3.27.70 \(\int \frac {\sqrt {1-2 x} (3+5 x)^{5/2}}{(2+3 x)^{11/2}} \, dx\) [2670]

Optimal. Leaf size=222 \[ -\frac {12934 \sqrt {1-2 x} \sqrt {3+5 x}}{138915 (2+3 x)^{5/2}}+\frac {568318 \sqrt {1-2 x} \sqrt {3+5 x}}{2917215 (2+3 x)^{3/2}}+\frac {27198452 \sqrt {1-2 x} \sqrt {3+5 x}}{20420505 \sqrt {2+3 x}}-\frac {118 \sqrt {1-2 x} (3+5 x)^{3/2}}{1323 (2+3 x)^{7/2}}-\frac {2 \sqrt {1-2 x} (3+5 x)^{5/2}}{27 (2+3 x)^{9/2}}-\frac {27198452 \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{20420505}-\frac {442868 \sqrt {\frac {11}{3}} F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{20420505} \]

[Out]

-27198452/61261515*EllipticE(1/7*21^(1/2)*(1-2*x)^(1/2),1/33*1155^(1/2))*33^(1/2)-442868/61261515*EllipticF(1/
7*21^(1/2)*(1-2*x)^(1/2),1/33*1155^(1/2))*33^(1/2)-118/1323*(3+5*x)^(3/2)*(1-2*x)^(1/2)/(2+3*x)^(7/2)-2/27*(3+
5*x)^(5/2)*(1-2*x)^(1/2)/(2+3*x)^(9/2)-12934/138915*(1-2*x)^(1/2)*(3+5*x)^(1/2)/(2+3*x)^(5/2)+568318/2917215*(
1-2*x)^(1/2)*(3+5*x)^(1/2)/(2+3*x)^(3/2)[phone]/20420505*(1-2*x)^(1/2)*(3+5*x)^(1/2)/(2+3*x)^(1/2)

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Rubi [A]
time = 0.05, antiderivative size = 222, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {99, 155, 157, 164, 114, 120} \begin {gather*} -\frac {442868 \sqrt {\frac {11}{3}} F\left (\text {ArcSin}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{20420505}-\frac {27198452 \sqrt {\frac {11}{3}} E\left (\text {ArcSin}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{20420505}-\frac {2 \sqrt {1-2 x} (5 x+3)^{5/2}}{27 (3 x+2)^{9/2}}-\frac {118 \sqrt {1-2 x} (5 x+3)^{3/2}}{1323 (3 x+2)^{7/2}}+\frac {27198452 \sqrt {1-2 x} \sqrt {5 x+3}}{20420505 \sqrt {3 x+2}}+\frac {568318 \sqrt {1-2 x} \sqrt {5 x+3}}{2917215 (3 x+2)^{3/2}}-\frac {12934 \sqrt {1-2 x} \sqrt {5 x+3}}{138915 (3 x+2)^{5/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(Sqrt[1 - 2*x]*(3 + 5*x)^(5/2))/(2 + 3*x)^(11/2),x]

[Out]

(-12934*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(138915*(2 + 3*x)^(5/2)) + (568318*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(2917215*
(2 + 3*x)^(3/2)) + (27198452*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(20420505*Sqrt[2 + 3*x]) - (118*Sqrt[1 - 2*x]*(3 + 5
*x)^(3/2))/(1323*(2 + 3*x)^(7/2)) - (2*Sqrt[1 - 2*x]*(3 + 5*x)^(5/2))/(27*(2 + 3*x)^(9/2)) - (27198452*Sqrt[11
/3]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/20420505 - (442868*Sqrt[11/3]*EllipticF[ArcSin[Sqrt[3/7
]*Sqrt[1 - 2*x]], 35/33])/20420505

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/(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 114

Int[Sqrt[(e_.) + (f_.)*(x_)]/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]), x_Symbol] :> Simp[(2/b)*Rt[-(b
*e - a*f)/d, 2]*EllipticE[ArcSin[Sqrt[a + b*x]/Rt[-(b*c - a*d)/d, 2]], f*((b*c - a*d)/(d*(b*e - a*f)))], x] /;
FreeQ[{a, b, c, d, e, f}, x] && GtQ[b/(b*c - a*d), 0] && GtQ[b/(b*e - a*f), 0] &&  !LtQ[-(b*c - a*d)/d, 0] &&
!(SimplerQ[c + d*x, a + b*x] && GtQ[-d/(b*c - a*d), 0] && GtQ[d/(d*e - c*f), 0] &&  !LtQ[(b*c - a*d)/b, 0])

