3.22.6 \(\int \frac {(2+3 x)^4}{(1-2 x)^{3/2} (3+5 x)} \, dx\) [2106]

Optimal. Leaf size=80 \[ \frac {2401}{88 \sqrt {1-2 x}}+\frac {34371 \sqrt {1-2 x}}{1000}-\frac {963}{200} (1-2 x)^{3/2}+\frac {81}{200} (1-2 x)^{5/2}-\frac {2 \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{1375 \sqrt {55}} \]

[Out]

-963/200*(1-2*x)^(3/2)+81/200*(1-2*x)^(5/2)-2/75625*arctanh(1/11*55^(1/2)*(1-2*x)^(1/2))*55^(1/2)+2401/88/(1-2
*x)^(1/2)+34371/1000*(1-2*x)^(1/2)

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Rubi [A]
time = 0.03, antiderivative size = 80, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 4, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {89, 45, 65, 212} \begin {gather*} \frac {81}{200} (1-2 x)^{5/2}-\frac {963}{200} (1-2 x)^{3/2}+\frac {34371 \sqrt {1-2 x}}{1000}+\frac {2401}{88 \sqrt {1-2 x}}-\frac {2 \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{1375 \sqrt {55}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(2 + 3*x)^4/((1 - 2*x)^(3/2)*(3 + 5*x)),x]

[Out]

2401/(88*Sqrt[1 - 2*x]) + (34371*Sqrt[1 - 2*x])/1000 - (963*(1 - 2*x)^(3/2))/200 + (81*(1 - 2*x)^(5/2))/200 -
(2*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/(1375*Sqrt[55])

Rule 45

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

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 89

Int[(((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_))/((a_.) + (b_.)*(x_)), x_Symbol] :> Int[ExpandIntegr
and[(e + f*x)^FractionalPart[p], (c + d*x)^n*((e + f*x)^IntegerPart[p]/(a + b*x)), x], x] /; FreeQ[{a, b, c, d
, e, f}, x] && IGtQ[n, 0] && LtQ[p, -1] && FractionQ[p]

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 {(2+3 x)^4}{(1-2 x)^{3/2} (3+5 x)} \, dx &=\int \left (\frac {2401}{88 (1-2 x)^{3/2}}-\frac {21951}{1000 \sqrt {1-2 x}}-\frac {2079 x}{100 \sqrt {1-2 x}}-\frac {81 x^2}{10 \sqrt {1-2 x}}+\frac {1}{1375 \sqrt {1-2 x} (3+5 x)}\right ) \, dx\\ &=\frac {2401}{88 \sqrt {1-2 x}}+\frac {21951 \sqrt {1-2 x}}{1000}+\frac {\int \frac {1}{\sqrt {1-2 x} (3+5 x)} \, dx}{1375}-\frac {81}{10} \int \frac {x^2}{\sqrt {1-2 x}} \, dx-\frac {2079}{100} \int \frac {x}{\sqrt {1-2 x}} \, dx\\ &=\frac {2401}{88 \sqrt {1-2 x}}+\frac {21951 \sqrt {1-2 x}}{1000}-\frac {\text {Subst}\left (\int \frac {1}{\frac {11}{2}-\frac {5 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )}{1375}-\frac {81}{10} \int \left (\frac {1}{4 \sqrt {1-2 x}}-\frac {1}{2} \sqrt {1-2 x}+\frac {1}{4} (1-2 x)^{3/2}\right ) \, dx-\frac {2079}{100} \int \left (\frac {1}{2 \sqrt {1-2 x}}-\frac {1}{2} \sqrt {1-2 x}\right ) \, dx\\ &=\frac {2401}{88 \sqrt {1-2 x}}+\frac {34371 \sqrt {1-2 x}}{1000}-\frac {963}{200} (1-2 x)^{3/2}+\frac {81}{200} (1-2 x)^{5/2}-\frac {2 \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{1375 \sqrt {55}}\\ \end {align*}

