Optimal. Leaf size=171 \[ \frac {2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{9 \left (c d^2-a e^2\right ) (d+e x)^7}+\frac {8 c d \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{63 \left (c d^2-a e^2\right )^2 (d+e x)^6}+\frac {16 c^2 d^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{315 \left (c d^2-a e^2\right )^3 (d+e x)^5} \]
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Rubi [A]
time = 0.06, antiderivative size = 171, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 37, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.054, Rules used = {672, 664}
\begin {gather*} \frac {16 c^2 d^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{315 (d+e x)^5 \left (c d^2-a e^2\right )^3}+\frac {8 c d \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{63 (d+e x)^6 \left (c d^2-a e^2\right )^2}+\frac {2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{9 (d+e x)^7 \left (c d^2-a e^2\right )} \end {gather*}
Antiderivative was successfully verified.
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Rule 664
Rule 672
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}}{(d+e x)^7} \, dx &=\frac {2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{9 \left (c d^2-a e^2\right ) (d+e x)^7}+\frac {(4 c d) \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{(d+e x)^6} \, dx}{9 \left (c d^2-a e^2\right )}\\ &=\frac {2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{9 \left (c d^2-a e^2\right ) (d+e x)^7}+\frac {8 c d \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{63 \left (c d^2-a e^2\right )^2 (d+e x)^6}+\frac {\left (8 c^2 d^2\right ) \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{(d+e x)^5} \, dx}{63 \left (c d^2-a e^2\right )^2}\\ &=\frac {2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{9 \left (c d^2-a e^2\right ) (d+e x)^7}+\frac {8 c d \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{63 \left (c d^2-a e^2\right )^2 (d+e x)^6}+\frac {16 c^2 d^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{315 \left (c d^2-a e^2\right )^3 (d+e x)^5}\\ \end {align*}
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Mathematica [A]
time = 0.14, size = 94, normalized size = 0.55 \begin {gather*} \frac {2 ((a e+c d x) (d+e x))^{5/2} \left (35 a^2 e^4-10 a c d e^2 (9 d+2 e x)+c^2 d^2 \left (63 d^2+36 d e x+8 e^2 x^2\right )\right )}{315 \left (c d^2-a e^2\right )^3 (d+e x)^7} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.71, size = 212, normalized size = 1.24
method | result | size |
gosper | \(-\frac {2 \left (c d x +a e \right ) \left (8 e^{2} x^{2} c^{2} d^{2}-20 a c d \,e^{3} x +36 c^{2} d^{3} e x +35 a^{2} e^{4}-90 a c \,d^{2} e^{2}+63 c^{2} d^{4}\right ) \left (c d e \,x^{2}+a \,e^{2} x +c \,d^{2} x +a d e \right )^{\frac {3}{2}}}{315 \left (e x +d \right )^{6} \left (e^{6} a^{3}-3 e^{4} d^{2} a^{2} c +3 d^{4} e^{2} c^{2} a -d^{6} c^{3}\right )}\) | \(146\) |