Rule 120

Int[1/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(e_) + (f_.)*(x_)]), x_Symbol] :> Simp[2*(Rt[-b/d,
2]/(b*Sqrt[(b*e - a*f)/b]))*EllipticF[ArcSin[Sqrt[a + b*x]/(Rt[-b/d, 2]*Sqrt[(b*c - a*d)/b])], f*((b*c - a*d)
/(d*(b*e - a*f)))], x] /; FreeQ[{a, b, c, d, e, f}, x] && GtQ[(b*c - a*d)/b, 0] && GtQ[(b*e - a*f)/b, 0] && Po
sQ[-b/d] &&  !(SimplerQ[c + d*x, a + b*x] && GtQ[(d*e - c*f)/d, 0] && GtQ[-d/b, 0]) &&  !(SimplerQ[c + d*x, a
+ b*x] && GtQ[((-b)*e + a*f)/f, 0] && GtQ[-f/b, 0]) &&  !(SimplerQ[e + f*x, a + b*x] && GtQ[((-d)*e + c*f)/f,
0] && GtQ[((-b)*e + a*f)/f, 0] && (PosQ[-f/d] || PosQ[-f/b]))

Rule 155

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] && IntegersQ[2*m, 2*n, 2*p]

Rule 157

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 164

Int[((g_.) + (h_.)*(x_))/(Sqrt[(a_.) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(e_) + (f_.)*(x_)]), x_Symbol]
:> Dist[h/f, Int[Sqrt[e + f*x]/(Sqrt[a + b*x]*Sqrt[c + d*x]), x], x] + Dist[(f*g - e*h)/f, Int[1/(Sqrt[a + b*
x]*Sqrt[c + d*x]*Sqrt[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f, g, h}, x] && SimplerQ[a + b*x, e + f*x] &&
SimplerQ[c + d*x, e + f*x]

Rubi steps

\begin {align*} \int \frac {\sqrt {1-2 x} (3+5 x)^{5/2}}{(2+3 x)^{11/2}} \, dx &=-\frac {2 \sqrt {1-2 x} (3+5 x)^{5/2}}{27 (2+3 x)^{9/2}}+\frac {2}{27} \int \frac {\left (\frac {19}{2}-30 x\right ) (3+5 x)^{3/2}}{\sqrt {1-2 x} (2+3 x)^{9/2}} \, dx\\ &=-\frac {118 \sqrt {1-2 x} (3+5 x)^{3/2}}{1323 (2+3 x)^{7/2}}-\frac {2 \sqrt {1-2 x} (3+5 x)^{5/2}}{27 (2+3 x)^{9/2}}+\frac {4 \int \frac {\left (\frac {207}{4}-\frac {4695 x}{2}\right ) \sqrt {3+5 x}}{\sqrt {1-2 x} (2+3 x)^{7/2}} \, dx}{3969}\\ &=-\frac {12934 \sqrt {1-2 x} \sqrt {3+5 x}}{138915 (2+3 x)^{5/2}}-\frac {118 \sqrt {1-2 x} (3+5 x)^{3/2}}{1323 (2+3 x)^{7/2}}-\frac {2 \sqrt {1-2 x} (3+5 x)^{5/2}}{27 (2+3 x)^{9/2}}+\frac {8 \int \frac {-\frac {423321}{8}-\frac {265305 x}{2}}{\sqrt {1-2 x} (2+3 x)^{5/2} \sqrt {3+5 x}} \, dx}{416745}\\ &=-\frac {12934 \sqrt {1-2 x} \sqrt {3+5 x}}{138915 (2+3 x)^{5/2}}+\frac {568318 \sqrt {1-2 x} \sqrt {3+5 x}}{2917215 (2+3 x)^{3/2}}-\frac {118 \sqrt {1-2 x} (3+5 x)^{3/2}}{1323 (2+3 x)^{7/2}}-\frac {2 \sqrt {1-2 x} (3+5 x)^{5/2}}{27 (2+3 x)^{9/2}}+\frac {16 \int \frac {\frac {3958023}{8}-\frac {4262385 x}{8}}{\sqrt {1-2 x} (2+3 x)^{3/2} \sqrt {3+5 x}} \, dx}{8751645}\\ &=-\frac {12934 \sqrt {1-2 x} \sqrt {3+5 x}}{138915 (2+3 x)^{5/2}}+\frac {568318 \sqrt {1-2 x} \sqrt {3+5 x}}{2917215 (2+3 x)^{3/2}}+\frac {27198452 \sqrt {1-2 x} \sqrt {3+5 x}}{20420505 \sqrt {2+3 x}}-\frac {118 \sqrt {1-2 x} (3+5 x)^{3/2}}{1323 (2+3 x)^{7/2}}-\frac {2 \sqrt {1-2 x} (3+5 x)^{5/2}}{27 (2+3 x)^{9/2}}+\frac {32 \int \frac {\frac {126046695}{16}+\frac {101994195 x}{8}}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{61261515}\\ &=-\frac {12934 \sqrt {1-2 x} \sqrt {3+5 x}}{138915 (2+3 x)^{5/2}}+\frac {568318 \sqrt {1-2 x} \sqrt {3+5 x}}{2917215 (2+3 x)^{3/2}}+\frac {27198452 \sqrt {1-2 x} \sqrt {3+5 x}}{20420505 \sqrt {2+3 x}}-\frac {118 \sqrt {1-2 x} (3+5 x)^{3/2}}{1323 (2+3 x)^{7/2}}-\frac {2 \sqrt {1-2 x} (3+5 x)^{5/2}}{27 (2+3 x)^{9/2}}+\frac {2435774 \int \frac {1}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{20420505}+\frac {27198452 \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx}{20420505}\\ &=-\frac {12934 \sqrt {1-2 x} \sqrt {3+5 x}}{138915 (2+3 x)^{5/2}}+\frac {568318 \sqrt {1-2 x} \sqrt {3+5 x}}{2917215 (2+3 x)^{3/2}}+\frac {27198452 \sqrt {1-2 x} \sqrt {3+5 x}}{20420505 \sqrt {2+3 x}}-\frac {118 \sqrt {1-2 x} (3+5 x)^{3/2}}{1323 (2+3 x)^{7/2}}-\frac {2 \sqrt {1-2 x} (3+5 x)^{5/2}}{27 (2+3 x)^{9/2}}-\frac {27198452 \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{20420505}-\frac {442868 \sqrt {\frac {11}{3}} F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{20420505}\\ \end {align*}

