3.2156 \(\int \frac{(3+5 x)^2}{(1-2 x)^{5/2} (2+3 x)^5} \, dx\)

Optimal. Leaf size=141 \[ \frac{20465}{201684 \sqrt{1-2 x}}-\frac{20465}{172872 \sqrt{1-2 x} (3 x+2)}-\frac{4093}{24696 \sqrt{1-2 x} (3 x+2)^2}-\frac{4093}{12348 \sqrt{1-2 x} (3 x+2)^3}-\frac{727}{588 \sqrt{1-2 x} (3 x+2)^4}+\frac{121}{42 (1-2 x)^{3/2} (3 x+2)^4}-\frac{20465 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{67228 \sqrt{21}} \]

[Out]

20465/(201684*Sqrt[1 - 2*x]) + 121/(42*(1 - 2*x)^(3/2)*(2 + 3*x)^4) - 727/(588*Sqrt[1 - 2*x]*(2 + 3*x)^4) - 40
93/(12348*Sqrt[1 - 2*x]*(2 + 3*x)^3) - 4093/(24696*Sqrt[1 - 2*x]*(2 + 3*x)^2) - 20465/(172872*Sqrt[1 - 2*x]*(2
+ 3*x)) - (20465*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/(67228*Sqrt[21])

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Rubi [A]  time = 0.048117, antiderivative size = 148, normalized size of antiderivative = 1.05, number of steps used = 8, number of rules used = 5, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.208, Rules used = {89, 78, 51, 63, 206} \[ -\frac{20465 \sqrt{1-2 x}}{134456 (3 x+2)}-\frac{20465 \sqrt{1-2 x}}{57624 (3 x+2)^2}-\frac{4093 \sqrt{1-2 x}}{4116 (3 x+2)^3}+\frac{4093}{2058 \sqrt{1-2 x} (3 x+2)^3}-\frac{727}{588 \sqrt{1-2 x} (3 x+2)^4}+\frac{121}{42 (1-2 x)^{3/2} (3 x+2)^4}-\frac{20465 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{67228 \sqrt{21}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

121/(42*(1 - 2*x)^(3/2)*(2 + 3*x)^4) - 727/(588*Sqrt[1 - 2*x]*(2 + 3*x)^4) + 4093/(2058*Sqrt[1 - 2*x]*(2 + 3*x
)^3) - (4093*Sqrt[1 - 2*x])/(4116*(2 + 3*x)^3) - (20465*Sqrt[1 - 2*x])/(57624*(2 + 3*x)^2) - (20465*Sqrt[1 - 2
*x])/(134456*(2 + 3*x)) - (20465*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/(67228*Sqrt[21])

