3.20.82 \(\int \frac {(1-2 x)^{5/2} (2+3 x)^2}{(3+5 x)^2} \, dx\) [1982]

Optimal. Leaf size=102 \[ \frac {1342 \sqrt {1-2 x}}{3125}+\frac {122 (1-2 x)^{3/2}}{1875}+\frac {122 (1-2 x)^{5/2}}{6875}-\frac {9}{175} (1-2 x)^{7/2}-\frac {(1-2 x)^{7/2}}{275 (3+5 x)}-\frac {1342 \sqrt {\frac {11}{5}} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{3125} \]

[Out]

122/1875*(1-2*x)^(3/2)+122/6875*(1-2*x)^(5/2)-9/175*(1-2*x)^(7/2)-1/275*(1-2*x)^(7/2)/(3+5*x)-1342/15625*arcta
nh(1/11*55^(1/2)*(1-2*x)^(1/2))*55^(1/2)+1342/3125*(1-2*x)^(1/2)

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Rubi [A]
time = 0.02, antiderivative size = 102, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {91, 81, 52, 65, 212} \begin {gather*} -\frac {(1-2 x)^{7/2}}{275 (5 x+3)}-\frac {9}{175} (1-2 x)^{7/2}+\frac {122 (1-2 x)^{5/2}}{6875}+\frac {122 (1-2 x)^{3/2}}{1875}+\frac {1342 \sqrt {1-2 x}}{3125}-\frac {1342 \sqrt {\frac {11}{5}} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{3125} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((1 - 2*x)^(5/2)*(2 + 3*x)^2)/(3 + 5*x)^2,x]

[Out]

(1342*Sqrt[1 - 2*x])/3125 + (122*(1 - 2*x)^(3/2))/1875 + (122*(1 - 2*x)^(5/2))/6875 - (9*(1 - 2*x)^(7/2))/175
- (1 - 2*x)^(7/2)/(275*(3 + 5*x)) - (1342*Sqrt[11/5]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/3125

Rule 52

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(a + b*x)^(m + 1)*((c + d*x)^n/(b*(
m + n + 1))), x] + Dist[n*((b*c - a*d)/(b*(m + n + 1))), Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a
, b, c, d}, x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !IntegerQ[n] || (G
tQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 0] && IntLinearQ[a, b, c, d, m, n, x]

Rule 65

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 81

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[b*(c + d*x)^
(n + 1)*((e + f*x)^(p + 1)/(d*f*(n + p + 2))), x] + Dist[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)))/(
d*f*(n + p + 2)), Int[(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && NeQ[n + p + 2,
0]

Rule 91

Int[((a_.) + (b_.)*(x_))^2*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(b*c - a*d
)^2*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(d^2*(d*e - c*f)*(n + 1))), x] - Dist[1/(d^2*(d*e - c*f)*(n + 1)), In
t[(c + d*x)^(n + 1)*(e + f*x)^p*Simp[a^2*d^2*f*(n + p + 2) + b^2*c*(d*e*(n + 1) + c*f*(p + 1)) - 2*a*b*d*(d*e*
(n + 1) + c*f*(p + 1)) - b^2*d*(d*e - c*f)*(n + 1)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && (LtQ
[n, -1] || (EqQ[n + p + 3, 0] && NeQ[n, -1] && (SumSimplerQ[n, 1] ||  !SumSimplerQ[p, 1])))

