3.2369 \(\int \frac{(1-2 x)^{3/2}}{(2+3 x)^5 \sqrt{3+5 x}} \, dx\)

Optimal. Leaf size=151 \[ \frac{4148797 \sqrt{1-2 x} \sqrt{5 x+3}}{28224 (3 x+2)}+\frac{39667 \sqrt{1-2 x} \sqrt{5 x+3}}{2016 (3 x+2)^2}+\frac{227 \sqrt{1-2 x} \sqrt{5 x+3}}{72 (3 x+2)^3}+\frac{7 \sqrt{1-2 x} \sqrt{5 x+3}}{12 (3 x+2)^4}-\frac{5274027 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{3136 \sqrt{7}} \]

[Out]

(7*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(12*(2 + 3*x)^4) + (227*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(72*(2 + 3*x)^3) + (39667
*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(2016*(2 + 3*x)^2) + (4148797*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(28224*(2 + 3*x)) - (
5274027*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(3136*Sqrt[7])

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Rubi [A]  time = 0.0511007, antiderivative size = 151, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.192, Rules used = {98, 151, 12, 93, 204} \[ \frac{4148797 \sqrt{1-2 x} \sqrt{5 x+3}}{28224 (3 x+2)}+\frac{39667 \sqrt{1-2 x} \sqrt{5 x+3}}{2016 (3 x+2)^2}+\frac{227 \sqrt{1-2 x} \sqrt{5 x+3}}{72 (3 x+2)^3}+\frac{7 \sqrt{1-2 x} \sqrt{5 x+3}}{12 (3 x+2)^4}-\frac{5274027 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{3136 \sqrt{7}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(7*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(12*(2 + 3*x)^4) + (227*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(72*(2 + 3*x)^3) + (39667
*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(2016*(2 + 3*x)^2) + (4148797*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(28224*(2 + 3*x)) - (
5274027*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(3136*Sqrt[7])

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 151

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 + 1)*(e + f*x)^(p + 1))/((m + 1)*(b*c - a*d)*(b*e - a*
f)), x] + Dist[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*Simp[(a*d*f*
g - b*(d*e + c*f)*g + b*c*e*h)*(m + 1) - (b*g - a*h)*(d*e*(n + 1) + c*f*(p + 1)) - d*f*(b*g - a*h)*(m + n + p
+ 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x] && LtQ[m, -1] && IntegerQ[m]

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

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)^{3/2}}{(2+3 x)^5 \sqrt{3+5 x}} \, dx &=\frac{7 \sqrt{1-2 x} \sqrt{3+5 x}}{12 (2+3 x)^4}+\frac{1}{12} \int \frac{\frac{271}{2}-194 x}{\sqrt{1-2 x} (2+3 x)^4 \sqrt{3+5 x}} \, dx\\ &=\frac{7 \sqrt{1-2 x} \sqrt{3+5 x}}{12 (2+3 x)^4}+\frac{227 \sqrt{1-2 x} \sqrt{3+5 x}}{72 (2+3 x)^3}+\frac{1}{252} \int \frac{\frac{50183}{4}-15890 x}{\sqrt{1-2 x} (2+3 x)^3 \sqrt{3+5 x}} \, dx\\ &=\frac{7 \sqrt{1-2 x} \sqrt{3+5 x}}{12 (2+3 x)^4}+\frac{227 \sqrt{1-2 x} \sqrt{3+5 x}}{72 (2+3 x)^3}+\frac{39667 \sqrt{1-2 x} \sqrt{3+5 x}}{2016 (2+3 x)^2}+\frac{\int \frac{\frac{5978273}{8}-\frac{1388345 x}{2}}{\sqrt{1-2 x} (2+3 x)^2 \sqrt{3+5 x}} \, dx}{3528}\\ &=\frac{7 \sqrt{1-2 x} \sqrt{3+5 x}}{12 (2+3 x)^4}+\frac{227 \sqrt{1-2 x} \sqrt{3+5 x}}{72 (2+3 x)^3}+\frac{39667 \sqrt{1-2 x} \sqrt{3+5 x}}{2016 (2+3 x)^2}+\frac{4148797 \sqrt{1-2 x} \sqrt{3+5 x}}{28224 (2+3 x)}+\frac{\int \frac{332263701}{16 \sqrt{1-2 x} (2+3 x) \sqrt{3+5 x}} \, dx}{24696}\\ &=\frac{7 \sqrt{1-2 x} \sqrt{3+5 x}}{12 (2+3 x)^4}+\frac{227 \sqrt{1-2 x} \sqrt{3+5 x}}{72 (2+3 x)^3}+\frac{39667 \sqrt{1-2 x} \sqrt{3+5 x}}{2016 (2+3 x)^2}+\frac{4148797 \sqrt{1-2 x} \sqrt{3+5 x}}{28224 (2+3 x)}+\frac{5274027 \int \frac{1}{\sqrt{1-2 x} (2+3 x) \sqrt{3+5 x}} \, dx}{6272}\\ &=\frac{7 \sqrt{1-2 x} \sqrt{3+5 x}}{12 (2+3 x)^4}+\frac{227 \sqrt{1-2 x} \sqrt{3+5 x}}{72 (2+3 x)^3}+\frac{39667 \sqrt{1-2 x} \sqrt{3+5 x}}{2016 (2+3 x)^2}+\frac{4148797 \sqrt{1-2 x} \sqrt{3+5 x}}{28224 (2+3 x)}+\frac{5274027 \operatorname{Subst}\left (\int \frac{1}{-7-x^2} \, dx,x,\frac{\sqrt{1-2 x}}{\sqrt{3+5 x}}\right )}{3136}\\ &=\frac{7 \sqrt{1-2 x} \sqrt{3+5 x}}{12 (2+3 x)^4}+\frac{227 \sqrt{1-2 x} \sqrt{3+5 x}}{72 (2+3 x)^3}+\frac{39667 \sqrt{1-2 x} \sqrt{3+5 x}}{2016 (2+3 x)^2}+\frac{4148797 \sqrt{1-2 x} \sqrt{3+5 x}}{28224 (2+3 x)}-\frac{5274027 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{3+5 x}}\right )}{3136 \sqrt{7}}\\ \end{align*}