Rule 89

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 78

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

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)^2}{(1-2 x)^{5/2} (2+3 x)^5} \, dx &=\frac{121}{42 (1-2 x)^{3/2} (2+3 x)^4}-\frac{1}{42} \int \frac{-1104+525 x}{(1-2 x)^{3/2} (2+3 x)^5} \, dx\\ &=\frac{121}{42 (1-2 x)^{3/2} (2+3 x)^4}-\frac{727}{588 \sqrt{1-2 x} (2+3 x)^4}+\frac{4093}{588} \int \frac{1}{(1-2 x)^{3/2} (2+3 x)^4} \, dx\\ &=\frac{121}{42 (1-2 x)^{3/2} (2+3 x)^4}-\frac{727}{588 \sqrt{1-2 x} (2+3 x)^4}+\frac{4093}{2058 \sqrt{1-2 x} (2+3 x)^3}+\frac{4093}{196} \int \frac{1}{\sqrt{1-2 x} (2+3 x)^4} \, dx\\ &=\frac{121}{42 (1-2 x)^{3/2} (2+3 x)^4}-\frac{727}{588 \sqrt{1-2 x} (2+3 x)^4}+\frac{4093}{2058 \sqrt{1-2 x} (2+3 x)^3}-\frac{4093 \sqrt{1-2 x}}{4116 (2+3 x)^3}+\frac{20465 \int \frac{1}{\sqrt{1-2 x} (2+3 x)^3} \, dx}{4116}\\ &=\frac{121}{42 (1-2 x)^{3/2} (2+3 x)^4}-\frac{727}{588 \sqrt{1-2 x} (2+3 x)^4}+\frac{4093}{2058 \sqrt{1-2 x} (2+3 x)^3}-\frac{4093 \sqrt{1-2 x}}{4116 (2+3 x)^3}-\frac{20465 \sqrt{1-2 x}}{57624 (2+3 x)^2}+\frac{20465 \int \frac{1}{\sqrt{1-2 x} (2+3 x)^2} \, dx}{19208}\\ &=\frac{121}{42 (1-2 x)^{3/2} (2+3 x)^4}-\frac{727}{588 \sqrt{1-2 x} (2+3 x)^4}+\frac{4093}{2058 \sqrt{1-2 x} (2+3 x)^3}-\frac{4093 \sqrt{1-2 x}}{4116 (2+3 x)^3}-\frac{20465 \sqrt{1-2 x}}{57624 (2+3 x)^2}-\frac{20465 \sqrt{1-2 x}}{134456 (2+3 x)}+\frac{20465 \int \frac{1}{\sqrt{1-2 x} (2+3 x)} \, dx}{134456}\\ &=\frac{121}{42 (1-2 x)^{3/2} (2+3 x)^4}-\frac{727}{588 \sqrt{1-2 x} (2+3 x)^4}+\frac{4093}{2058 \sqrt{1-2 x} (2+3 x)^3}-\frac{4093 \sqrt{1-2 x}}{4116 (2+3 x)^3}-\frac{20465 \sqrt{1-2 x}}{57624 (2+3 x)^2}-\frac{20465 \sqrt{1-2 x}}{134456 (2+3 x)}-\frac{20465 \operatorname{Subst}\left (\int \frac{1}{\frac{7}{2}-\frac{3 x^2}{2}} \, dx,x,\sqrt{1-2 x}\right )}{134456}\\ &=\frac{121}{42 (1-2 x)^{3/2} (2+3 x)^4}-\frac{727}{588 \sqrt{1-2 x} (2+3 x)^4}+\frac{4093}{2058 \sqrt{1-2 x} (2+3 x)^3}-\frac{4093 \sqrt{1-2 x}}{4116 (2+3 x)^3}-\frac{20465 \sqrt{1-2 x}}{57624 (2+3 x)^2}-\frac{20465 \sqrt{1-2 x}}{134456 (2+3 x)}-\frac{20465 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{67228 \sqrt{21}}\\ \end{align*}

Mathematica [C]  time = 0.0203561, size = 60, normalized size = 0.43 \[ \frac{65488 (1-2 x) (3 x+2)^4 \, _2F_1\left (-\frac{1}{2},4;\frac{1}{2};\frac{3}{7}-\frac{6 x}{7}\right )+1745527 (2 x-1)+4067294}{1411788 (1-2 x)^{3/2} (3 x+2)^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(4067294 + 1745527*(-1 + 2*x) + 65488*(1 - 2*x)*(2 + 3*x)^4*Hypergeometric2F1[-1/2, 4, 1/2, 3/7 - (6*x)/7])/(1
411788*(1 - 2*x)^(3/2)*(2 + 3*x)^4)

