3.2167 \(\int \frac{(3+5 x)^3}{(1-2 x)^{5/2} (2+3 x)^5} \, dx\)

Optimal. Leaf size=140 \[ \frac{11 (5 x+3)^2}{21 (1-2 x)^{3/2} (3 x+2)^4}+\frac{85754 x+57069}{4116 \sqrt{1-2 x} (3 x+2)^4}-\frac{177635 \sqrt{1-2 x}}{403368 (3 x+2)}-\frac{177635 \sqrt{1-2 x}}{172872 (3 x+2)^2}-\frac{35527 \sqrt{1-2 x}}{12348 (3 x+2)^3}-\frac{177635 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{201684 \sqrt{21}} \]

[Out]

(-35527*Sqrt[1 - 2*x])/(12348*(2 + 3*x)^3) - (177635*Sqrt[1 - 2*x])/(172872*(2 + 3*x)^2) - (177635*Sqrt[1 - 2*
x])/(403368*(2 + 3*x)) + (11*(3 + 5*x)^2)/(21*(1 - 2*x)^(3/2)*(2 + 3*x)^4) + (57069 + 85754*x)/(4116*Sqrt[1 -
2*x]*(2 + 3*x)^4) - (177635*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/(201684*Sqrt[21])

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Rubi [A]  time = 0.0408503, antiderivative size = 140, normalized size of antiderivative = 1., 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 = {98, 144, 51, 63, 206} \[ \frac{11 (5 x+3)^2}{21 (1-2 x)^{3/2} (3 x+2)^4}+\frac{85754 x+57069}{4116 \sqrt{1-2 x} (3 x+2)^4}-\frac{177635 \sqrt{1-2 x}}{403368 (3 x+2)}-\frac{177635 \sqrt{1-2 x}}{172872 (3 x+2)^2}-\frac{35527 \sqrt{1-2 x}}{12348 (3 x+2)^3}-\frac{177635 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{201684 \sqrt{21}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-35527*Sqrt[1 - 2*x])/(12348*(2 + 3*x)^3) - (177635*Sqrt[1 - 2*x])/(172872*(2 + 3*x)^2) - (177635*Sqrt[1 - 2*
x])/(403368*(2 + 3*x)) + (11*(3 + 5*x)^2)/(21*(1 - 2*x)^(3/2)*(2 + 3*x)^4) + (57069 + 85754*x)/(4116*Sqrt[1 -
2*x]*(2 + 3*x)^4) - (177635*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/(201684*Sqrt[21])

Rule 98

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

Rule 144

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_) + (f_.)*(x_))*((g_.) + (h_.)*(x_)), x_Symbol] :>
Simp[((b^2*c*d*e*g*(n + 1) + a^2*c*d*f*h*(n + 1) + a*b*(d^2*e*g*(m + 1) + c^2*f*h*(m + 1) - c*d*(f*g + e*h)*(
m + n + 2)) + (a^2*d^2*f*h*(n + 1) - a*b*d^2*(f*g + e*h)*(n + 1) + b^2*(c^2*f*h*(m + 1) - c*d*(f*g + e*h)*(m +
1) + d^2*e*g*(m + n + 2)))*x)*(a + b*x)^(m + 1)*(c + d*x)^(n + 1))/(b*d*(b*c - a*d)^2*(m + 1)*(n + 1)), x] -
Dist[(a^2*d^2*f*h*(2 + 3*n + n^2) + a*b*d*(n + 1)*(2*c*f*h*(m + 1) - d*(f*g + e*h)*(m + n + 3)) + b^2*(c^2*f*h
*(2 + 3*m + m^2) - c*d*(f*g + e*h)*(m + 1)*(m + n + 3) + d^2*e*g*(6 + m^2 + 5*n + n^2 + m*(2*n + 5))))/(b*d*(b
*c - a*d)^2*(m + 1)*(n + 1)), Int[(a + b*x)^(m + 1)*(c + d*x)^(n + 1), x], x] /; FreeQ[{a, b, c, d, e, f, g, h
}, x] && LtQ[m, -1] && LtQ[n, -1]

