Optimal. Leaf size=515 \[ \frac {5 \left (3 a^2 e^4+6 a c d^2 e^2+7 c^2 d^4\right ) \tanh ^{-1}\left (\frac {a e^2+c d^2+2 c d e x}{2 \sqrt {c} \sqrt {d} \sqrt {e} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{8 c^{7/2} d^{7/2} e^{9/2}}-\frac {2 x^2 \left (a d e \left (c d^2-a e^2\right ) \left (-3 a^2 e^4-12 a c d^2 e^2+7 c^2 d^4\right )+x \left (c d^2-a e^2\right ) \left (-3 a^3 e^6-a^2 c d^2 e^4-11 a c^2 d^4 e^2+7 c^3 d^6\right )\right )}{3 c d e^2 \left (c d^2-a e^2\right )^4 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}-\frac {\left (-45 a^4 e^8+30 a^3 c d^2 e^6+36 a^2 c^2 d^4 e^4-2 c d e x \left (-15 a^3 e^6+9 a^2 c d^2 e^4-61 a c^2 d^4 e^2+35 c^3 d^6\right )-190 a c^3 d^6 e^2+105 c^4 d^8\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{12 c^3 d^3 e^4 \left (c d^2-a e^2\right )^3}-\frac {2 d x^4 \left (c d x \left (c d^2-a e^2\right )+a e \left (c d^2-a e^2\right )\right )}{3 e \left (c d^2-a e^2\right )^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}} \]
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Rubi [A] time = 0.62, antiderivative size = 515, 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 = {849, 818, 779, 621, 206} \begin {gather*} -\frac {2 x^2 \left (x \left (c d^2-a e^2\right ) \left (-a^2 c d^2 e^4-3 a^3 e^6-11 a c^2 d^4 e^2+7 c^3 d^6\right )+a d e \left (c d^2-a e^2\right ) \left (-3 a^2 e^4-12 a c d^2 e^2+7 c^2 d^4\right )\right )}{3 c d e^2 \left (c d^2-a e^2\right )^4 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}-\frac {\left (-2 c d e x \left (9 a^2 c d^2 e^4-15 a^3 e^6-61 a c^2 d^4 e^2+35 c^3 d^6\right )+36 a^2 c^2 d^4 e^4+30 a^3 c d^2 e^6-45 a^4 e^8-190 a c^3 d^6 e^2+105 c^4 d^8\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{12 c^3 d^3 e^4 \left (c d^2-a e^2\right )^3}+\frac {5 \left (3 a^2 e^4+6 a c d^2 e^2+7 c^2 d^4\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 )}{8 c^{7/2} d^{7/2} e^{9/2}}-\frac {2 d x^4 \left (c d x \left (c d^2-a e^2\right )+a e \left (c d^2-a e^2\right )\right )}{3 e \left (c d^2-a e^2\right )^2 \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 206
Rule 621
Rule 779
Rule 818
Rule 849
Rubi steps
\begin {align*} \int \frac {x^5}{(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 {x^5 (a e+c d x)}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}} \, dx\\ &=-\frac {2 d x^4 \left (a e \left (c d^2-a e^2\right )+c d \left (c d^2-a e^2\right ) x\right )}{3 e \left (c d^2-a e^2\right )^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 {x^3 \left (4 a c d^2 e \left (c d^2-a e^2\right )+\frac {1}{2} c d \left (7 c d^2-3 a e^2\right ) \left (c d^2-a e^2\right ) x\right )}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx}{3 c d e \left (c d^2-a e^2\right )^2}\\ &=-\frac {2 d x^4 \left (a e \left (c d^2-a e^2\right )+c d \left (c d^2-a e^2\right ) x\right )}{3 e \left (c d^2-a e^2\right )^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}-\frac {2 x^2 \left (a d e \left (c d^2-a e^2\right ) \left (7 c^2 d^4-12 a c d^2 e^2-3 a^2 e^4\right )+\left (c d^2-a e^2\right ) \left (7 c^3 d^6-11 a c^2 d^4 e^2-a^2 c d^2 e^4-3 a^3 e^6\right ) x\right )}{3 c d e^2 \left (c d^2-a e^2\right )^4 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac {4 \int \frac {x \left (a c d^2 e \left (c d^2-a e^2\right ) \left (7 c^2 d^4-12 a c d^2 e^2-3 a^2 e^4\right )+\frac {1}{4} c d \left (c d^2-a e^2\right ) \left (35 c^3 d^6-61 a c^2 d^4 e^2+9 a^2 c d^2 e^4-15 a^3 e^6\right ) x\right )}{\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx}{3 c^2 d^2 e^2 \left (c d^2-a e^2\right )^4}\\ &=-\frac {2 d x^4 \left (a e \left (c d^2-a e^2\right )+c d \left (c d^2-a e^2\right ) x\right )}{3 e \left (c d^2-a e^2\right )^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}-\frac {2 x^2 \left (a d e \left (c d^2-a e^2\right ) \left (7 c^2 d^4-12 a c d^2 e^2-3 a^2 e^4\right )+\left (c d^2-a e^2\right ) \left (7 c^3 d^6-11 a c^2 d^4 e^2-a^2 c d^2 e^4-3 a^3 e^6\right ) x\right )}{3 c d e^2 \left (c d^2-a e^2\right )^4 