3.99.30 \(\int \frac {-40 x+60 x^5+\frac {(-10+15 x^4)^5 (-100 x-1350 x^5)}{e}}{-8+12 x^4+\frac {(-10+15 x^4)^5 (-40+60 x^4)}{e}+\frac {(-10+15 x^4)^{10} (-50+75 x^4)}{e^2}} \, dx\)

Optimal. Leaf size=24 \[ \frac {x^2}{\frac {2}{5}+\frac {3125 \left (-2+3 x^4\right )^5}{e}} \]

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Rubi [F]  time = 3.58, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {-40 x+60 x^5+\frac {\left (-10+15 x^4\right )^5 \left (-100 x-1350 x^5\right )}{e}}{-8+12 x^4+\frac {\left (-10+15 x^4\right )^5 \left (-40+60 x^4\right )}{e}+\frac {\left (-10+15 x^4\right )^{10} \left (-50+75 x^4\right )}{e^2}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-40*x + 60*x^5 + ((-10 + 15*x^4)^5*(-100*x - 1350*x^5))/E)/(-8 + 12*x^4 + ((-10 + 15*x^4)^5*(-40 + 60*x^4
))/E + ((-10 + 15*x^4)^10*(-50 + 75*x^4))/E^2),x]

[Out]

(3*Sqrt[3*(50 + 5^(4/5)*(-2*E)^(1/5))]*E*ArcTanh[(5*x^2)/Sqrt[(50 + 5^(4/5)*(-2*E)^(1/5))/3]])/(10*(5*5^(1/5)*
(-2*E)^(4/5) - E)) + (3*Sqrt[3/(-1 + 5*2^(4/5)*(5/E)^(1/5))]*E^(1/10)*ArcTanh[(5^(3/5)*Sqrt[3/(-1 + 5*2^(4/5)*
(5/E)^(1/5))]*x^2)/(2*E)^(1/10)])/(2^(9/10)*5^(3/5)) - (3*E*Sqrt[3*(50 - (-5)^(4/5)*(2*E)^(1/5))]*ArcTanh[(5*x
^2)/Sqrt[(50 - (-5)^(4/5)*(2*E)^(1/5))/3]])/(10*(E + 5*(-5)^(1/5)*(2*E)^(4/5))) - (3*E*Sqrt[3*(50 - (-1)^(2/5)
*5^(4/5)*(2*E)^(1/5))]*ArcTanh[(5*x^2)/Sqrt[(50 - (-1)^(2/5)*5^(4/5)*(2*E)^(1/5))/3]])/(10*(E + 5*(-1)^(3/5)*5
^(1/5)*(2*E)^(4/5))) - (3*E*Sqrt[3*(50 + (-1)^(3/5)*5^(4/5)*(2*E)^(1/5))]*ArcTanh[(5*x^2)/Sqrt[(50 + (-1)^(3/5
)*5^(4/5)*(2*E)^(1/5))/3]])/(10*(E - 5*(-1)^(2/5)*5^(1/5)*(2*E)^(4/5))) - 100*(250000 - E)*E*Defer[Subst][Defe
r[Int][(2*E + 15625*(-2 + 3*x^2)^5)^(-2), x], x, x^2] + 150000000*E*Defer[Subst][Defer[Int][x^2/(2*E + 15625*(
-2 + 3*x^2)^5)^2, x], x, x^2] - 337500000*E*Defer[Subst][Defer[Int][x^4/(2*E + 15625*(-2 + 3*x^2)^5)^2, x], x,
x^2] + 337500000*E*Defer[Subst][Defer[Int][x^6/(2*E + 15625*(-2 + 3*x^2)^5)^2, x], x, x^2] - 126562500*E*Defe
r[Subst][Defer[Int][x^8/(2*E + 15625*(-2 + 3*x^2)^5)^2, x], x, x^2]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {\left (-5000000 e+20 e^2\right ) x-37500000 e x^5+337500000 e x^9-843750000 e x^{13}[phone] e x^{17}-341718750 e x^{21}}{250000000000-2000000 e+4 e^2+(-3750000000000+15000000 e) x^4+(25312500000000-45000000 e) x^8+(-101250000000000+67500000 e) x^{12}+(265781250000000-50625000 