3.1954 \(\int \frac{(1-2 x)^{5/2} (3+5 x)^2}{(2+3 x)^8} \, dx\)

Optimal. Leaf size=168 \[ \frac{23 (1-2 x)^{7/2}}{882 (3 x+2)^6}-\frac{(1-2 x)^{7/2}}{441 (3 x+2)^7}-\frac{467 (1-2 x)^{5/2}}{2646 (3 x+2)^5}+\frac{2335 (1-2 x)^{3/2}}{31752 (3 x+2)^4}+\frac{2335 \sqrt{1-2 x}}{3111696 (3 x+2)}+\frac{2335 \sqrt{1-2 x}}{1333584 (3 x+2)^2}-\frac{2335 \sqrt{1-2 x}}{95256 (3 x+2)^3}+\frac{2335 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{1555848 \sqrt{21}} \]

[Out]

-(1 - 2*x)^(7/2)/(441*(2 + 3*x)^7) + (23*(1 - 2*x)^(7/2))/(882*(2 + 3*x)^6) - (467*(1 - 2*x)^(5/2))/(2646*(2 +
3*x)^5) + (2335*(1 - 2*x)^(3/2))/(31752*(2 + 3*x)^4) - (2335*Sqrt[1 - 2*x])/(95256*(2 + 3*x)^3) + (2335*Sqrt[
1 - 2*x])/(1333584*(2 + 3*x)^2) + (2335*Sqrt[1 - 2*x])/(3111696*(2 + 3*x)) + (2335*ArcTanh[Sqrt[3/7]*Sqrt[1 -
2*x]])/(1555848*Sqrt[21])

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Rubi [A]  time = 0.0599715, antiderivative size = 168, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 6, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {89, 78, 47, 51, 63, 206} \[ \frac{23 (1-2 x)^{7/2}}{882 (3 x+2)^6}-\frac{(1-2 x)^{7/2}}{441 (3 x+2)^7}-\frac{467 (1-2 x)^{5/2}}{2646 (3 x+2)^5}+\frac{2335 (1-2 x)^{3/2}}{31752 (3 x+2)^4}+\frac{2335 \sqrt{1-2 x}}{3111696 (3 x+2)}+\frac{2335 \sqrt{1-2 x}}{1333584 (3 x+2)^2}-\frac{2335 \sqrt{1-2 x}}{95256 (3 x+2)^3}+\frac{2335 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{1555848 \sqrt{21}} \]

Antiderivative was successfully verified.

[In]

Int[((1 - 2*x)^(5/2)*(3 + 5*x)^2)/(2 + 3*x)^8,x]

[Out]

-(1 - 2*x)^(7/2)/(441*(2 + 3*x)^7) + (23*(1 - 2*x)^(7/2))/(882*(2 + 3*x)^6) - (467*(1 - 2*x)^(5/2))/(2646*(2 +
3*x)^5) + (2335*(1 - 2*x)^(3/2))/(31752*(2 + 3*x)^4) - (2335*Sqrt[1 - 2*x])/(95256*(2 + 3*x)^3) + (2335*Sqrt[
1 - 2*x])/(1333584*(2 + 3*x)^2) + (2335*Sqrt[1 - 2*x])/(3111696*(2 + 3*x)) + (2335*ArcTanh[Sqrt[3/7]*Sqrt[1 -
2*x]])/(1555848*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 47