Mathematica [A]  time = 0.0538495, size = 79, normalized size = 0.52 \[ \frac{\frac{7 \sqrt{1-2 x} \sqrt{5 x+3} \left (12446391 x^3+25448120 x^2+17365300 x+3956240\right )}{(3 x+2)^4}-5274027 \sqrt{7} \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{21952} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((7*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(3956240 + 17365300*x + 25448120*x^2 + 12446391*x^3))/(2 + 3*x)^4 - 5274027*Sq
rt[7]*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/21952

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Maple [B]  time = 0.013, size = 250, normalized size = 1.7 \begin{align*}{\frac{1}{43904\, \left ( 2+3\,x \right ) ^{4}}\sqrt{1-2\,x}\sqrt{3+5\,x} \left ( 427196187\,\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+356273680\,{x}^{2}\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 ) [phone]\,x\sqrt{-10\,{x}^{2}-x+3}[phone]\,\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)^(3/2)/(2+3*x)^5/(3+5*x)^(1/2),x)

[Out]

1/43904*(1-2*x)^(1/2)*(3+5*x)^(1/2)*(427196187*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^4+
1139189832*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^3+1139189832*7^(1/2)*arctan(1/14*(37*x
+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^2+174249474*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+356273680*x^2*(-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))[phone]*x*(-10*x^2-x+3)^(1/2)[phone]*(-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 = 3.06177, size = 193, normalized size = 1.28 \begin{align*} \frac{5274027}{43904} \, \sqrt{7} \arcsin \left (\frac{37 \, x}{11 \,{\left | 3 \, x + 2 \right |}} + \frac{20}{11 \,{\left | 3 \, x + 2 \right |}}\right ) + \frac{7 \, \sqrt{-10 \, x^{2} - x + 3}}{12 \,{\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )}} + \frac{227 \, \sqrt{-10 \, x^{2} - x + 3}}{72 \,{\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}} + \frac{39667 \, \sqrt{-10 \, x^{2} - x + 3}}{2016 \,{\left (9 \, x^{2} + 12 \, x + 4\right )}} + \frac{4148797 \, \sqrt{-10 \, x^{2} - x + 3}}{28224 \,{\left (3 \, x + 2\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

5274027/43904*sqrt(7)*arcsin(37/11*x/abs(3*x + 2) + 20/11/abs(3*x + 2)) + 7/12*sqrt(-10*x^2 - x + 3)/(81*x^4 +
216*x^3 + 216*x^2 + 96*x + 16) + 227/72*sqrt(-10*x^2 - x + 3)/(27*x^3 + 54*x^2 + 36*x + 8) + 39667/2016*sqrt(
-10*x^2 - x + 3)/(9*x^2 + 12*x + 4) + 4148797/28224*sqrt(-10*x^2 - x + 3)/(3*x + 2)

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Fricas [A]  time = 1.53635, size = 373, normalized size = 2.47 \begin{align*} -\frac{5274027 \, \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 (12446391 \, x^{3} + 25448120 \, x^{2} + 17365300 \, x + 3956240\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}{43904 \,{\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((1-2*x)^(3/2)/(2+3*x)^5/(3+5*x)^(1/2),x, algorithm="fricas")

[Out]

-1/43904*(5274027*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*(12446391*x^3 + 25448120*x^2 + 17365300*x + 3956240)*sqrt(5*x + 3)*sq
rt(-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((1-2*x)**(3/2)/(2+3*x)**5/(3+5*x)**(1/2),x)

[Out]

Timed out

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Giac [B]  time = 3.06385, size = 512, normalized size = 3.39 \begin{align*} \frac{5274027}{439040} \, \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{121 \,{\left (113213 \, \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} + 59365880 \, \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} + 12529809600 \, \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} + 956821824000 \, \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 )}}{1568 \,{\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((1-2*x)^(3/2)/(2+3*x)^5/(3+5*x)^(1/2),x, algorithm="giac")

[Out]

5274027/439040*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)))) + 121/1568*(113213*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 + 59365880*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
+ 12529809600*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 + 956821824000*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