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Mathematica [A]
time = 0.07, size = 56, normalized size = 0.70 \begin {gather*} \frac {-\frac {55 \left (-78712+71379 x+19800 x^2+4455 x^3\right )}{\sqrt {1-2 x}}-2 \sqrt {55} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{75625} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(2 + 3*x)^4/((1 - 2*x)^(3/2)*(3 + 5*x)),x]

[Out]

((-55*(-78712 + 71379*x + 19800*x^2 + 4455*x^3))/Sqrt[1 - 2*x] - 2*Sqrt[55]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])
/75625

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Maple [A]
time = 0.11, size = 56, normalized size = 0.70

method	result	size
risch	\(-\frac {4455 x^{3}+19800 x^{2}+71379 x -78712}{1375 \sqrt {1-2 x}}-\frac {2 \arctanh \left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{75625}\)	\(44\)
derivativedivides	\(-\frac {963 \left (1-2 x \right )^{\frac {3}{2}}}{200}+\frac {81 \left (1-2 x \right )^{\frac {5}{2}}}{200}-\frac {2 \arctanh \left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{75625}+\frac {2401}{88 \sqrt {1-2 x}}+\frac {34371 \sqrt {1-2 x}}{1000}\)	\(56\)
default	\(-\frac {963 \left (1-2 x \right )^{\frac {3}{2}}}{200}+\frac {81 \left (1-2 x \right )^{\frac {5}{2}}}{200}-\frac {2 \arctanh \left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{75625}+\frac {2401}{88 \sqrt {1-2 x}}+\frac {34371 \sqrt {1-2 x}}{1000}\)	\(56\)
trager	\(\frac {\left (4455 x^{3}+19800 x^{2}+71379 x -78712\right ) \sqrt {1-2 x}}{-1375+2750 x}+\frac {\RootOf \left (\textit {\_Z}^{2}-55\right ) \ln \left (\frac {5 \RootOf \left (\textit {\_Z}^{2}-55\right ) x +55 \sqrt {1-2 x}-8 \RootOf \left (\textit {\_Z}^{2}-55\right )}{3+5 x}\right )}{75625}\)	\(77\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2+3*x)^4/(1-2*x)^(3/2)/(3+5*x),x,method=_RETURNVERBOSE)

[Out]

-963/200*(1-2*x)^(3/2)+81/200*(1-2*x)^(5/2)-2/75625*arctanh(1/11*55^(1/2)*(1-2*x)^(1/2))*55^(1/2)+2401/88/(1-2
*x)^(1/2)+34371/1000*(1-2*x)^(1/2)

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Maxima [A]
time = 0.49, size = 73, normalized size = 0.91 \begin {gather*} \frac {81}{200} \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} - \frac {963}{200} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + \frac {1}{75625} \, \sqrt {55} \log \left (-\frac {\sqrt {55} - 5 \, \sqrt {-2 \, x + 1}}{\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}}\right ) + \frac {34371}{1000} \, \sqrt {-2 \, x + 1} + \frac {2401}{88 \, \sqrt {-2 \, x + 1}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2+3*x)^4/(1-2*x)^(3/2)/(3+5*x),x, algorithm="maxima")

[Out]

81/200*(-2*x + 1)^(5/2) - 963/200*(-2*x + 1)^(3/2) + 1/75625*sqrt(55)*log(-(sqrt(55) - 5*sqrt(-2*x + 1))/(sqrt
(55) + 5*sqrt(-2*x + 1))) + 34371/1000*sqrt(-2*x + 1) + 2401/88/sqrt(-2*x + 1)

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Fricas [A]
time = 0.70, size = 68, normalized size = 0.85 \begin {gather*} \frac {\sqrt {55} {\left (2 \, x - 1\right )} \log \left (\frac {5 \, x + \sqrt {55} \sqrt {-2 \, x + 1} - 8}{5 \, x + 3}\right ) + 55 \, {\left (4455 \, x^{3} + 19800 \, x^{2} + 71379 \, x - 78712\right )} \sqrt {-2 \, x + 1}}{75625 \, {\left (2 \, x - 1\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2+3*x)^4/(1-2*x)^(3/2)/(3+5*x),x, algorithm="fricas")

