3.2326 \(\int \frac{\sqrt{1-2 x}}{(2+3 x)^4 (3+5 x)^{5/2}} \, dx\)

Optimal. Leaf size=166 \[ \frac{63678595 \sqrt{1-2 x}}{12936 \sqrt{5 x+3}}-\frac{638165 \sqrt{1-2 x}}{1176 (5 x+3)^{3/2}}+\frac{25441 \sqrt{1-2 x}}{392 (3 x+2) (5 x+3)^{3/2}}+\frac{313 \sqrt{1-2 x}}{84 (3 x+2)^2 (5 x+3)^{3/2}}+\frac{\sqrt{1-2 x}}{3 (3 x+2)^3 (5 x+3)^{3/2}}-\frac{13246251 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{392 \sqrt{7}} \]

[Out]

(-638165*Sqrt[1 - 2*x])/(1176*(3 + 5*x)^(3/2)) + Sqrt[1 - 2*x]/(3*(2 + 3*x)^3*(3 + 5*x)^(3/2)) + (313*Sqrt[1 -
2*x])/(84*(2 + 3*x)^2*(3 + 5*x)^(3/2)) + (25441*Sqrt[1 - 2*x])/(392*(2 + 3*x)*(3 + 5*x)^(3/2)) + (63678595*Sq
rt[1 - 2*x])/(12936*Sqrt[3 + 5*x]) - (13246251*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(392*Sqrt[7])

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Rubi [A]  time = 0.0595912, antiderivative size = 166, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231, Rules used = {99, 151, 152, 12, 93, 204} \[ \frac{63678595 \sqrt{1-2 x}}{12936 \sqrt{5 x+3}}-\frac{638165 \sqrt{1-2 x}}{1176 (5 x+3)^{3/2}}+\frac{25441 \sqrt{1-2 x}}{392 (3 x+2) (5 x+3)^{3/2}}+\frac{313 \sqrt{1-2 x}}{84 (3 x+2)^2 (5 x+3)^{3/2}}+\frac{\sqrt{1-2 x}}{3 (3 x+2)^3 (5 x+3)^{3/2}}-\frac{13246251 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{392 \sqrt{7}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-638165*Sqrt[1 - 2*x])/(1176*(3 + 5*x)^(3/2)) + Sqrt[1 - 2*x]/(3*(2 + 3*x)^3*(3 + 5*x)^(3/2)) + (313*Sqrt[1 -
2*x])/(84*(2 + 3*x)^2*(3 + 5*x)^(3/2)) + (25441*Sqrt[1 - 2*x])/(392*(2 + 3*x)*(3 + 5*x)^(3/2)) + (63678595*Sq
rt[1 - 2*x])/(12936*Sqrt[3 + 5*x]) - (13246251*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(392*Sqrt[7])

Rule 99

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[1/((m + 1)*(b*e - a*f)), Int[(a +
b*x)^(m + 1)*(c + d*x)^(n - 1)*(e + f*x)^p*Simp[d*e*n + c*f*(m + p + 2) + d*f*(m + n + p + 2)*x, x], x], x] /;
FreeQ[{a, b, c, d, e, f, p}, x] && LtQ[m, -1] && GtQ[n, 0] && (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 152

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] && IntegersQ[2*m, 2*n, 2*p]