e) x^{16}+(-478406250000000+15187500 e) x^{20}[phone] x^{24}-512578125000000 x^{28}[phone] x^{32}-96108398437500 x^{36}[phone] x^{40}} \, dx\\ &=\int \left (\frac {200 e x \left (-250000 \left (1-\frac {e}{250000}\right )+1500000 x^4-3375000 x^8+3375000 x^{12}-1265625 x^{16}\right )}{\left (500000 \left (1-\frac {e}{250000}\right )-3750000 x^4+11250000 x^8-16875000 x^{12}[phone] x^{16}-3796875 x^{20}\right )^2}+\frac {90 e x}{500000 \left (1-\frac {e}{250000}\right )-3750000 x^4+11250000 x^8-16875000 x^{12}[phone] x^{16}-3796875 x^{20}}\right ) \, dx\\ &=(90 e) \int \frac {x}{500000 \left (1-\frac {e}{250000}\right )-3750000 x^4+11250000 x^8-16875000 x^{12}[phone] x^{16}-3796875 x^{20}} \, dx+(200 e) \int \frac {x \left (-250000 \left (1-\frac {e}{250000}\right )+1500000 x^4-3375000 x^8+3375000 x^{12}-1265625 x^{16}\right )}{\left (500000 \left (1-\frac {e}{250000}\right )-3750000 x^4+11250000 x^8-16875000 x^{12}[phone] x^{16}-3796875 x^{20}\right )^2} \, dx\\ &=(90 e) \int \frac {x}{-2 e-15625 \left (-2+3 x^4\right )^5} \, dx+(200 e) \int \frac {x \left (e-15625 \left (2-3 x^4\right )^4\right )}{\left (2 e+15625 \left (-2+3 x^4\right )^5\right )^2} \, dx\\ &=(45 e) \operatorname {Subst}\left (\int \frac {1}{-2 e-15625 \left (-2+3 x^2\right )^5} \, dx,x,x^2\right )+(100 e) \operatorname {Subst}\left (\int \frac {e-15625 \left (2-3 x^2\right )^4}{\left (2 e+15625 \left (-2+3 x^2\right )^5\right )^2} \, dx,x,x^2\right )\\ &=-\left (\frac {9}{2} \operatorname {Subst}\left (\int -\frac {2}{-2-10\ 2^{4/5} \sqrt [5]{-\frac {5}{e}}+15\ 2^{4/5} \sqrt [5]{-\frac {5}{e}} x^2} \, dx,x,x^2\right )\right )-\frac {9}{2} \operatorname {Subst}\left (\int \frac {2}{2-10 (-2)^{4/5} \sqrt [5]{\frac {5}{e}}+15 (-2)^{4/5} \sqrt [5]{\frac {5}{e}} x^2} \, dx,x,x^2\right )-\frac {9}{2} \operatorname {Subst}\left (\int \frac {2}{2-10\ 2^{4/5} \sqrt [5]{\frac {5}{e}}+15\ 2^{4/5} \sqrt [5]{\frac {5}{e}} x^2} \, dx,x,x^2\right )-\frac {9}{2} \operatorname {Subst}\left (\int \frac {2}{2-10 (-1)^{2/5} 2^{4/5} \sqrt [5]{\frac {5}{e}}+15 (-1)^{2/5} 2^{4/5} \sqrt [5]{\frac {5}{e}} x^2} \, dx,x,x^2\right )-\frac {9}{2} \operatorname {Subst}\left (\int -\frac {2}{-2-10 (-1)^{3/5} 2^{4/5} \sqrt [5]{\frac {5}{e}}+15 (-1)^{3/5} 2^{4/5} \sqrt [5]{\frac {5}{e}} x^2} \, dx,x,x^2\right )+(100 e) \operatorname {Subst}\left (\int \left (-\frac {250000 \left (1-\frac {e}{250000}\right )}{\left (500000 \left (1-\frac {e}{250000}\right )-3750000 x^2+11250000 x^4-16875000 x^6+12656250 x^8-3796875 x^{10}\right )^2}+\frac {1500000 x^2}{\left (500000 \left (1-\frac {e}{250000}\right )-3750000 x^2+11250000 x^4-16875000 x^6+12656250 x^8-3796875 x^{10}\right )^2}-\frac {3375000 x^4}{\left (500000 \left (1-\frac {e}{250000}\right )-3750000 x^2+11250000 x^4-16875000 x^6+12656250 x^8-3796875 x^{10}\right )^2}+\frac {3375000 x^6}{\left (500000 \left (1-\frac {e}{250000}\right )-3750000 x^2+11250000 x^4-16875000 x^6+12656250 x^8-3796875 x^{10}\right )^2}-\frac {1265625 x^8}{\left (500000 \left (1-\frac {e}{250000}\right )-3750000 x^2+11250000 x^4-16875000 x^6+12656250 x^8-3796875 x^{10}\right )^2}\right ) \, dx,x,x^2\right )\\ &=9 \operatorname {Subst}\left (\int \frac {1}{-2-10\ 2^{4/5} \sqrt [5]{-\frac {5}{e}}+15\ 2^{4/5} \sqrt [5]{-\frac {5}{e}} x^2} \, dx,x,x^2\right )-9 \operatorname {Subst}\left (\int \frac {1}{2-10 (-2)^{4/5} \sqrt [5]{\frac {5}{e}}+15 (-2)^{4/5} \sqrt [5]{\frac {5}{e}} x^2} \, dx,x,x^2\right )-9 \operatorname {Subst}\left (\int \frac {1}{2-10\ 2^{4/5} \sqrt [5]{\frac {5}{e}}+15\ 2^{4/5} \sqrt [5]{\frac {5}{e}} x^2} \, dx,x,x^2\right )-9 \operatorname {Subst}\left (\int \frac {1}{2-10 (-1)^{2/5} 2^{4/5} \sqrt [5]{\frac {5}{e}}+15 (-1)^{2/5} 2^{4/5} \sqrt [5]{\frac {5}{e}} x^2} \, dx,x,x^2\right )+9 \operatorname {Subst}\left (\int \frac {1}{-2-10 (-1)^{3/5} 2^{4/5} \sqrt [5]{\frac {5}{e}}+15 (-1)^{3/5} 2^{4/5} \sqrt [5]{\frac {5}{e}} x^2} \, dx,x,x^2\right )-(126562500 e) \operatorname {Subst}\left (\int \frac {x^8}{\left (500000 \left (1-\frac {e}{250000}\right )-3750000 x^2+11250000 x^4-16875000 x^6+12656250 x^8-3796875 x^{10}\right )^2} \, dx,x,x^2\right )+(150000000 e) \operatorname {Subst}\left (\int \frac {x^2}{\left (500000 \left (1-\frac {e}{250000}\right )-3750000 x^2+11250000 x^4-16875000 x^6+12656250 x^8-3796875 x^{10}\right )^2} \, dx,x,x^2\right )-(337500000 e) \operatorname {Subst}\left (\int \frac {x^4}{\left (500000 \left (1-\frac {e}{250000}\right )-3750000 x^2+11250000 x^4-16875000 x^6+12656250 x^8-3796875 x^{10}\right )^2} \, dx,x,x^2\right )+(337500000 e) \operatorname {Subst}\left (\int \frac {x^6}{\left (500000 \left (1-\frac {e}{250000}\right )-3750000 x^2+11250000 x^4-16875000 x^6+12656250 x^8-3796875 x^{10}\right )^2} \, dx,x,x^2\right )-(100 (250000-e) e) \operatorname {Subst}\left (\int \frac {1}{\left (500000 \left (1-\frac {e}{250000}\right )-3750000 x^2+11250000 x^4-16875000 x^6+12656250 x^8-3796875 x^{10}\right )^2} \, dx,x,x^2\right )\\ &=-\frac {3 \sqrt {3 \left (50+5^{4/5} \sqrt [5]{-2 e}\right )} \tanh ^{-1}\left (\frac {5 x^2}{\sqrt {\frac {1}{3} \left (50+5^{4/5} \sqrt [5]{-2 e}\right )}}\right )}{10 \left (1-5 (-2)^{4/5} \sqrt [5]{\frac {5}{e}}\right )}+\frac {3 \sqrt {\frac {3}{-1+5\ 2^{4/5} \sqrt [5]{\frac {5}{e}}}} \sqrt [10]{e} \tanh ^{-1}\left (\frac {5^{3/5} \sqrt {\frac {3}{-1+5\ 2^{4/5} \sqrt [5]{\frac {5}{e}}}} x^2}{\sqrt [10]{2 e}}\right )}{2^{9/10} 5^{3/5}}-\frac {3 \sqrt {3 \left (50-(-5)^{4/5} \sqrt [5]{2 e}\right )} \tanh ^{-1}\left (\frac {5 x^2}{\sqrt {\frac {1}{3} \left (50-(-5)^{4/5} \sqrt [5]{2 e}\right )}}\right )}{10 \left (1+5\ 2^{4/5} \sqrt [5]{-\frac {5}{e}}\right )}-\frac {3 \sqrt {3 \left (50-(-1)^{2/5} 5^{4/5} \sqrt [5]{2 e}\right )} \tanh ^{-1}\left (\frac {5 x^2}{\sqrt {\frac {1}{3} \left (50-(-1)^{2/5} 5^{4/5} \sqrt [5]{2 e}\right )}}\right )}{10 \left (1+5 (-1)^{3/5} 2^{4/5} \sqrt [5]{\frac {5}{e}}\right )}-\frac {3 \sqrt {3 \left (50+(-1)^{3/5} 5^{4/5} \sqrt [5]{2 e}\right )} \tanh ^{-1}\left (\frac {5 x^2}{\sqrt {\frac {1}{3} \left (50+(-1)^{3/5} 5^{4/5} \sqrt [5]{2 e}\right )}}\right )}{10 \left (1-5 (-1)^{2/5} 2^{4/5} \sqrt [5]{\frac {5}{e}}\right )}-(126562500 e) \operatorname {Subst}\left (\int \frac {x^8}{\left (2 e+15625 \left (-2+3 x^2\right )^5\right )^2} \, dx,x,x^2\right )+(150000000 e) \operatorname {Subst}\left (\int \frac {x^2}{\left (2 e+15625 \left (-2+3 x^2\right )^5\right )^2} \, dx,x,x^2\right )-(337500000 e) \operatorname {Subst}\left (\int \frac {x^4}{\left (2 e+15625 \left (-2+3 x^2\right )^5\right )^2} \, dx,x,x^2\right )+(337500000 e) \operatorname {Subst}\left (\int \frac {x^6}{\left (2 e+15625 \left (-2+3 x^2\right )^5\right )^2} \, dx,x,x^2\right )-(100 (250000-e) e) \operatorname {Subst}\left (\int \frac {1}{\left (2 e+15625 \left (-2+3 x^2\right )^5\right )^2} \, dx,x,x^2\right )\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.04, size = 23, normalized size = 0.96 \begin {gather*} \frac {10 e x^2}{4 e+31250 \left (-2+3 x^4\right )^5} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-40*x + 60*x^5 + ((-10 + 15*x^4)^5*(-100*x - 1350*x^5))/E)/(-8 + 12*x^4 + ((-10 + 15*x^4)^5*(-40 +
60*x^4))/E + ((-10 + 15*x^4)^10*(-50 + 75*x^4))/E^2),x]