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}-\frac {\left (105 c^4 d^8-190 a c^3 d^6 e^2+36 a^2 c^2 d^4 e^4+30 a^3 c d^2 e^6-45 a^4 e^8-2 c d e \left (35 c^3 d^6-61 a c^2 d^4 e^2+9 a^2 c d^2 e^4-15 a^3 e^6\right ) x\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{12 c^3 d^3 e^4 \left (c d^2-a e^2\right )^3}+\frac {\left (5 \left (7 c^2 d^4+6 a c d^2 e^2+3 a^2 e^4\right )\right ) \int \frac {1}{\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx}{8 c^3 d^3 e^4}\\ &=-\frac {2 d x^4 \left (a e \left (c d^2-a e^2\right )+c d \left (c d^2-a e^2\right ) x\right )}{3 e \left (c d^2-a e^2\right )^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}-\frac {2 x^2 \left (a d e \left (c d^2-a e^2\right ) \left (7 c^2 d^4-12 a c d^2 e^2-3 a^2 e^4\right )+\left (c d^2-a e^2\right ) \left (7 c^3 d^6-11 a c^2 d^4 e^2-a^2 c d^2 e^4-3 a^3 e^6\right ) x\right )}{3 c d e^2 \left (c d^2-a e^2\right )^4 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}-\frac {\left (105 c^4 d^8-190 a c^3 d^6 e^2+36 a^2 c^2 d^4 e^4+30 a^3 c d^2 e^6-45 a^4 e^8-2 c d e \left (35 c^3 d^6-61 a c^2 d^4 e^2+9 a^2 c d^2 e^4-15 a^3 e^6\right ) x\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{12 c^3 d^3 e^4 \left (c d^2-a e^2\right )^3}+\frac {\left (5 \left (7 c^2 d^4+6 a c d^2 e^2+3 a^2 e^4\right )\right ) \operatorname {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 )}{4 c^3 d^3 e^4}\\ &=-\frac {2 d x^4 \left (a e \left (c d^2-a e^2\right )+c d \left (c d^2-a e^2\right ) x\right )}{3 e \left (c d^2-a e^2\right )^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}-\frac {2 x^2 \left (a d e \left (c d^2-a e^2\right ) \left (7 c^2 d^4-12 a c d^2 e^2-3 a^2 e^4\right )+\left (c d^2-a e^2\right ) \left (7 c^3 d^6-11 a c^2 d^4 e^2-a^2 c d^2 e^4-3 a^3 e^6\right ) x\right )}{3 c d e^2 \left (c d^2-a e^2\right )^4 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}-\frac {\left (105 c^4 d^8-190 a c^3 d^6 e^2+36 a^2 c^2 d^4 e^4+30 a^3 c d^2 e^6-45 a^4 e^8-2 c d e \left (35 c^3 d^6-61 a c^2 d^4 e^2+9 a^2 c d^2 e^4-15 a^3 e^6\right ) x\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{12 c^3 d^3 e^4 \left (c d^2-a e^2\right )^3}+\frac {5 \left (7 c^2 d^4+6 a c d^2 e^2+3 a^2 e^4\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 )}{8 c^{7/2} d^{7/2} e^{9/2}}\\ \end {align*}
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Mathematica [A] time = 5.63, size = 296, normalized size = 0.57 \begin {gather*} \frac {\frac {2 (d+e x)^2 (a e+c d x)^2 \left (\frac {24 a^5 e^9}{c^3 \left (c d^2-a e^2\right )^3 (a e+c d x)}-\frac {3 \left (7 a e^2+11 c d^2\right )}{c^3}+\frac {8 d^8}{(d+e x)^2 \left (c d^2-a e^2\right )^2}+\frac {40 \left (3 a d^7 e^2-2 c d^9\right )}{(d+e x) \left (c d^2-a e^2\right )^3}+\frac {6 d e x}{c^2}\right )}{3 d^3 e^4}+\frac {5 (d+e x)^{3/2} \left (3 a^2 e^4+6 a c d^2 e^2+7 c^2 d^4\right ) (a e+c d x)^{3/2} \log \left (2 \sqrt {c} \sqrt {d} \sqrt {e} \sqrt {d+e x} \sqrt {a e+c d x}+a e^2+c d (d+2 e x)\right )}{c^{7/2} d^{7/2} e^{9/2}}}{8 ((d+e x) (a e+c d x))^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 180.03, size = 0, normalized size = 0.00 \begin {gather*} \text {\$Aborted} \end {gather*}
Verification is not applicable to the result.
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fricas [B] time = 6.65, size = 2120, normalized size = 4.12
result too large to display
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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maple [B] time = 0.03, size = 1680, normalized size = 3.26
result too large to display
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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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {x^5}{\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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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{5}}{\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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