Optimal. Leaf size=438 \[ -\frac {2 x \left (a d e \left (c d^2-a e^2\right ) \left (-3 a^2 e^4-10 a c d^2 e^2+5 c^2 d^4\right )+x \left (c d^2-a e^2\right ) \left (-3 a^3 e^6-a^2 c d^2 e^4-9 a c^2 d^4 e^2+5 c^3 d^6\right )\right )}{3 c d e^2 \left (c d^2-a e^2\right )^4 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}+\frac {\left (-9 a^3 e^6+9 a^2 c d^2 e^4-31 a c^2 d^4 e^2+15 c^3 d^6\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{3 c^2 d^2 e^3 \left (c d^2-a e^2\right )^3}-\frac {\left (3 a e^2+5 c d^2\right ) \tanh ^{-1}\left (\frac {a e^2+c d^2+2 c d e x}{2 \sqrt {c} \sqrt {d} \sqrt {e} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{2 c^{5/2} d^{5/2} e^{7/2}}-\frac {2 d x^3 \left (c d x \left (c d^2-a e^2\right )+a e \left (c d^2-a e^2\right )\right )}{3 e \left (c d^2-a e^2\right )^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}} \]
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Rubi [A] time = 0.54, antiderivative size = 438, 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, 640, 621, 206} \begin {gather*} -\frac {2 x \left (x \left (c d^2-a e^2\right ) \left (-a^2 c d^2 e^4-3 a^3 e^6-9 a c^2 d^4 e^2+5 c^3 d^6\right )+a d e \left (c d^2-a e^2\right ) \left (-3 a^2 e^4-10 a c d^2 e^2+5 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 (9 a^2 c d^2 e^4-9 a^3 e^6-31 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}}{3 c^2 d^2 e^3 \left (c d^2-a e^2\right )^3}-\frac {\left (3 a e^2+5 c d^2\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 )}{2 c^{5/2} d^{5/2} e^{7/2}}-\frac {2 d x^3 \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 640
Rule 818
Rule 849
Rubi steps
\begin {align*} \int \frac {x^4}{(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^4 (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^3 \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^2 \left (3 a c d^2 e \left (c d^2-a e^2\right )+\frac {1}{2} c d \left (5 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^3 \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 \left (a d e \left (c d^2-a e^2\right ) \left (5 c^2 d^4-10 a c d^2 e^2-3 a^2 e^4\right )+\left (c d^2-a e^2\right ) \left (5 c^3 d^6-9 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 {\frac {1}{2} a c d^2 e \left (c d^2-a e^2\right ) \left (5 c^2 d^4-10 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 (15 c^3 d^6-31 a c^2 d^4 e^2+9 a^2 c d^2 e^4-9 a^3 e^6\right ) x}{\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^3 \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 \left (a d e \left (c d^2-a e^2\right ) \left (5 c^2 d^4-10 a c d^2 e^2-3 a^2 e^4\right )+\left (c d^2-a e^2\right ) \left (5 c^3 d^6-9 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 (15 c^3 d^6-31 a c^2 d^4 e^2+9 a^2 c d^2 e^4-9 a^3 e^6\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{3 c^2 d^2 e^3 \left (c d^2-a e^2\right )^3}-\frac {\left (5 c d^2+3 a e^2\right ) \int \frac {1}{\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx}{2 c^2 d^2 e^3}\\ &=-\frac {2 d x^3 \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 \left (a d e \left (c d^2-a e^2\right ) \left (5 c^2 d^4-10 a c d^2 e^2-3 a^2 e^4\right )+\left (c d^2-a e^2\right ) \left (5 c^3 d^6-9 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 (15 c^3 d^6-31 a c^2 d^4 e^2+9 a^2 c d^2 e^4-9 a^3 e^6\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{3 c^2 d^2 e^3 \left (c d^2-a e^2\right )^3}-\frac {\left (5 c d^2+3 a e^2\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 )}{c^2 d^2 e^3}\\ &=-\frac {2 d x^3 \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 \left (a d e \left (c d^2-a e^2\right ) \left (5 c^2 d^4-10 a c d^2 e^2-3 a^2 e^4\right )+\left (c d^2-a e^2\right ) \left (5 c^3 d^6-9 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 (15 c^3 d^6-31 a c^2 d^4 e^2+9 a^2 c d^2 e^4-9 a^3 e^6\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{3 c^2 d^2 e^3 \left (c d^2-a e^2\right )^3}-\frac {\left (5 c d^2+3 a e^2\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 )}{2 c^{5/2} d^{5/2} e^{7/2}}\\ \end {align*}
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Mathematica [A] time = 1.34, size = 387, normalized size = 0.88 \begin {gather*} \frac {(a e+c d x) \left (-\frac {a e \left (3 a e^2-c d^2\right ) \left (a^2 e^2 \left (8 d^2+12 d e x+3 e^2 x^2\right )+2 a c d^2 e x (2 d+3 e x)-c^2 d^4 x^2\right )}{c d \left (c d^2-a e^2\right )^3}-\frac {\left (3 a e^2+5 c d^2\right ) \sqrt {a e+c d x} \left (c^{3/2} d^{7/2} \sqrt {e} \left (c d^2-a e^2\right ) \sqrt {a e+c d x}-(d+e x) \left (2 c^{3/2} d^{5/2} \sqrt {e} \left (2 c d^2-3 a e^2\right ) \sqrt {a e+c d x}-3 \sqrt {c d} \left (c d^2-a e^2\right )^{5/2} \sqrt {\frac {c d (d+e x)}{c d^2-a e^2}} \sinh ^{-1}\left (\frac {\sqrt {c} \sqrt {d} \sqrt {e} \sqrt {a e+c d x}}{\sqrt {c d} \sqrt {c d^2-a e^2}}\right )\right )\right )}{c^{5/2} d^{5/2} e^{5/2} \left (c d^2-a e^2\right )^2}+3 x^3\right )}{3 c d e ((d+e x) (a e+c d x))^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 180.17, 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 = 2.66, size = 1782, normalized size = 4.07
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 = 0.01, size = 1266, normalized size = 2.89
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^4}{\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^{4}}{\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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