Rule 51

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

Rule 63

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 206

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

Rubi steps

\begin{align*} \int \frac{(3+5 x)^3}{(1-2 x)^{5/2} (2+3 x)^5} \, dx &=\frac{11 (3+5 x)^2}{21 (1-2 x)^{3/2} (2+3 x)^4}-\frac{1}{21} \int \frac{(-167-315 x) (3+5 x)}{(1-2 x)^{3/2} (2+3 x)^5} \, dx\\ &=\frac{11 (3+5 x)^2}{21 (1-2 x)^{3/2} (2+3 x)^4}+\frac{57069+85754 x}{4116 \sqrt{1-2 x} (2+3 x)^4}+\frac{35527}{588} \int \frac{1}{\sqrt{1-2 x} (2+3 x)^4} \, dx\\ &=-\frac{35527 \sqrt{1-2 x}}{12348 (2+3 x)^3}+\frac{11 (3+5 x)^2}{21 (1-2 x)^{3/2} (2+3 x)^4}+\frac{57069+85754 x}{4116 \sqrt{1-2 x} (2+3 x)^4}+\frac{177635 \int \frac{1}{\sqrt{1-2 x} (2+3 x)^3} \, dx}{12348}\\ &=-\frac{35527 \sqrt{1-2 x}}{12348 (2+3 x)^3}-\frac{177635 \sqrt{1-2 x}}{172872 (2+3 x)^2}+\frac{11 (3+5 x)^2}{21 (1-2 x)^{3/2} (2+3 x)^4}+\frac{57069+85754 x}{4116 \sqrt{1-2 x} (2+3 x)^4}+\frac{177635 \int \frac{1}{\sqrt{1-2 x} (2+3 x)^2} \, dx}{57624}\\ &=-\frac{35527 \sqrt{1-2 x}}{12348 (2+3 x)^3}-\frac{177635 \sqrt{1-2 x}}{172872 (2+3 x)^2}-\frac{177635 \sqrt{1-2 x}}{403368 (2+3 x)}+\frac{11 (3+5 x)^2}{21 (1-2 x)^{3/2} (2+3 x)^4}+\frac{57069+85754 x}{4116 \sqrt{1-2 x} (2+3 x)^4}+\frac{177635 \int \frac{1}{\sqrt{1-2 x} (2+3 x)} \, dx}{403368}\\ &=-\frac{35527 \sqrt{1-2 x}}{12348 (2+3 x)^3}-\frac{177635 \sqrt{1-2 x}}{172872 (2+3 x)^2}-\frac{177635 \sqrt{1-2 x}}{403368 (2+3 x)}+\frac{11 (3+5 x)^2}{21 (1-2 x)^{3/2} (2+3 x)^4}+\frac{57069+85754 x}{4116 \sqrt{1-2 x} (2+3 x)^4}-\frac{177635 \operatorname{Subst}\left (\int \frac{1}{\frac{7}{2}-\frac{3 x^2}{2}} \, dx,x,\sqrt{1-2 x}\right )}{403368}\\ &=-\frac{35527 \sqrt{1-2 x}}{12348 (2+3 x)^3}-\frac{177635 \sqrt{1-2 x}}{172872 (2+3 x)^2}-\frac{177635 \sqrt{1-2 x}}{403368 (2+3 x)}+\frac{11 (3+5 x)^2}{21 (1-2 x)^{3/2} (2+3 x)^4}+\frac{57069+85754 x}{4116 \sqrt{1-2 x} (2+3 x)^4}-\frac{177635 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{201684 \sqrt{21}}\\ \end{align*}

Mathematica [C]  time = 0.0497831, size = 61, normalized size = 0.44 \[ \frac{875 x^2-3779 x+4000}{147 (1-2 x)^{3/2} (3 x+2)^4}-\frac{1136864 \sqrt{1-2 x} \, _2F_1\left (\frac{1}{2},5;\frac{3}{2};\frac{3}{7}-\frac{6 x}{7}\right )}{2470629} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(4000 - 3779*x + 875*x^2)/(147*(1 - 2*x)^(3/2)*(2 + 3*x)^4) - (1136864*Sqrt[1 - 2*x]*Hypergeometric2F1[1/2, 5,
3/2, 3/7 - (6*x)/7])/2470629

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Maple [A]  time = 0.015, size = 84, normalized size = 0.6 \begin{align*}{\frac{324}{117649\, \left ( -6\,x-4 \right ) ^{4}} \left ({\frac{198005}{144} \left ( 1-2\,x \right ) ^{{\frac{7}{2}}}}-{\frac{11953249}{1296} \left ( 1-2\,x \right ) ^{{\frac{5}{2}}}}+{\frac{80180905}{3888} \left ( 1-2\,x \right ) ^{{\frac{3}{2}}}}-{\frac{59762605}{3888}\sqrt{1-2\,x}} \right ) }-{\frac{177635\,\sqrt{21}}{4235364}{\it Artanh} \left ({\frac{\sqrt{21}}{7}\sqrt{1-2\,x}} \right ) }+{\frac{5324}{50421} \left ( 1-2\,x \right ) ^{-{\frac{3}{2}}}}+{\frac{29040}{117649}{\frac{1}{\sqrt{1-2\,x}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