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{\sqrt{1-2 x}}{(2+3 x)^4 (3+5 x)^{5/2}} \, dx &=\frac{\sqrt{1-2 x}}{3 (2+3 x)^3 (3+5 x)^{3/2}}-\frac{1}{3} \int \frac{-\frac{51}{2}+40 x}{\sqrt{1-2 x} (2+3 x)^3 (3+5 x)^{5/2}} \, dx\\ &=\frac{\sqrt{1-2 x}}{3 (2+3 x)^3 (3+5 x)^{3/2}}+\frac{313 \sqrt{1-2 x}}{84 (2+3 x)^2 (3+5 x)^{3/2}}-\frac{1}{42} \int \frac{-\frac{12921}{4}+4695 x}{\sqrt{1-2 x} (2+3 x)^2 (3+5 x)^{5/2}} \, dx\\ &=\frac{\sqrt{1-2 x}}{3 (2+3 x)^3 (3+5 x)^{3/2}}+\frac{313 \sqrt{1-2 x}}{84 (2+3 x)^2 (3+5 x)^{3/2}}+\frac{25441 \sqrt{1-2 x}}{392 (2+3 x) (3+5 x)^{3/2}}-\frac{1}{294} \int \frac{-\frac{2380137}{8}+381615 x}{\sqrt{1-2 x} (2+3 x) (3+5 x)^{5/2}} \, dx\\ &=-\frac{638165 \sqrt{1-2 x}}{1176 (3+5 x)^{3/2}}+\frac{\sqrt{1-2 x}}{3 (2+3 x)^3 (3+5 x)^{3/2}}+\frac{313 \sqrt{1-2 x}}{84 (2+3 x)^2 (3+5 x)^{3/2}}+\frac{25441 \sqrt{1-2 x}}{392 (2+3 x) (3+5 x)^{3/2}}+\frac{\int \frac{-\frac{268650723}{16}+\frac{63178335 x}{4}}{\sqrt{1-2 x} (2+3 x) (3+5 x)^{3/2}} \, dx}{4851}\\ &=-\frac{638165 \sqrt{1-2 x}}{1176 (3+5 x)^{3/2}}+\frac{\sqrt{1-2 x}}{3 (2+3 x)^3 (3+5 x)^{3/2}}+\frac{313 \sqrt{1-2 x}}{84 (2+3 x)^2 (3+5 x)^{3/2}}+\frac{25441 \sqrt{1-2 x}}{392 (2+3 x) (3+5 x)^{3/2}}+\frac{63678595 \sqrt{1-2 x}}{12936 \sqrt{3+5 x}}-\frac{2 \int -\frac{14425167339}{32 \sqrt{1-2 x} (2+3 x) \sqrt{3+5 x}} \, dx}{53361}\\ &=-\frac{638165 \sqrt{1-2 x}}{1176 (3+5 x)^{3/2}}+\frac{\sqrt{1-2 x}}{3 (2+3 x)^3 (3+5 x)^{3/2}}+\frac{313 \sqrt{1-2 x}}{84 (2+3 x)^2 (3+5 x)^{3/2}}+\frac{25441 \sqrt{1-2 x}}{392 (2+3 x) (3+5 x)^{3/2}}+\frac{63678595 \sqrt{1-2 x}}{12936 \sqrt{3+5 x}}+\frac{13246251}{784} \int \frac{1}{\sqrt{1-2 x} (2+3 x) \sqrt{3+5 x}} \, dx\\ &=-\frac{638165 \sqrt{1-2 x}}{1176 (3+5 x)^{3/2}}+\frac{\sqrt{1-2 x}}{3 (2+3 x)^3 (3+5 x)^{3/2}}+\frac{313 \sqrt{1-2 x}}{84 (2+3 x)^2 (3+5 x)^{3/2}}+\frac{25441 \sqrt{1-2 x}}{392 (2+3 x) (3+5 x)^{3/2}}+\frac{63678595 \sqrt{1-2 x}}{12936 \sqrt{3+5 x}}+\frac{13246251}{392} \operatorname{Subst}\left (\int \frac{1}{-7-x^2} \, dx,x,\frac{\sqrt{1-2 x}}{\sqrt{3+5 x}}\right )\\ &=-\frac{638165 \sqrt{1-2 x}}{1176 (3+5 x)^{3/2}}+\frac{\sqrt{1-2 x}}{3 (2+3 x)^3 (3+5 x)^{3/2}}+\frac{313 \sqrt{1-2 x}}{84 (2+3 x)^2 (3+5 x)^{3/2}}+\frac{25441 \sqrt{1-2 x}}{392 (2+3 x) (3+5 x)^{3/2}}+\frac{63678595 \sqrt{1-2 x}}{12936 \sqrt{3+5 x}}-\frac{13246251 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{3+5 x}}\right )}{392 \sqrt{7}}\\ \end{align*}

Mathematica [A]  time = 0.0787976, size = 84, normalized size = 0.51 \[ \frac{\frac{7 \sqrt{1-2 x} \left (8596610325 x^4+22161651840 x^3+21406565457 x^2+9181937962 x+1475586688\right )}{(3 x+2)^3 (5 x+3)^{3/2}}-437126283 \sqrt{7} \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{90552} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((7*Sqrt[1 - 2*x]*(1475586688 + 9181937962*x + 21406565457*x^2 + 22161651840*x^3 + 8596610325*x^4))/((2 + 3*x)
^3*(3 + 5*x)^(3/2)) - 437126283*Sqrt[7]*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/90552

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Maple [B]  time = 0.014, size = 298, normalized size = 1.8 \begin{align*}{\frac{1}{181104\, \left ( 2+3\,x \right ) ^{3}} \left ( 295060241025\,\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) \sqrt{7}{x}^{5}[phone]\,\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]\,{x}^{4}\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}^{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+299691916398\,{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 ) \sqrt{1-2\,x}{\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}} \left ( 3+5\,x \right ) ^{-{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/181104*(295060241025*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*7^(1/2)*x^5+944192771280*7^(1/2)*arc
tan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^4+1207779919929*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x
^2-x+3)^(1/2))*x^3+120352544550*x^4*(-10*x^2-x+3)^(1/2)[phone]*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-1
0*x^2-x+3)^(1/2))*x^2+310263125760*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+299691916398*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))*(1-2*x)^(1/2)/(2+3*x)
^3/(-10*x^2-x+3)^(1/2)/(3+5*x)^(3/2)

