3.2283 \(\int \frac{\sqrt{1-2 x} (3+5 x)^{3/2}}{(2+3 x)^5} \, dx\)

Optimal. Leaf size=151 \[ \frac{115 \sqrt{1-2 x} (5 x+3)^{5/2}}{168 (3 x+2)^3}+\frac{3 (1-2 x)^{3/2} (5 x+3)^{5/2}}{28 (3 x+2)^4}-\frac{1265 \sqrt{1-2 x} (5 x+3)^{3/2}}{4704 (3 x+2)^2}-\frac{13915 \sqrt{1-2 x} \sqrt{5 x+3}}{21952 (3 x+2)}-\frac{153065 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{21952 \sqrt{7}} \]

[Out]

(-13915*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(21952*(2 + 3*x)) - (1265*Sqrt[1 - 2*x]*(3 + 5*x)^(3/2))/(4704*(2 + 3*x)^
2) + (3*(1 - 2*x)^(3/2)*(3 + 5*x)^(5/2))/(28*(2 + 3*x)^4) + (115*Sqrt[1 - 2*x]*(3 + 5*x)^(5/2))/(168*(2 + 3*x)
^3) - (153065*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(21952*Sqrt[7])

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Rubi [A]  time = 0.0430622, antiderivative size = 151, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {96, 94, 93, 204} \[ \frac{115 \sqrt{1-2 x} (5 x+3)^{5/2}}{168 (3 x+2)^3}+\frac{3 (1-2 x)^{3/2} (5 x+3)^{5/2}}{28 (3 x+2)^4}-\frac{1265 \sqrt{1-2 x} (5 x+3)^{3/2}}{4704 (3 x+2)^2}-\frac{13915 \sqrt{1-2 x} \sqrt{5 x+3}}{21952 (3 x+2)}-\frac{153065 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{21952 \sqrt{7}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-13915*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(21952*(2 + 3*x)) - (1265*Sqrt[1 - 2*x]*(3 + 5*x)^(3/2))/(4704*(2 + 3*x)^
2) + (3*(1 - 2*x)^(3/2)*(3 + 5*x)^(5/2))/(28*(2 + 3*x)^4) + (115*Sqrt[1 - 2*x]*(3 + 5*x)^(5/2))/(168*(2 + 3*x)
^3) - (153065*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(21952*Sqrt[7])

Rule 96

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(b*(a +
b*x)^(m + 1)*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/((m + 1)*(b*c - a*d)*(b*e - a*f)), x] + Dist[(a*d*f*(m + 1)
+ b*c*f*(n + 1) + b*d*e*(p + 1))/((m + 1)*(b*c - a*d)*(b*e - a*f)), Int[(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*
x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && EqQ[Simplify[m + n + p + 3], 0] && (LtQ[m, -1] || Sum
SimplerQ[m, 1])

Rule 94

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 + 1))/((m + 1)*(b*e - a*f)), x] - Dist[(n*(d*e - c*f))/((m + 1)*(b*e - a*
f)), Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, m, p}, x] && EqQ[
m + n + p + 2, 0] && GtQ[n, 0] &&  !(SumSimplerQ[p, 1] &&  !SumSimplerQ[m, 1])

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{\sqrt{1-2 x} (3+5 x)^{3/2}}{(2+3 x)^5} \, dx &=\frac{3 (1-2 x)^{3/2} (3+5 x)^{5/2}}{28 (2+3 x)^4}+\frac{115}{56} \int \frac{\sqrt{1-2 x} (3+5 x)^{3/2}}{(2+3 x)^4} \, dx\\ &=\frac{3 (1-2 x)^{3/2} (3+5 x)^{5/2}}{28 (2+3 x)^4}+\frac{115 \sqrt{1-2 x} (3+5 x)^{5/2}}{168 (2+3 x)^3}+\frac{1265}{336} \int \frac{(3+5 x)^{3/2}}{\sqrt{1-2 x} (2+3 x)^3} \, dx\\ &=-\frac{1265 \sqrt{1-2 x} (3+5 x)^{3/2}}{4704 (2+3 x)^2}+\frac{3 (1-2 x)^{3/2} (3+5 x)^{5/2}}{28 (2+3 x)^4}+\frac{115 \sqrt{1-2 x} (3+5 x)^{5/2}}{168 (2+3 x)^3}+\frac{13915 \int \frac{\sqrt{3+5 x}}{\sqrt{1-2 x} (2+3 x)^2} \, dx}{3136}\\ &=-\frac{13915 \sqrt{1-2 x} \sqrt{3+5 x}}{21952 (2+3 x)}-\frac{1265 \sqrt{1-2 x} (3+5 x)^{3/2}}{4704 (2+3 x)^2}+\frac{3 (1-2 x)^{3/2} (3+5 x)^{5/2}}{28 (2+3 x)^4}+\frac{115 \sqrt{1-2 x} (3+5 x)^{5/2}}{168 (2+3 x)^3}+\frac{153065 \int \frac{1}{\sqrt{1-2 x} (2+3 x) \sqrt{3+5 x}} \, dx}{43904}\\ &=-\frac{13915 \sqrt{1-2 x} \sqrt{3+5 x}}{21952 (2+3 x)}-\frac{1265 \sqrt{1-2 x} (3+5 x)^{3/2}}{4704 (2+3 x)^2}+\frac{3 (1-2 x)^{3/2} (3+5 x)^{5/2}}{28 (2+3 x)^4}+\frac{115 \sqrt{1-2 x} (3+5 x)^{5/2}}{168 (2+3 x)^3}+\frac{153065 \operatorname{Subst}\left (\int \frac{1}{-7-x^2} \, dx,x,\frac{\sqrt{1-2 x}}{\sqrt{3+5 x}}\right )}{21952}\\ &=-\frac{13915 \sqrt{1-2 x} \sqrt{3+5 x}}{21952 (2+3 x)}-\frac{1265 \sqrt{1-2 x} (3+5 x)^{3/2}}{4704 (2+3 x)^2}+\frac{3 (1-2 x)^{3/2} (3+5 x)^{5/2}}{28 (2+3 x)^4}+\frac{115 \sqrt{1-2 x} (3+5 x)^{5/2}}{168 (2+3 x)^3}-\frac{153065 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{3+5 x}}\right )}{21952 \sqrt{7}}\\ \end{align*}