324/117649*(198005/144*(1-2*x)^(7/2)-11953249/1296*(1-2*x)^(5/2)[phone]/3888*(1-2*x)^(3/2)-59762605/3888*(1-
2*x)^(1/2))/(-6*x-4)^4-177635/4235364*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)+5324/50421/(1-2*x)^(3/2)+29
040/117649/(1-2*x)^(1/2)

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Maxima [A]  time = 1.59577, size = 173, normalized size = 1.24 \begin{align*} \frac{177635}{8470728} \, \sqrt{21} \log \left (-\frac{\sqrt{21} - 3 \, \sqrt{-2 \, x + 1}}{\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}}\right ) - \frac{4796145 \,{\left (2 \, x - 1\right )}^{5} + 41033685 \,{\left (2 \, x - 1\right )}^{4} + 127080079 \,{\left (2 \, x - 1\right )}^{3} + 157094539 \,{\left (2 \, x - 1\right )}^{2} + 63748608 \, x - 83006000}{201684 \,{\left (81 \,{\left (-2 \, x + 1\right )}^{\frac{11}{2}} - 756 \,{\left (-2 \, x + 1\right )}^{\frac{9}{2}} + 2646 \,{\left (-2 \, x + 1\right )}^{\frac{7}{2}} - 4116 \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} + 2401 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

177635/8470728*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) - 1/201684*(4796145*
(2*x - 1)^5 + 41033685*(2*x - 1)^4 + 127080079*(2*x - 1)^3 + 157094539*(2*x - 1)^2 + 63748608*x - 83006000)/(8
1*(-2*x + 1)^(11/2) - 756*(-2*x + 1)^(9/2) + 2646*(-2*x + 1)^(7/2) - 4116*(-2*x + 1)^(5/2) + 2401*(-2*x + 1)^(
3/2))

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Fricas [A]  time = 1.66742, size = 412, normalized size = 2.94 \begin{align*} \frac{177635 \, \sqrt{21}{\left (324 \, x^{6} + 540 \, x^{5} + 81 \, x^{4} - 264 \, x^{3} - 104 \, x^{2} + 32 \, x + 16\right )} \log \left (\frac{3 \, x + \sqrt{21} \sqrt{-2 \, x + 1} - 5}{3 \, x + 2}\right ) - 21 \,{\left (19184580 \, x^{5} + 34105920 \, x^{4} + 10906789 \, x^{3} - 12952519 \, x^{2} - 10307138 \, x - 2094250\right )} \sqrt{-2 \, x + 1}}{8470728 \,{\left (324 \, x^{6} + 540 \, x^{5} + 81 \, x^{4} - 264 \, x^{3} - 104 \, x^{2} + 32 \, x + 16\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8470728*(177635*sqrt(21)*(324*x^6 + 540*x^5 + 81*x^4 - 264*x^3 - 104*x^2 + 32*x + 16)*log((3*x + sqrt(21)*sq
rt(-2*x + 1) - 5)/(3*x + 2)) - 21*(19184580*x^5 + 34105920*x^4 + 10906789*x^3 - 12952519*x^2 - 10307138*x - 20
94250)*sqrt(-2*x + 1))/(324*x^6 + 540*x^5 + 81*x^4 - 264*x^3 - 104*x^2 + 32*x + 16)

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Sympy [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Giac [A]  time = 2.719, size = 163, normalized size = 1.16 \begin{align*} \frac{177635}{8470728} \, \sqrt{21} \log \left (\frac{{\left | -2 \, \sqrt{21} + 6 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}\right )}}\right ) + \frac{484 \,{\left (360 \, x - 257\right )}}{352947 \,{\left (2 \, x - 1\right )} \sqrt{-2 \, x + 1}} - \frac{5346135 \,{\left (2 \, x - 1\right )}^{3} \sqrt{-2 \, x + 1} + 35859747 \,{\left (2 \, x - 1\right )}^{2} \sqrt{-2 \, x + 1} - 80180905 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + 59762605 \, \sqrt{-2 \, x + 1}}{22588608 \,{\left (3 \, x + 2\right )}^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

177635/8470728*sqrt(21)*log(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) + 484/35294
7*(360*x - 257)/((2*x - 1)*sqrt(-2*x + 1)) - 1/22588608*(5346135*(2*x - 1)^3*sqrt(-2*x + 1) + 35859747*(2*x -
1)^2*sqrt(-2*x + 1) - 80180905*(-2*x + 1)^(3/2) + 59762605*sqrt(-2*x + 1))/(3*x + 2)^4