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

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{(1-2 x)^{5/2} (3+5 x)^2}{(2+3 x)^8} \, dx &=-\frac{(1-2 x)^{7/2}}{441 (2+3 x)^7}+\frac{1}{441} \int \frac{(1-2 x)^{5/2} (1967+3675 x)}{(2+3 x)^7} \, dx\\ &=-\frac{(1-2 x)^{7/2}}{441 (2+3 x)^7}+\frac{23 (1-2 x)^{7/2}}{882 (2+3 x)^6}+\frac{2335}{882} \int \frac{(1-2 x)^{5/2}}{(2+3 x)^6} \, dx\\ &=-\frac{(1-2 x)^{7/2}}{441 (2+3 x)^7}+\frac{23 (1-2 x)^{7/2}}{882 (2+3 x)^6}-\frac{467 (1-2 x)^{5/2}}{2646 (2+3 x)^5}-\frac{2335 \int \frac{(1-2 x)^{3/2}}{(2+3 x)^5} \, dx}{2646}\\ &=-\frac{(1-2 x)^{7/2}}{441 (2+3 x)^7}+\frac{23 (1-2 x)^{7/2}}{882 (2+3 x)^6}-\frac{467 (1-2 x)^{5/2}}{2646 (2+3 x)^5}+\frac{2335 (1-2 x)^{3/2}}{31752 (2+3 x)^4}+\frac{2335 \int \frac{\sqrt{1-2 x}}{(2+3 x)^4} \, dx}{10584}\\ &=-\frac{(1-2 x)^{7/2}}{441 (2+3 x)^7}+\frac{23 (1-2 x)^{7/2}}{882 (2+3 x)^6}-\frac{467 (1-2 x)^{5/2}}{2646 (2+3 x)^5}+\frac{2335 (1-2 x)^{3/2}}{31752 (2+3 x)^4}-\frac{2335 \sqrt{1-2 x}}{95256 (2+3 x)^3}-\frac{2335 \int \frac{1}{\sqrt{1-2 x} (2+3 x)^3} \, dx}{95256}\\ &=-\frac{(1-2 x)^{7/2}}{441 (2+3 x)^7}+\frac{23 (1-2 x)^{7/2}}{882 (2+3 x)^6}-\frac{467 (1-2 x)^{5/2}}{2646 (2+3 x)^5}+\frac{2335 (1-2 x)^{3/2}}{31752 (2+3 x)^4}-\frac{2335 \sqrt{1-2 x}}{95256 (2+3 x)^3}+\frac{2335 \sqrt{1-2 x}}{1333584 (2+3 x)^2}-\frac{2335 \int \frac{1}{\sqrt{1-2 x} (2+3 x)^2} \, dx}{444528}\\ &=-\frac{(1-2 x)^{7/2}}{441 (2+3 x)^7}+\frac{23 (1-2 x)^{7/2}}{882 (2+3 x)^6}-\frac{467 (1-2 x)^{5/2}}{2646 (2+3 x)^5}+\frac{2335 (1-2 x)^{3/2}}{31752 (2+3 x)^4}-\frac{2335 \sqrt{1-2 x}}{95256 (2+3 x)^3}+\frac{2335 \sqrt{1-2 x}}{1333584 (2+3 x)^2}+\frac{2335 \sqrt{1-2 x}}{3111696 (2+3 x)}-\frac{2335 \int \frac{1}{\sqrt{1-2 x} (2+3 x)} \, dx}{3111696}\\ &=-\frac{(1-2 x)^{7/2}}{441 (2+3 x)^7}+\frac{23 (1-2 x)^{7/2}}{882 (2+3 x)^6}-\frac{467 (1-2 x)^{5/2}}{2646 (2+3 x)^5}+\frac{2335 (1-2 x)^{3/2}}{31752 (2+3 x)^4}-\frac{2335 \sqrt{1-2 x}}{95256 (2+3 x)^3}+\frac{2335 \sqrt{1-2 x}}{1333584 (2+3 x)^2}+\frac{2335 \sqrt{1-2 x}}{3111696 (2+3 x)}+\frac{2335 \operatorname{Subst}\left (\int \frac{1}{\frac{7}{2}-\frac{3 x^2}{2}} \, dx,x,\sqrt{1-2 x}\right )}{3111696}\\ &=-\frac{(1-2 x)^{7/2}}{441 (2+3 x)^7}+\frac{23 (1-2 x)^{7/2}}{882 (2+3 x)^6}-\frac{467 (1-2 x)^{5/2}}{2646 (2+3 x)^5}+\frac{2335 (1-2 x)^{3/2}}{31752 (2+3 x)^4}-\frac{2335 \sqrt{1-2 x}}{95256 (2+3 x)^3}+\frac{2335 \sqrt{1-2 x}}{1333584 (2+3 x)^2}+\frac{2335 \sqrt{1-2 x}}{3111696 (2+3 x)}+\frac{2335 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{1555848 \sqrt{21}}\\ \end{align*}

