3.2395 \(\int \frac{(1-2 x)^{5/2} \sqrt{3+5 x}}{(2+3 x)^4} \, dx\)

Optimal. Leaf size=149 \[ -\frac{\sqrt{5 x+3} (1-2 x)^{5/2}}{9 (3 x+2)^3}+\frac{5 \sqrt{5 x+3} (1-2 x)^{3/2}}{12 (3 x+2)^2}+\frac{925 \sqrt{5 x+3} \sqrt{1-2 x}}{216 (3 x+2)}-\frac{8}{81} \sqrt{10} \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )-\frac{32765 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{648 \sqrt{7}} \]

[Out]

-((1 - 2*x)^(5/2)*Sqrt[3 + 5*x])/(9*(2 + 3*x)^3) + (5*(1 - 2*x)^(3/2)*Sqrt[3 + 5*x])/(12*(2 + 3*x)^2) + (925*S
qrt[1 - 2*x]*Sqrt[3 + 5*x])/(216*(2 + 3*x)) - (8*Sqrt[10]*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/81 - (32765*ArcTan
[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(648*Sqrt[7])

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Rubi [A]  time = 0.0505666, antiderivative size = 149, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.269, Rules used = {97, 149, 157, 54, 216, 93, 204} \[ -\frac{\sqrt{5 x+3} (1-2 x)^{5/2}}{9 (3 x+2)^3}+\frac{5 \sqrt{5 x+3} (1-2 x)^{3/2}}{12 (3 x+2)^2}+\frac{925 \sqrt{5 x+3} \sqrt{1-2 x}}{216 (3 x+2)}-\frac{8}{81} \sqrt{10} \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )-\frac{32765 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{648 \sqrt{7}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((1 - 2*x)^(5/2)*Sqrt[3 + 5*x])/(9*(2 + 3*x)^3) + (5*(1 - 2*x)^(3/2)*Sqrt[3 + 5*x])/(12*(2 + 3*x)^2) + (925*S
qrt[1 - 2*x]*Sqrt[3 + 5*x])/(216*(2 + 3*x)) - (8*Sqrt[10]*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/81 - (32765*ArcTan
[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(648*Sqrt[7])

Rule 97

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

Rule 149

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)), x_Symb
ol] :> Simp[((b*g - a*h)*(a + b*x)^(m + 1)*(c + d*x)^n*(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 - 1)*(e + f*x)^p*Simp[b*c*(f*g - e*h)*(m + 1) + (
b*g - a*h)*(d*e*n + c*f*(p + 1)) + d*(b*(f*g - e*h)*(m + 1) + f*(b*g - a*h)*(n + p + 1))*x, x], x], x] /; Free
Q[{a, b, c, d, e, f, g, h, p}, x] && LtQ[m, -1] && GtQ[n, 0] && IntegerQ[m]

Rule 157

Int[(((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)))/((a_.) + (b_.)*(x_)), x_Symbol]
:> Dist[h/b, Int[(c + d*x)^n*(e + f*x)^p, x], x] + Dist[(b*g - a*h)/b, Int[((c + d*x)^n*(e + f*x)^p)/(a + b*x
), x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x]

Rule 54

Int[1/(Sqrt[(a_.) + (b_.)*(x_)]*Sqrt[(c_.) + (d_.)*(x_)]), x_Symbol] :> Dist[2/Sqrt[b], Subst[Int[1/Sqrt[b*c -
a*d + d*x^2], x], x, Sqrt[a + b*x]], x] /; FreeQ[{a, b, c, d}, x] && GtQ[b*c - a*d, 0] && GtQ[b, 0]

Rule 216

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

Rule 93

Int[(((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_))/((e_.) + (f_.)*(x_)), x_Symbol] :> With[{q = Denomin
ator[m]}, Dist[q, Subst[Int[x^(q*(m + 1) - 1)/(b*e - a*f - (d*e - c*f)*x^q), x], x, (a + b*x)^(1/q)/(c + d*x)^
(1/q)], x]] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[m + n + 1, 0] && RationalQ[n] && LtQ[-1, m, 0] && SimplerQ[
a + b*x, c + d*x]