Rule 212

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rubi steps

\begin {align*} \int \frac {(1-2 x)^{5/2} (2+3 x)^2}{(3+5 x)^2} \, dx &=-\frac {(1-2 x)^{7/2}}{275 (3+5 x)}+\frac {1}{275} \int \frac {(1-2 x)^{5/2} (358+495 x)}{3+5 x} \, dx\\ &=-\frac {9}{175} (1-2 x)^{7/2}-\frac {(1-2 x)^{7/2}}{275 (3+5 x)}+\frac {61}{275} \int \frac {(1-2 x)^{5/2}}{3+5 x} \, dx\\ &=\frac {122 (1-2 x)^{5/2}}{6875}-\frac {9}{175} (1-2 x)^{7/2}-\frac {(1-2 x)^{7/2}}{275 (3+5 x)}+\frac {61}{125} \int \frac {(1-2 x)^{3/2}}{3+5 x} \, dx\\ &=\frac {122 (1-2 x)^{3/2}}{1875}+\frac {122 (1-2 x)^{5/2}}{6875}-\frac {9}{175} (1-2 x)^{7/2}-\frac {(1-2 x)^{7/2}}{275 (3+5 x)}+\frac {671}{625} \int \frac {\sqrt {1-2 x}}{3+5 x} \, dx\\ &=\frac {1342 \sqrt {1-2 x}}{3125}+\frac {122 (1-2 x)^{3/2}}{1875}+\frac {122 (1-2 x)^{5/2}}{6875}-\frac {9}{175} (1-2 x)^{7/2}-\frac {(1-2 x)^{7/2}}{275 (3+5 x)}+\frac {7381 \int \frac {1}{\sqrt {1-2 x} (3+5 x)} \, dx}{3125}\\ &=\frac {1342 \sqrt {1-2 x}}{3125}+\frac {122 (1-2 x)^{3/2}}{1875}+\frac {122 (1-2 x)^{5/2}}{6875}-\frac {9}{175} (1-2 x)^{7/2}-\frac {(1-2 x)^{7/2}}{275 (3+5 x)}-\frac {7381 \text {Subst}\left (\int \frac {1}{\frac {11}{2}-\frac {5 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )}{3125}\\ &=\frac {1342 \sqrt {1-2 x}}{3125}+\frac {122 (1-2 x)^{3/2}}{1875}+\frac {122 (1-2 x)^{5/2}}{6875}-\frac {9}{175} (1-2 x)^{7/2}-\frac {(1-2 x)^{7/2}}{275 (3+5 x)}-\frac {1342 \sqrt {\frac {11}{5}} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{3125}\\ \end {align*}

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Mathematica [A]
time = 0.12, size = 68, normalized size = 0.67 \begin {gather*} \frac {\frac {5 \sqrt {1-2 x} \left (90486+173795 x-75130 x^2-96300 x^3+135000 x^4\right )}{3+5 x}-28182 \sqrt {55} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{328125} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((1 - 2*x)^(5/2)*(2 + 3*x)^2)/(3 + 5*x)^2,x]

[Out]

((5*Sqrt[1 - 2*x]*(90486 + 173795*x - 75130*x^2 - 96300*x^3 + 135000*x^4))/(3 + 5*x) - 28182*Sqrt[55]*ArcTanh[
Sqrt[5/11]*Sqrt[1 - 2*x]])/328125

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Maple [A]
time = 0.10, size = 72, normalized size = 0.71

method	result	size
risch	\(-\frac {270000 x^{5}-327600 x^{4}-53960 x^{3}+422720 x^{2}+7177 x -90486}{65625 \left (3+5 x \right ) \sqrt {1-2 x}}-\frac {1342 \arctanh \left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{15625}\)	\(61\)
derivativedivides	\(-\frac {9 \left (1-2 x \right )^{\frac {7}{2}}}{175}+\frac {12 \left (1-2 x \right )^{\frac {5}{2}}}{625}+\frac {128 \left (1-2 x \right )^{\frac {3}{2}}}{1875}+\frac {1364 \sqrt {1-2 x}}{3125}+\frac {242 \sqrt {1-2 x}}{15625 \left (-\frac {6}{5}-2 x \right )}-\frac {1342 \arctanh \left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{15625}\)	\(72\)
default	\(-\frac {9 \left (1-2 x \right )^{\frac {7}{2}}}{175}+\frac {12 \left (1-2 x \right )^{\frac {5}{2}}}{625}+\frac {128 \left (1-2 x \right )^{\frac {3}{2}}}{1875}+\frac {1364 \sqrt {1-2 x}}{3125}+\frac {242 \sqrt {1-2 x}}{15625 \left (-\frac {6}{5}-2 x \right )}-\frac {1342 \arctanh \left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{15625}\)	\(72\)
trager	\(\frac {\left (135000 x^{4}-96300 x^{3}-75130 x^{2}+173795 x +90486\right ) \sqrt {1-2 x}}{196875+328125 x}+\frac {671 \RootOf \left (\textit {\_Z}^{2}-55\right ) \ln \left (\frac {5 \RootOf \left (\textit {\_Z}^{2}-55\right ) x +55 \sqrt {1-2 x}-8 \RootOf \left (\textit {\_Z}^{2}-55\right )}{3+5 x}\right )}{15625}\)	\(82\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1-2*x)^(5/2)*(2+3*x)^2/(3+5*x)^2,x,method=_RETURNVERBOSE)

[Out]

-9/175*(1-2*x)^(7/2)+12/625*(1-2*x)^(5/2)+128/1875*(1-2*x)^(3/2)+1364/3125*(1-2*x)^(1/2)+242/15625*(1-2*x)^(1/
2)/(-6/5-2*x)-1342/15625*arctanh(1/11*55^(1/2)*(1-2*x)^(1/2))*55^(1/2)

