{2} d^{4}\right )}{3 \sqrt {e x +d}\, \left (c d x +a e \right ) c^{3} d^{3}}\) | \(102\) |
gosper | \(-\frac {2 \left (c d x +a e \right ) \left (-e^{2} x^{2} c^{2} d^{2}+4 a c d \,e^{3} x -6 c^{2} d^{3} e x +8 a^{2} e^{4}-12 a c \,d^{2} e^{2}+3 c^{2} d^{4}\right ) \left (e x +d \right )^{\frac {3}{2}}}{3 c^{3} d^{3} \left (c d e \,x^{2}+a \,e^{2} x +c \,d^{2} x +a d e \right )^{\frac {3}{2}}}\) | \(110\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.30, size = 77, normalized size = 0.45 \begin {gather*} \frac {2 \, {\left (c^{2} d^{2} x^{2} e^{2} - 3 \, c^{2} d^{4} + 12 \, a c d^{2} e^{2} - 8 \, a^{2} e^{4} + 2 \, {\left (3 \, c^{2} d^{3} e - 2 \, a c d e^{3}\right )} x\right )}}{3 \, \sqrt {c d x + a e} c^{3} d^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 2.74, size = 137, normalized size = 0.80 \begin {gather*} \frac {2 \, {\left (6 \, c^{2} d^{3} x e - 3 \, c^{2} d^{4} - 4 \, a c d x e^{3} - 8 \, a^{2} e^{4} + {\left (c^{2} d^{2} x^{2} + 12 \, a c d^{2}\right )} e^{2}\right )} \sqrt {c d^{2} x + a x e^{2} + {\left (c d x^{2} + a d\right )} e} \sqrt {x e + d}}{3 \, {\left (c^{4} d^{5} x + a c^{3} d^{3} x e^{2} + {\left (c^{4} d^{4} x^{2} + a c^{3} d^{4}\right )} e\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.41, size = 227, normalized size = 1.33 \begin {gather*} -\frac {2 \, {\left (c^{2} d^{4} e - 2 \, a c d^{2} e^{3} + a^{2} e^{5}\right )}}{\sqrt {{\left (x e + d\right )} c d e - c d^{2} e + a e^{3}} c^{3} d^{3}} + \frac {16 \, {\left (c^{2} d^{4} e - 2 \, a c d^{2} e^{3} + a^{2} e^{5}\right )}}{3 \, \sqrt {-c d^{2} e + a e^{3}} c^{3} d^{3}} + \frac {2 \, {\left (6 \, \sqrt {{\left (x e + d\right )} c d e - c d^{2} e + a e^{3}} c^{7} d^{8} e^{3} - 6 \, \sqrt {{\left (x e + d\right )} c d e - c d^{2} e + a e^{3}} a c^{6} d^{6} e^{5} + {\left ({\left (x e + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {3}{2}} c^{6} d^{6} e^{2}\right )} e^{\left (-3\right )}}{3 \, c^{9} d^{9}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.10, size = 174, normalized size = 1.02 \begin {gather*} \frac {\left (\frac {2\,e\,x^2\,\sqrt {d+e\,x}}{3\,c^2\,d^2}-\frac {\sqrt {d+e\,x}\,\left (16\,a^2\,e^4-24\,a\,c\,d^2\,e^2+6\,c^2\,d^4\right )}{3\,c^4\,d^4\,e}+\frac {x\,\left (12\,c^2\,d^3\,e-8\,a\,c\,d\,e^3\right )\,\sqrt {d+e\,x}}{3\,c^4\,d^4\,e}\right )\,\sqrt {c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e}}{\frac {a}{c}+x^2+\frac {x\,\left (3\,c^4\,d^5+3\,a\,c^3\,d^3\,e^2\right )}{3\,c^4\,d^4\,e}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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