Mathematica [C]  time = 0.0274476, size = 47, normalized size = 0.28 \[ \frac{(1-2 x)^{7/2} \left (\frac{823543 (69 x+44)}{(3 x+2)^7}-149440 \, _2F_1\left (\frac{7}{2},6;\frac{9}{2};\frac{3}{7}-\frac{6 x}{7}\right )\right )}{726364926} \]

Antiderivative was successfully verified.

[In]

Integrate[((1 - 2*x)^(5/2)*(3 + 5*x)^2)/(2 + 3*x)^8,x]

[Out]

((1 - 2*x)^(7/2)*((823543*(44 + 69*x))/(2 + 3*x)^7 - 149440*Hypergeometric2F1[7/2, 6, 9/2, 3/7 - (6*x)/7]))/72
6364926

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Maple [A]  time = 0.01, size = 93, normalized size = 0.6 \begin{align*} -139968\,{\frac{1}{ \left ( -6\,x-4 \right ) ^{7}} \left ({\frac{2335\, \left ( 1-2\,x \right ) ^{13/2}}{298722816}}-{\frac{11675\, \left ( 1-2\,x \right ) ^{11/2}}{96018048}}+{\frac{6721\, \left ( 1-2\,x \right ) ^{9/2}}{164602368}}+{\frac{571\, \left ( 1-2\,x \right ) ^{7/2}}{321489}}-{\frac{132161\, \left ( 1-2\,x \right ) ^{5/2}}{30233088}}+{\frac{81725\, \left ( 1-2\,x \right ) ^{3/2}}{22674816}}-{\frac{114415\,\sqrt{1-2\,x}}{90699264}} \right ) }+{\frac{2335\,\sqrt{21}}{32672808}{\it Artanh} \left ({\frac{\sqrt{21}}{7}\sqrt{1-2\,x}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1-2*x)^(5/2)*(3+5*x)^2/(2+3*x)^8,x)

[Out]

-139968*(2335/298722816*(1-2*x)^(13/2)-11675/96018048*(1-2*x)^(11/2)+6721/164602368*(1-2*x)^(9/2)+571/321489*(
1-2*x)^(7/2)-132161/30233088*(1-2*x)^(5/2)+81725/22674816*(1-2*x)^(3/2)-114415/90699264*(1-2*x)^(1/2))/(-6*x-4
)^7+2335/32672808*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)

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Maxima [A]  time = 4.05187, size = 221, normalized size = 1.32 \begin{align*} -\frac{2335}{65345616} \, \sqrt{21} \log \left (-\frac{\sqrt{21} - 3 \, \sqrt{-2 \, x + 1}}{\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}}\right ) + \frac{1702215 \,{\left (-2 \, x + 1\right )}^{\frac{13}{2}} - 26478900 \,{\left (-2 \, x + 1\right )}^{\frac{11}{2}} + 8891883 \,{\left (-2 \, x + 1\right )}^{\frac{9}{2}} + 386781696 \,{\left (-2 \, x + 1\right )}^{\frac{7}{2}} - 951955683 \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} + 784886900 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 274710415 \, \sqrt{-2 \, x + 1}}{1555848 \,{\left (2187 \,{\left (2 \, x - 1\right )}^{7} + 35721 \,{\left (2 \, x - 1\right )}^{6} + 250047 \,{\left (2 \, x - 1\right )}^{5} + 972405 \,{\left (2 \, x - 1\right )}^{4} + 2268945 \,{\left (2 \, x - 1\right )}^{3} + 3176523 \,{\left (2 \, x - 1\right )}^{2} + 4941258 \, x - 1647086\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-2*x)^(5/2)*(3+5*x)^2/(2+3*x)^8,x, algorithm="maxima")

[Out]