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Maxima [A]
time = 0.49, size = 89, normalized size = 0.87 \begin {gather*} -\frac {9}{175} \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} + \frac {12}{625} \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} + \frac {128}{1875} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + \frac {671}{15625} \, \sqrt {55} \log \left (-\frac {\sqrt {55} - 5 \, \sqrt {-2 \, x + 1}}{\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}}\right ) + \frac {1364}{3125} \, \sqrt {-2 \, x + 1} - \frac {121 \, \sqrt {-2 \, x + 1}}{3125 \, {\left (5 \, x + 3\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-2*x)^(5/2)*(2+3*x)^2/(3+5*x)^2,x, algorithm="maxima")

[Out]

-9/175*(-2*x + 1)^(7/2) + 12/625*(-2*x + 1)^(5/2) + 128/1875*(-2*x + 1)^(3/2) + 671/15625*sqrt(55)*log(-(sqrt(
55) - 5*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) + 1364/3125*sqrt(-2*x + 1) - 121/3125*sqrt(-2*x + 1)/(5
*x + 3)

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Fricas [A]
time = 1.27, size = 80, normalized size = 0.78 \begin {gather*} \frac {14091 \, \sqrt {11} \sqrt {5} {\left (5 \, x + 3\right )} \log \left (\frac {\sqrt {11} \sqrt {5} \sqrt {-2 \, x + 1} + 5 \, x - 8}{5 \, x + 3}\right ) + 5 \, {\left (135000 \, x^{4} - 96300 \, x^{3} - 75130 \, x^{2} + 173795 \, x + 90486\right )} \sqrt {-2 \, x + 1}}{328125 \, {\left (5 \, x + 3\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-2*x)^(5/2)*(2+3*x)^2/(3+5*x)^2,x, algorithm="fricas")

[Out]

1/328125*(14091*sqrt(11)*sqrt(5)*(5*x + 3)*log((sqrt(11)*sqrt(5)*sqrt(-2*x + 1) + 5*x - 8)/(5*x + 3)) + 5*(135
000*x^4 - 96300*x^3 - 75130*x^2 + 173795*x + 90486)*sqrt(-2*x + 1))/(5*x + 3)

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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.

[In]

integrate((1-2*x)**(5/2)*(2+3*x)**2/(3+5*x)**2,x)

[Out]

Timed out

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Giac [A]
time = 0.72, size = 106, normalized size = 1.04 \begin {gather*} \frac {9}{175} \, {\left (2 \, x - 1\right )}^{3} \sqrt {-2 \, x + 1} + \frac {12}{625} \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} + \frac {128}{1875} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + \frac {671}{15625} \, \sqrt {55} \log \left (\frac {{\left | -2 \, \sqrt {55} + 10 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}\right )}}\right ) + \frac {1364}{3125} \, \sqrt {-2 \, x + 1} - \frac {121 \, \sqrt {-2 \, x + 1}}{3125 \, {\left (5 \, x + 3\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-2*x)^(5/2)*(2+3*x)^2/(3+5*x)^2,x, algorithm="giac")

[Out]

9/175*(2*x - 1)^3*sqrt(-2*x + 1) + 12/625*(2*x - 1)^2*sqrt(-2*x + 1) + 128/1875*(-2*x + 1)^(3/2) + 671/15625*s
qrt(55)*log(1/2*abs(-2*sqrt(55) + 10*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) + 1364/3125*sqrt(-2*x + 1)
- 121/3125*sqrt(-2*x + 1)/(5*x + 3)

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Mupad [B]
time = 1.18, size = 73, normalized size = 0.72 \begin {gather*} \frac {1364\,\sqrt {1-2\,x}}{3125}-\frac {242\,\sqrt {1-2\,x}}{15625\,\left (2\,x+\frac {6}{5}\right )}+\frac {128\,{\left (1-2\,x\right )}^{3/2}}{1875}+\frac {12\,{\left (1-2\,x\right )}^{5/2}}{625}-\frac {9\,{\left (1-2\,x\right )}^{7/2}}{175}+\frac {\sqrt {55}\,\mathrm {atan}\left (\frac {\sqrt {55}\,\sqrt {1-2\,x}\,1{}\mathrm {i}}{11}\right )\,1342{}\mathrm {i}}{15625} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((1 - 2*x)^(5/2)*(3*x + 2)^2)/(5*x + 3)^2,x)

[Out]

(55^(1/2)*atan((55^(1/2)*(1 - 2*x)^(1/2)*1i)/11)*1342i)/15625 - (242*(1 - 2*x)^(1/2))/(15625*(2*x + 6/5)) + (1
364*(1 - 2*x)^(1/2))/3125 + (128*(1 - 2*x)^(3/2))/1875 + (12*(1 - 2*x)^(5/2))/625 - (9*(1 - 2*x)^(7/2))/175

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