[Out]

1/75625*(sqrt(55)*(2*x - 1)*log((5*x + sqrt(55)*sqrt(-2*x + 1) - 8)/(5*x + 3)) + 55*(4455*x^3 + 19800*x^2 + 71
379*x - 78712)*sqrt(-2*x + 1))/(2*x - 1)

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Sympy [A]
time = 40.83, size = 107, normalized size = 1.34 \begin {gather*} \frac {81 \left (1 - 2 x\right )^{\frac {5}{2}}}{200} - \frac {963 \left (1 - 2 x\right )^{\frac {3}{2}}}{200} + \frac {34371 \sqrt {1 - 2 x}}{1000} + \frac {2 \left (\begin {cases} - \frac {\sqrt {55} \operatorname {acoth}{\left (\frac {\sqrt {55} \sqrt {1 - 2 x}}{11} \right )}}{55} & \text {for}\: x < - \frac {3}{5} \\- \frac {\sqrt {55} \operatorname {atanh}{\left (\frac {\sqrt {55} \sqrt {1 - 2 x}}{11} \right )}}{55} & \text {for}\: x > - \frac {3}{5} \end {cases}\right )}{1375} + \frac {2401}{88 \sqrt {1 - 2 x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2+3*x)**4/(1-2*x)**(3/2)/(3+5*x),x)

[Out]

81*(1 - 2*x)**(5/2)/200 - 963*(1 - 2*x)**(3/2)/200 + 34371*sqrt(1 - 2*x)/1000 + 2*Piecewise((-sqrt(55)*acoth(s
qrt(55)*sqrt(1 - 2*x)/11)/55, x < -3/5), (-sqrt(55)*atanh(sqrt(55)*sqrt(1 - 2*x)/11)/55, x > -3/5))/1375 + 240
1/(88*sqrt(1 - 2*x))

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Giac [A]
time = 1.60, size = 83, normalized size = 1.04 \begin {gather*} \frac {81}{200} \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} - \frac {963}{200} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + \frac {1}{75625} \, \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 {34371}{1000} \, \sqrt {-2 \, x + 1} + \frac {2401}{88 \, \sqrt {-2 \, x + 1}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2+3*x)^4/(1-2*x)^(3/2)/(3+5*x),x, algorithm="giac")

[Out]

81/200*(2*x - 1)^2*sqrt(-2*x + 1) - 963/200*(-2*x + 1)^(3/2) + 1/75625*sqrt(55)*log(1/2*abs(-2*sqrt(55) + 10*s
qrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) + 34371/1000*sqrt(-2*x + 1) + 2401/88/sqrt(-2*x + 1)

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Mupad [B]
time = 0.05, size = 57, normalized size = 0.71 \begin {gather*} \frac {2401}{88\,\sqrt {1-2\,x}}+\frac {34371\,\sqrt {1-2\,x}}{1000}-\frac {963\,{\left (1-2\,x\right )}^{3/2}}{200}+\frac {81\,{\left (1-2\,x\right )}^{5/2}}{200}+\frac {\sqrt {55}\,\mathrm {atan}\left (\frac {\sqrt {55}\,\sqrt {1-2\,x}\,1{}\mathrm {i}}{11}\right )\,2{}\mathrm {i}}{75625} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((3*x + 2)^4/((1 - 2*x)^(3/2)*(5*x + 3)),x)

[Out]

(55^(1/2)*atan((55^(1/2)*(1 - 2*x)^(1/2)*1i)/11)*2i)/75625 + 2401/(88*(1 - 2*x)^(1/2)) + (34371*(1 - 2*x)^(1/2
))/1000 - (963*(1 - 2*x)^(3/2))/200 + (81*(1 - 2*x)^(5/2))/200

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