-2335/65345616*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) + 1/1555848*(1702215
*(-2*x + 1)^(13/2) - 26478900*(-2*x + 1)^(11/2) + 8891883*(-2*x + 1)^(9/2) + 386781696*(-2*x + 1)^(7/2) - 9519
55683*(-2*x + 1)^(5/2) + 784886900*(-2*x + 1)^(3/2) - 274710415*sqrt(-2*x + 1))/(2187*(2*x - 1)^7 + 35721*(2*x
- 1)^6 + 250047*(2*x - 1)^5 + 972405*(2*x - 1)^4 + 2268945*(2*x - 1)^3 + 3176523*(2*x - 1)^2 + 4941258*x - 16
47086)

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Fricas [A]  time = 1.24402, size = 490, normalized size = 2.92 \begin{align*} \frac{2335 \, \sqrt{21}{\left (2187 \, x^{7} + 10206 \, x^{6} + 20412 \, x^{5} + 22680 \, x^{4} + 15120 \, x^{3} + 6048 \, x^{2} + 1344 \, x + 128\right )} \log \left (\frac{3 \, x - \sqrt{21} \sqrt{-2 \, x + 1} - 5}{3 \, x + 2}\right ) + 21 \,{\left (1702215 \, x^{6} + 8132805 \, x^{5} - 24492348 \, x^{4} - 23950566 \, x^{3} + 1405308 \, x^{2} + 1415408 \, x - 1107536\right )} \sqrt{-2 \, x + 1}}{65345616 \,{\left (2187 \, x^{7} + 10206 \, x^{6} + 20412 \, x^{5} + 22680 \, x^{4} + 15120 \, x^{3} + 6048 \, x^{2} + 1344 \, x + 128\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-2*x)^(5/2)*(3+5*x)^2/(2+3*x)^8,x, algorithm="fricas")

[Out]

1/65345616*(2335*sqrt(21)*(2187*x^7 + 10206*x^6 + 20412*x^5 + 22680*x^4 + 15120*x^3 + 6048*x^2 + 1344*x + 128)
*log((3*x - sqrt(21)*sqrt(-2*x + 1) - 5)/(3*x + 2)) + 21*(1702215*x^6 + 8132805*x^5 - 24492348*x^4 - 23950566*
x^3 + 1405308*x^2 + 1415408*x - 1107536)*sqrt(-2*x + 1))/(2187*x^7 + 10206*x^6 + 20412*x^5 + 22680*x^4 + 15120
*x^3 + 6048*x^2 + 1344*x + 128)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-2*x)**(5/2)*(3+5*x)**2/(2+3*x)**8,x)

[Out]

Timed out

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Giac [A]  time = 1.73076, size = 200, normalized size = 1.19 \begin{align*} -\frac{2335}{65345616} \, \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{1702215 \,{\left (2 \, x - 1\right )}^{6} \sqrt{-2 \, x + 1} + 26478900 \,{\left (2 \, x - 1\right )}^{5} \sqrt{-2 \, x + 1} + 8891883 \,{\left (2 \, x - 1\right )}^{4} \sqrt{-2 \, x + 1} - 386781696 \,{\left (2 \, x - 1\right )}^{3} \sqrt{-2 \, x + 1} - 951955683 \,{\left (2 \, x - 1\right )}^{2} \sqrt{-2 \, x + 1} + 784886900 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 274710415 \, \sqrt{-2 \, x + 1}}{199148544 \,{\left (3 \, x + 2\right )}^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-2*x)^(5/2)*(3+5*x)^2/(2+3*x)^8,x, algorithm="giac")

[Out]

-2335/65345616*sqrt(21)*log(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) + 1/1991485
44*(1702215*(2*x - 1)^6*sqrt(-2*x + 1) + 26478900*(2*x - 1)^5*sqrt(-2*x + 1) + 8891883*(2*x - 1)^4*sqrt(-2*x +
1) - 386781696*(2*x - 1)^3*sqrt(-2*x + 1) - 951955683*(2*x - 1)^2*sqrt(-2*x + 1) + 784886900*(-2*x + 1)^(3/2)
- 274710415*sqrt(-2*x + 1))/(3*x + 2)^7