default | \(\frac {-\frac {2 \left (c d e \left (x +\frac {d}{e}\right )^{2}+\left (e^{2} a -c \,d^{2}\right ) \left (x +\frac {d}{e}\right )\right )^{\frac {5}{2}}}{9 \left (e^{2} a -c \,d^{2}\right ) \left (x +\frac {d}{e}\right )^{7}}-\frac {4 c d e \left (-\frac {2 \left (c d e \left (x +\frac {d}{e}\right )^{2}+\left (e^{2} a -c \,d^{2}\right ) \left (x +\frac {d}{e}\right )\right )^{\frac {5}{2}}}{7 \left (e^{2} a -c \,d^{2}\right ) \left (x +\frac {d}{e}\right )^{6}}+\frac {4 c d e \left (c d e \left (x +\frac {d}{e}\right )^{2}+\left (e^{2} a -c \,d^{2}\right ) \left (x +\frac {d}{e}\right )\right )^{\frac {5}{2}}}{35 \left (e^{2} a -c \,d^{2}\right )^{2} \left (x +\frac {d}{e}\right )^{5}}\right )}{9 \left (e^{2} a -c \,d^{2}\right )}}{e^{7}}\) | \(212\) |
trager | \(-\frac {2 \left (8 c^{4} d^{4} e^{2} x^{4}-4 a \,c^{3} d^{3} e^{3} x^{3}+36 c^{4} d^{5} e \,x^{3}+3 a^{2} c^{2} d^{2} e^{4} x^{2}-18 a \,c^{3} d^{4} e^{2} x^{2}+63 c^{4} d^{6} x^{2}+50 d \,e^{5} c \,a^{3} x -144 a^{2} c^{2} d^{3} e^{3} x +126 a \,c^{3} d^{5} e x +35 a^{4} e^{6}-90 a^{3} c \,d^{2} e^{4}+63 a^{2} c^{2} d^{4} e^{2}\right ) \sqrt {c d e \,x^{2}+a \,e^{2} x +c \,d^{2} x +a d e}}{315 \left (e^{6} a^{3}-3 e^{4} d^{2} a^{2} c +3 d^{4} e^{2} c^{2} a -d^{6} c^{3}\right ) \left (e x +d \right )^{5}}\) | \(234\) |
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 [B] Leaf count of result is larger than twice the leaf count of optimal. 470 vs.
\(2 (162) = 324\).
time = 28.64, size = 470, normalized size = 2.75 \begin {gather*} \frac {2 \, {\left (63 \, c^{4} d^{6} x^{2} + 50 \, a^{3} c d x e^{5} + 35 \, a^{4} e^{6} + 3 \, {\left (a^{2} c^{2} d^{2} x^{2} - 30 \, a^{3} c d^{2}\right )} e^{4} - 4 \, {\left (a c^{3} d^{3} x^{3} + 36 \, a^{2} c^{2} d^{3} x\right )} e^{3} + {\left (8 \, c^{4} d^{4} x^{4} - 18 \, a c^{3} d^{4} x^{2} + 63 \, a^{2} c^{2} d^{4}\right )} e^{2} + 18 \, {\left (2 \, c^{4} d^{5} x^{3} + 7 \, a c^{3} d^{5} x\right )} e\right )} \sqrt {c d^{2} x + a x e^{2} + {\left (c d x^{2} + a d\right )} e}}{315 \, {\left (5 \, c^{3} d^{10} x e + c^{3} d^{11} - a^{3} x^{5} e^{11} - 5 \, a^{3} d x^{4} e^{10} + {\left (3 \, a^{2} c d^{2} x^{5} - 10 \, a^{3} d^{2} x^{3}\right )} e^{9} + 5 \, {\left (3 \, a^{2} c d^{3} x^{4} - 2 \, a^{3} d^{3} x^{2}\right )} e^{8} - {\left (3 \, a c^{2} d^{4} x^{5} - 30 \, a^{2} c d^{4} x^{3} + 5 \, a^{3} d^{4} x\right )} e^{7} - {\left (15 \, a c^{2} d^{5} x^{4} - 30 \, a^{2} c d^{5} x^{2} + a^{3} d^{5}\right )} e^{6} + {\left (c^{3} d^{6} x^{5} - 30 \, a c^{2} d^{6} x^{3} + 15 \, a^{2} c d^{6} x\right )} e^{5} + {\left (5 \, c^{3} d^{7} x^{4} - 30 \, a c^{2} d^{7} x^{2} + 3 \, a^{2} c d^{7}\right )} e^{4} + 5 \, {\left (2 \, c^{3} d^{8} x^{3} - 3 \, a c^{2} d^{8} x\right )} e^{3} + {\left (10 \, c^{3} d^{9} x^{2} - 3 \, a c^{2} d^{9}\right )} e^{2}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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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.
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Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 3.85, size = 2067, normalized size = 12.09 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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