Rule 204

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

Rubi steps

\begin{align*} \int \frac{(1-2 x)^{5/2} \sqrt{3+5 x}}{(2+3 x)^4} \, dx &=-\frac{(1-2 x)^{5/2} \sqrt{3+5 x}}{9 (2+3 x)^3}+\frac{1}{9} \int \frac{\left (-\frac{25}{2}-30 x\right ) (1-2 x)^{3/2}}{(2+3 x)^3 \sqrt{3+5 x}} \, dx\\ &=-\frac{(1-2 x)^{5/2} \sqrt{3+5 x}}{9 (2+3 x)^3}+\frac{5 (1-2 x)^{3/2} \sqrt{3+5 x}}{12 (2+3 x)^2}-\frac{1}{54} \int \frac{\left (-\frac{1245}{4}-120 x\right ) \sqrt{1-2 x}}{(2+3 x)^2 \sqrt{3+5 x}} \, dx\\ &=-\frac{(1-2 x)^{5/2} \sqrt{3+5 x}}{9 (2+3 x)^3}+\frac{5 (1-2 x)^{3/2} \sqrt{3+5 x}}{12 (2+3 x)^2}+\frac{925 \sqrt{1-2 x} \sqrt{3+5 x}}{216 (2+3 x)}+\frac{1}{162} \int \frac{\frac{31485}{8}-240 x}{\sqrt{1-2 x} (2+3 x) \sqrt{3+5 x}} \, dx\\ &=-\frac{(1-2 x)^{5/2} \sqrt{3+5 x}}{9 (2+3 x)^3}+\frac{5 (1-2 x)^{3/2} \sqrt{3+5 x}}{12 (2+3 x)^2}+\frac{925 \sqrt{1-2 x} \sqrt{3+5 x}}{216 (2+3 x)}-\frac{40}{81} \int \frac{1}{\sqrt{1-2 x} \sqrt{3+5 x}} \, dx+\frac{32765 \int \frac{1}{\sqrt{1-2 x} (2+3 x) \sqrt{3+5 x}} \, dx}{1296}\\ &=-\frac{(1-2 x)^{5/2} \sqrt{3+5 x}}{9 (2+3 x)^3}+\frac{5 (1-2 x)^{3/2} \sqrt{3+5 x}}{12 (2+3 x)^2}+\frac{925 \sqrt{1-2 x} \sqrt{3+5 x}}{216 (2+3 x)}+\frac{32765}{648} \operatorname{Subst}\left (\int \frac{1}{-7-x^2} \, dx,x,\frac{\sqrt{1-2 x}}{\sqrt{3+5 x}}\right )-\frac{1}{81} \left (16 \sqrt{5}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{11-2 x^2}} \, dx,x,\sqrt{3+5 x}\right )\\ &=-\frac{(1-2 x)^{5/2} \sqrt{3+5 x}}{9 (2+3 x)^3}+\frac{5 (1-2 x)^{3/2} \sqrt{3+5 x}}{12 (2+3 x)^2}+\frac{925 \sqrt{1-2 x} \sqrt{3+5 x}}{216 (2+3 x)}-\frac{8}{81} \sqrt{10} \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{3+5 x}\right )-\frac{32765 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{3+5 x}}\right )}{648 \sqrt{7}}\\ \end{align*}

Mathematica [A]  time = 0.156476, size = 126, normalized size = 0.85 \[ \frac{21 \sqrt{5 x+3} \left (-15378 x^3-14523 x^2+3394 x+3856\right )+448 \sqrt{10-20 x} (3 x+2)^3 \sin ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )-32765 \sqrt{7-14 x} (3 x+2)^3 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{4536 \sqrt{1-2 x} (3 x+2)^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(21*Sqrt[3 + 5*x]*(3856 + 3394*x - 14523*x^2 - 15378*x^3) + 448*Sqrt[10 - 20*x]*(2 + 3*x)^3*ArcSin[Sqrt[5/11]*
Sqrt[1 - 2*x]] - 32765*Sqrt[7 - 14*x]*(2 + 3*x)^3*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(4536*Sqrt[1
- 2*x]*(2 + 3*x)^3)

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Maple [B]  time = 0.013, size = 253, normalized size = 1.7 \begin{align*}{\frac{1}{9072\, \left ( 2+3\,x \right ) ^{3}}\sqrt{1-2\,x}\sqrt{3+5\,x} \left ( 884655\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{3}-12096\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ){x}^{3}+1769310\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{2}-24192\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ){x}^{2}+1179540\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) x-16128\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) x+322938\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}+262120\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) -3584\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) +466452\,x\sqrt{-10\,{x}^{2}-x+3}+161952\,\sqrt{-10\,{x}^{2}-x+3} \right ){\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/9072*(1-2*x)^(1/2)*(3+5*x)^(1/2)*(884655*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^3-1209
6*10^(1/2)*arcsin(20/11*x+1/11)*x^3+1769310*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^2-241
92*10^(1/2)*arcsin(20/11*x+1/11)*x^2+1179540*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x-1612
8*10^(1/2)*arcsin(20/11*x+1/11)*x+322938*x^2*(-10*x^2-x+3)^(1/2)+262120*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/
(-10*x^2-x+3)^(1/2))-3584*10^(1/2)*arcsin(20/11*x+1/11)+466452*x*(-10*x^2-x+3)^(1/2)+161952*(-10*x^2-x+3)^(1/2
))/(-10*x^2-x+3)^(1/2)/(2+3*x)^3