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Maple [A]  time = 0.014, size = 84, normalized size = 0.6 \begin{align*}{\frac{648}{117649\, \left ( -6\,x-4 \right ) ^{4}} \left ({\frac{42935}{96} \left ( 1-2\,x \right ) ^{{\frac{7}{2}}}}-{\frac{2847691}{864} \left ( 1-2\,x \right ) ^{{\frac{5}{2}}}}+{\frac{20832595}{2592} \left ( 1-2\,x \right ) ^{{\frac{3}{2}}}}-{\frac{5609765}{864}\sqrt{1-2\,x}} \right ) }-{\frac{20465\,\sqrt{21}}{1411788}{\it Artanh} \left ({\frac{\sqrt{21}}{7}\sqrt{1-2\,x}} \right ) }+{\frac{968}{50421} \left ( 1-2\,x \right ) ^{-{\frac{3}{2}}}}+{\frac{8360}{117649}{\frac{1}{\sqrt{1-2\,x}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

648/117649*(42935/96*(1-2*x)^(7/2)-2847691/864*(1-2*x)^(5/2)[phone]/2592*(1-2*x)^(3/2)-5609765/864*(1-2*x)^(
1/2))/(-6*x-4)^4-20465/1411788*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)+968/50421/(1-2*x)^(3/2)+8360/11764
9/(1-2*x)^(1/2)

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Maxima [A]  time = 1.97979, size = 173, normalized size = 1.23 \begin{align*} \frac{20465}{2823576} \, \sqrt{21} \log \left (-\frac{\sqrt{21} - 3 \, \sqrt{-2 \, x + 1}}{\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}}\right ) - \frac{1657665 \,{\left (2 \, x - 1\right )}^{5} + 14182245 \,{\left (2 \, x - 1\right )}^{4} + 43921983 \,{\left (2 \, x - 1\right )}^{3} + 55955403 \,{\left (2 \, x - 1\right )}^{2} + 36945216 \, x - 27769280}{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)^2/(1-2*x)^(5/2)/(2+3*x)^5,x, algorithm="maxima")

[Out]

20465/2823576*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) - 1/201684*(1657665*(
2*x - 1)^5 + 14182245*(2*x - 1)^4 + 43921983*(2*x - 1)^3 + 55955403*(2*x - 1)^2 + 36945216*x - 27769280)/(81*(
-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 = 2.12767, size = 402, normalized size = 2.85 \begin{align*} \frac{20465 \, \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 ) - 7 \,{\left (6630660 \, x^{5} + 11787840 \, x^{4} + 3769653 \, x^{3} - 3646863 \, x^{2} - 2528226 \, x - 401410\right )} \sqrt{-2 \, x + 1}}{2823576 \,{\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)^2/(1-2*x)^(5/2)/(2+3*x)^5,x, algorithm="fricas")

[Out]

1/2823576*(20465*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)*sqr
t(-2*x + 1) - 5)/(3*x + 2)) - 7*(6630660*x^5 + 11787840*x^4 + 3769653*x^3 - 3646863*x^2 - 2528226*x - 401410)*
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)**2/(1-2*x)**(5/2)/(2+3*x)**5,x)

[Out]

Exception raised: ValueError

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Giac [A]  time = 2.18068, size = 163, normalized size = 1.16 \begin{align*} \frac{20465}{2823576} \, \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{176 \,{\left (285 \, x - 181\right )}}{352947 \,{\left (2 \, x - 1\right )} \sqrt{-2 \, x + 1}} - \frac{1159245 \,{\left (2 \, x - 1\right )}^{3} \sqrt{-2 \, x + 1} + 8543073 \,{\left (2 \, x - 1\right )}^{2} \sqrt{-2 \, x + 1} - 20832595 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + 16829295 \, \sqrt{-2 \, x + 1}}{7529536 \,{\left (3 \, x + 2\right )}^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

20465/2823576*sqrt(21)*log(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) + 176/352947
*(285*x - 181)/((2*x - 1)*sqrt(-2*x + 1)) - 1/7529536*(1159245*(2*x - 1)^3*sqrt(-2*x + 1) + 8543073*(2*x - 1)^
2*sqrt(-2*x + 1) - 20832595*(-2*x + 1)^(3/2) + 16829295*sqrt(-2*x + 1))/(3*x + 2)^4