[Out]

(10*E*x^2)/(4*E + 31250*(-2 + 3*x^4)^5)

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fricas [A]  time = 0.89, size = 40, normalized size = 1.67 \begin {gather*} \frac {5 \, x^{2} e}{3796875 \, x^{20} - 12656250 \, x^{16} + 16875000 \, x^{12} - 11250000 \, x^{8} + 3750000 \, x^{4} + 2 \, e - 500000} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-1350*x^5-100*x)*exp(5*log(15*x^4-10)-1)+60*x^5-40*x)/((75*x^4-50)*exp(5*log(15*x^4-10)-1)^2+(60*x
^4-40)*exp(5*log(15*x^4-10)-1)+12*x^4-8),x, algorithm="fricas")

[Out]

5*x^2*e/(3796875*x^20 - 12656250*x^16 + 16875000*x^12 - 11250000*x^8 + 3750000*x^4 + 2*e - 500000)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \mathit {undef} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-1350*x^5-100*x)*exp(5*log(15*x^4-10)-1)+60*x^5-40*x)/((75*x^4-50)*exp(5*log(15*x^4-10)-1)^2+(60*x
^4-40)*exp(5*log(15*x^4-10)-1)+12*x^4-8),x, algorithm="giac")

[Out]

undef

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maple [B]  time = 1.74, size = 48, normalized size = 2.00

method	result	size
risch	\(\frac {x^{2}}{759375 \,{\mathrm e}^{-1} x^{20}-2531250 \,{\mathrm e}^{-1} x^{16}+3375000 \,{\mathrm e}^{-1} x^{12}-2250000 \,{\mathrm e}^{-1} x^{8}+750000 \,{\mathrm e}^{-1} x^{4}-100000 \,{\mathrm e}^{-1}+\frac {2}{5}}\)	\(48\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-1350*x^5-100*x)*exp(5*ln(15*x^4-10)-1)+60*x^5-40*x)/((75*x^4-50)*exp(5*ln(15*x^4-10)-1)^2+(60*x^4-40)*e
xp(5*ln(15*x^4-10)-1)+12*x^4-8),x,method=_RETURNVERBOSE)

[Out]

1/759375*x^2/(exp(-1)*x^20-10/3*exp(-1)*x^16+40/9*exp(-1)*x^12-80/27*exp(-1)*x^8+80/81*exp(-1)*x^4-32/243*exp(
-1)+2/3796875)

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maxima [A]  time = 0.46, size = 40, normalized size = 1.67 \begin {gather*} \frac {5 \, x^{2} e}{3796875 \, x^{20} - 12656250 \, x^{16} + 16875000 \, x^{12} - 11250000 \, x^{8} + 3750000 \, x^{4} + 2 \, e - 500000} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-1350*x^5-100*x)*exp(5*log(15*x^4-10)-1)+60*x^5-40*x)/((75*x^4-50)*exp(5*log(15*x^4-10)-1)^2+(60*x
^4-40)*exp(5*log(15*x^4-10)-1)+12*x^4-8),x, algorithm="maxima")

[Out]

5*x^2*e/(3796875*x^20 - 12656250*x^16 + 16875000*x^12 - 11250000*x^8 + 3750000*x^4 + 2*e - 500000)

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mupad [B]  time = 6.39, size = 40, normalized size = 1.67 \begin {gather*} \frac {5\,x^2\,\mathrm {e}}{3796875\,x^{20}-12656250\,x^{16}[phone]\,x^{12}-11250000\,x^8+3750000\,x^4+2\,\mathrm {e}-500000} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(40*x - 60*x^5 + exp(5*log(15*x^4 - 10) - 1)*(100*x + 1350*x^5))/(exp(5*log(15*x^4 - 10) - 1)*(60*x^4 - 4
0) + exp(10*log(15*x^4 - 10) - 2)*(75*x^4 - 50) + 12*x^4 - 8),x)

[Out]

(5*x^2*exp(1))/(2*exp(1) + 3750000*x^4 - 11250000*x^8 + 16875000*x^12 - 12656250*x^16 + 3796875*x^20 - 500000)

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sympy [B]  time = 6.40, size = 39, normalized size = 1.62 \begin {gather*} \frac {5 e x^{2}}{3796875 x^{20} - 12656250 x^{16} + 16875000 x^{12} - 11250000 x^{8} + 3750000 x^{4} - 500000 + 2 e} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-1350*x**5-100*x)*exp(5*ln(15*x**4-10)-1)+60*x**5-40*x)/((75*x**4-50)*exp(5*ln(15*x**4-10)-1)**2+(
60*x**4-40)*exp(5*ln(15*x**4-10)-1)+12*x**4-8),x)

[Out]

5*E*x**2/(3796875*x**20 - 12656250*x**16 + 16875000*x**12 - 11250000*x**8 + 3750000*x**4 - 500000 + 2*E)

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