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Maxima [A]  time = 2.80191, size = 178, normalized size = 1.19 \begin{align*} -\frac{4}{81} \, \sqrt{10} \arcsin \left (\frac{20}{11} \, x + \frac{1}{11}\right ) + \frac{32765}{9072} \, \sqrt{7} \arcsin \left (\frac{37 \, x}{11 \,{\left | 3 \, x + 2 \right |}} + \frac{20}{11 \,{\left | 3 \, x + 2 \right |}}\right ) + \frac{145}{54} \, \sqrt{-10 \, x^{2} - x + 3} + \frac{7 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}}{9 \,{\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}} + \frac{29 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}}{12 \,{\left (9 \, x^{2} + 12 \, x + 4\right )}} - \frac{1105 \, \sqrt{-10 \, x^{2} - x + 3}}{216 \,{\left (3 \, x + 2\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-4/81*sqrt(10)*arcsin(20/11*x + 1/11) + 32765/9072*sqrt(7)*arcsin(37/11*x/abs(3*x + 2) + 20/11/abs(3*x + 2)) +
145/54*sqrt(-10*x^2 - x + 3) + 7/9*(-10*x^2 - x + 3)^(3/2)/(27*x^3 + 54*x^2 + 36*x + 8) + 29/12*(-10*x^2 - x
+ 3)^(3/2)/(9*x^2 + 12*x + 4) - 1105/216*sqrt(-10*x^2 - x + 3)/(3*x + 2)

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Fricas [A]  time = 1.57382, size = 468, normalized size = 3.14 \begin{align*} -\frac{32765 \, \sqrt{7}{\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )} \arctan \left (\frac{\sqrt{7}{\left (37 \, x + 20\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}{14 \,{\left (10 \, x^{2} + x - 3\right )}}\right ) - 448 \, \sqrt{10}{\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )} \arctan \left (\frac{\sqrt{10}{\left (20 \, x + 1\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}{20 \,{\left (10 \, x^{2} + x - 3\right )}}\right ) - 42 \,{\left (7689 \, x^{2} + 11106 \, x + 3856\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}{9072 \,{\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/9072*(32765*sqrt(7)*(27*x^3 + 54*x^2 + 36*x + 8)*arctan(1/14*sqrt(7)*(37*x + 20)*sqrt(5*x + 3)*sqrt(-2*x +
1)/(10*x^2 + x - 3)) - 448*sqrt(10)*(27*x^3 + 54*x^2 + 36*x + 8)*arctan(1/20*sqrt(10)*(20*x + 1)*sqrt(5*x + 3)
*sqrt(-2*x + 1)/(10*x^2 + x - 3)) - 42*(7689*x^2 + 11106*x + 3856)*sqrt(5*x + 3)*sqrt(-2*x + 1))/(27*x^3 + 54*
x^2 + 36*x + 8)

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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)**(1/2)/(2+3*x)**4,x)

[Out]

Timed out

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Giac [B]  time = 3.14792, size = 520, normalized size = 3.49 \begin{align*} \frac{6553}{18144} \, \sqrt{70} \sqrt{10}{\left (\pi + 2 \, \arctan \left (-\frac{\sqrt{70} \sqrt{5 \, x + 3}{\left (\frac{{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}\right )\right )} - \frac{4}{81} \, \sqrt{10}{\left (\pi + 2 \, \arctan \left (-\frac{\sqrt{5 \, x + 3}{\left (\frac{{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{4 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}\right )\right )} - \frac{11 \,{\left (989 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{5} - 795200 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{3} - 72520000 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}\right )}}{108 \,{\left ({\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{2} + 280\right )}^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

6553/18144*sqrt(70)*sqrt(10)*(pi + 2*arctan(-1/140*sqrt(70)*sqrt(5*x + 3)*((sqrt(2)*sqrt(-10*x + 5) - sqrt(22)
)^2/(5*x + 3) - 4)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22)))) - 4/81*sqrt(10)*(pi + 2*arctan(-1/4*sqrt(5*x + 3)*((
sqrt(2)*sqrt(-10*x + 5) - sqrt(22))^2/(5*x + 3) - 4)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22)))) - 11/108*(989*sqrt
(10)*((sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) - 4*sqrt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22)
))^5 - 795200*sqrt(10)*((sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) - 4*sqrt(5*x + 3)/(sqrt(2)*sqrt(-10
*x + 5) - sqrt(22)))^3 - 72520000*sqrt(10)*((sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) - 4*sqrt(5*x +
3)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))))/(((sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) - 4*sqrt(5*x +
3)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22)))^2 + 280)^3