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Maxima [A]  time = 3.94544, size = 324, normalized size = 1.95 \begin{align*} \frac{13246251}{5488} \, \sqrt{7} \arcsin \left (\frac{37 \, x}{11 \,{\left | 3 \, x + 2 \right |}} + \frac{20}{11 \,{\left | 3 \, x + 2 \right |}}\right ) - \frac{63678595 \, x}{6468 \, \sqrt{-10 \, x^{2} - x + 3}} + \frac{66486521}{12936 \, \sqrt{-10 \, x^{2} - x + 3}} + \frac{207835 \, x}{84 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}} + \frac{49}{27 \,{\left (27 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} x^{3} + 54 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} x^{2} + 36 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} x + 8 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}\right )}} + \frac{77}{4 \,{\left (9 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} x^{2} + 12 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} x + 4 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}\right )}} + \frac{24617}{72 \,{\left (3 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} x + 2 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}\right )}} - \frac{2020657}{1512 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

13246251/5488*sqrt(7)*arcsin(37/11*x/abs(3*x + 2) + 20/11/abs(3*x + 2)) - 63678595/6468*x/sqrt(-10*x^2 - x + 3
) + 66486521/12936/sqrt(-10*x^2 - x + 3) + 207835/84*x/(-10*x^2 - x + 3)^(3/2) + 49/27/(27*(-10*x^2 - x + 3)^(
3/2)*x^3 + 54*(-10*x^2 - x + 3)^(3/2)*x^2 + 36*(-10*x^2 - x + 3)^(3/2)*x + 8*(-10*x^2 - x + 3)^(3/2)) + 77/4/(
9*(-10*x^2 - x + 3)^(3/2)*x^2 + 12*(-10*x^2 - x + 3)^(3/2)*x + 4*(-10*x^2 - x + 3)^(3/2)) + 24617/72/(3*(-10*x
^2 - x + 3)^(3/2)*x + 2*(-10*x^2 - x + 3)^(3/2)) - 2020657/1512/(-10*x^2 - x + 3)^(3/2)

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Fricas [A]  time = 1.57472, size = 455, normalized size = 2.74 \begin{align*} -\frac{437126283 \, \sqrt{7}{\left (675 \, x^{5} + 2160 \, x^{4} + 2763 \, x^{3} + 1766 \, x^{2} + 564 \, x + 72\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 (8596610325 \, x^{4} + 22161651840 \, x^{3} + 21406565457 \, x^{2} + 9181937962 \, x + 1475586688\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}{181104 \,{\left (675 \, x^{5} + 2160 \, x^{4} + 2763 \, x^{3} + 1766 \, x^{2} + 564 \, x + 72\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/181104*(437126283*sqrt(7)*(675*x^5 + 2160*x^4 + 2763*x^3 + 1766*x^2 + 564*x + 72)*arctan(1/14*sqrt(7)*(37*x
+ 20)*sqrt(5*x + 3)*sqrt(-2*x + 1)/(10*x^2 + x - 3)) - 14*(8596610325*x^4 + 22161651840*x^3 + 21406565457*x^2
+ 9181937962*x + 1475586688)*sqrt(5*x + 3)*sqrt(-2*x + 1))/(675*x^5 + 2160*x^4 + 2763*x^3 + 1766*x^2 + 564*x
+ 72)

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

[Out]

Timed out

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Giac [B]  time = 3.64985, size = 587, normalized size = 3.54 \begin{align*} -\frac{1}{1811040} \, \sqrt{5}{\left (85750 \, \sqrt{2}{\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} - 437126283 \, \sqrt{70} \sqrt{2}{\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 )} - 271656000 \, \sqrt{2}{\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 )} - \frac{2744280 \, \sqrt{2}{\left (22317 \,{\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} + 10704960 \,{\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} + \frac{1323627200 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}{\sqrt{5 \, x + 3}} - \frac{5294508800 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}}{{\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}}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/1811040*sqrt(5)*(85750*sqrt(2)*((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 - 437126283*sqrt(70)*sqrt(2)*(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)))) - 271656000*sqrt(2
)*((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)))
- 2744280*sqrt(2)*(22317*((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 + 10704960*((sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) - 4*sqrt(5*x + 3)/(sqr
t(2)*sqrt(-10*x + 5) - sqrt(22)))^3 + 1323627200*(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) - 52945088
00*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)