Mathematica [A]  time = 0.0538825, size = 79, normalized size = 0.52 \[ \frac{\frac{7 \sqrt{1-2 x} \sqrt{5 x+3} \left (1104135 x^3+2269240 x^2+1512052 x+328464\right )}{(3 x+2)^4}-459195 \sqrt{7} \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{460992} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((7*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(328464 + 1512052*x + 2269240*x^2 + 1104135*x^3))/(2 + 3*x)^4 - 459195*Sqrt[7]
*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/460992

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Maple [B]  time = 0.011, size = 250, normalized size = 1.7 \begin{align*}{\frac{1}{921984\, \left ( 2+3\,x \right ) ^{4}}\sqrt{1-2\,x}\sqrt{3+5\,x} \left ( 37194795\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{4}[phone]\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{3}[phone]\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{2}[phone]\,{x}^{3}\sqrt{-10\,{x}^{2}-x+3}[phone]\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) x+31769360\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}+7347120\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) [phone]\,x\sqrt{-10\,{x}^{2}-x+3}+4598496\,\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((3+5*x)^(3/2)*(1-2*x)^(1/2)/(2+3*x)^5,x)

[Out]

1/921984*(1-2*x)^(1/2)*(3+5*x)^(1/2)*(37194795*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^4+
99186120*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^3+99186120*7^(1/2)*arctan(1/14*(37*x+20)
*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^2+15457890*x^3*(-10*x^2-x+3)^(1/2)[phone]*7^(1/2)*arctan(1/14*(37*x+20)*7^(1
/2)/(-10*x^2-x+3)^(1/2))*x+31769360*x^2*(-10*x^2-x+3)^(1/2)+7347120*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10
*x^2-x+3)^(1/2))[phone]*x*(-10*x^2-x+3)^(1/2)+4598496*(-10*x^2-x+3)^(1/2))/(-10*x^2-x+3)^(1/2)/(2+3*x)^4

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Maxima [A]  time = 3.86987, size = 212, normalized size = 1.4 \begin{align*} \frac{153065}{307328} \, \sqrt{7} \arcsin \left (\frac{37 \, x}{11 \,{\left | 3 \, x + 2 \right |}} + \frac{20}{11 \,{\left | 3 \, x + 2 \right |}}\right ) + \frac{6325}{16464} \, \sqrt{-10 \, x^{2} - x + 3} - \frac{{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}}{28 \,{\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )}} + \frac{95 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}}{1176 \,{\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}} + \frac{3795 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}}{10976 \,{\left (9 \, x^{2} + 12 \, x + 4\right )}} - \frac{46805 \, \sqrt{-10 \, x^{2} - x + 3}}{65856 \,{\left (3 \, x + 2\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

153065/307328*sqrt(7)*arcsin(37/11*x/abs(3*x + 2) + 20/11/abs(3*x + 2)) + 6325/16464*sqrt(-10*x^2 - x + 3) - 1
/28*(-10*x^2 - x + 3)^(3/2)/(81*x^4 + 216*x^3 + 216*x^2 + 96*x + 16) + 95/1176*(-10*x^2 - x + 3)^(3/2)/(27*x^3
+ 54*x^2 + 36*x + 8) + 3795/10976*(-10*x^2 - x + 3)^(3/2)/(9*x^2 + 12*x + 4) - 46805/65856*sqrt(-10*x^2 - x +
3)/(3*x + 2)

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Fricas [A]  time = 1.7763, size = 367, normalized size = 2.43 \begin{align*} -\frac{459195 \, \sqrt{7}{\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\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 ) - 14 \,{\left (1104135 \, x^{3} + 2269240 \, x^{2} + 1512052 \, x + 328464\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}{921984 \,{\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/921984*(459195*sqrt(7)*(81*x^4 + 216*x^3 + 216*x^2 + 96*x + 16)*arctan(1/14*sqrt(7)*(37*x + 20)*sqrt(5*x +
3)*sqrt(-2*x + 1)/(10*x^2 + x - 3)) - 14*(1104135*x^3 + 2269240*x^2 + 1512052*x + 328464)*sqrt(5*x + 3)*sqrt(-
2*x + 1))/(81*x^4 + 216*x^3 + 216*x^2 + 96*x + 16)

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

[Out]

Timed out

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Giac [B]  time = 3.76127, size = 512, normalized size = 3.39 \begin{align*} \frac{30613}{614656} \, \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{6655 \,{\left (69 \, \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 )}^{7} + 70840 \, \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} - 15821120 \, \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} - 1514688000 \, \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 )}}{32928 \,{\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 )}^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

30613/614656*sqrt(70)*sqrt(10)*(pi + 2*arctan(-1/140*sqrt(70)*sqrt(5*x + 3)*((sqrt(2)*sqrt(-10*x + 5) - sqrt(2
2))^2/(5*x + 3) - 4)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22)))) - 6655/32928*(69*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)))^7 + 70840*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 - 15821
120*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 - 1514688000*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)^4