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Mathematica [A]
time = 5.07, size = 110, normalized size = 0.50 \begin {gather*} \frac {\frac {24 \sqrt {1-2 x} \sqrt {3+5 x} \left (217427099+1325733891 x+3003721227 x^2+2991138867 x^3+1101537306 x^4\right )}{(2+3 x)^{9/2}}+8 \sqrt {2} \left (13599226 E\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )|-\frac {33}{2}\right )-9945565 F\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )|-\frac {33}{2}\right )\right )}{245046060} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(Sqrt[1 - 2*x]*(3 + 5*x)^(5/2))/(2 + 3*x)^(11/2),x]

[Out]

((24*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(217427099 + 1325733891*x + 3003721227*x^2 + 2991138867*x^3 + 1101537306*x^4)
)/(2 + 3*x)^(9/2) + 8*Sqrt[2]*(13599226*EllipticE[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2] - 9945565*EllipticF
[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2]))/245046060

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(493\) vs. \(2(162)=324\).
time = 0.09, size = 494, normalized size = 2.23

method	result	size
elliptic	\(\frac {\sqrt {-\left (3+5 x \right ) \left (-1+2 x \right ) \left (2+3 x \right )}\, \left (-\frac {186502 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{11252115 \left (\frac {2}{3}+x \right )^{3}}+\frac {568318 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{26254935 \left (\frac {2}{3}+x \right )^{2}}+\frac {-\frac {54396904}{4084101} x^{2}-\frac {27198452}{20420505} x +\frac {27198452}{6806835}}{\sqrt {\left (\frac {2}{3}+x \right ) \left (-30 x^{2}-3 x +9\right )}}+\frac {16806226 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{85766121 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}+\frac {27198452 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \left (-\frac {\EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{15}-\frac {3 \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{5}\right )}{85766121 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}-\frac {2 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{59049 \left (\frac {2}{3}+x \right )^{5}}+\frac {1334 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{964467 \left (\frac {2}{3}+x \right )^{4}}\right )}{\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}\)	\(297\)
default	\(-\frac {2 \left (295946541 \sqrt {2}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x^{4} \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}-1101537306 \sqrt {2}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x^{4} \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}[phone] \sqrt {2}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x^{3} \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}-2937432816 \sqrt {2}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x^{3} \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}[phone] \sqrt {2}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x^{2} \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}-2937432816 \sqrt {2}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x^{2} \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}[phone] \sqrt {2}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}-1305525696 \sqrt {2}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}-33046119180 x^{6}[phone] \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )-217587616 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )-93038777928 x^{5}-89171217657 x^{4}-21862930608 x^{3}[phone] x^{2}[phone] x [phone]\right ) \sqrt {1-2 x}\, \sqrt {3+5 x}}{61261515 \left (10 x^{2}+x -3\right ) \left (2+3 x \right )^{\frac {9}{2}}}\)	\(494\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((3+5*x)^(5/2)*(1-2*x)^(1/2)/(2+3*x)^(11/2),x,method=_RETURNVERBOSE)

[Out]

-2/61261515*(295946541*2^(1/2)*EllipticF(1/7*(28+42*x)^(1/2),1/2*70^(1/2))*x^4*(2+3*x)^(1/2)*(-3-5*x)^(1/2)*(1
-2*x)^(1/2)-1101537306*2^(1/2)*EllipticE(1/7*(28+42*x)^(1/2),1/2*70^(1/2))*x^4*(2+3*x)^(1/2)*(-3-5*x)^(1/2)*(1
-2*x)^(1/2)[phone]*2^(1/2)*EllipticF(1/7*(28+42*x)^(1/2),1/2*70^(1/2))*x^3*(2+3*x)^(1/2)*(-3-5*x)^(1/2)*(1-
2*x)^(1/2)-2937432816*2^(1/2)*EllipticE(1/7*(28+42*x)^(1/2),1/2*70^(1/2))*x^3*(2+3*x)^(1/2)*(-3-5*x)^(1/2)*(1-
2*x)^(1/2)[phone]*2^(1/2)*EllipticF(1/7*(28+42*x)^(1/2),1/2*70^(1/2))*x^2*(2+3*x)^(1/2)*(-3-5*x)^(1/2)*(1-2
*x)^(1/2)-2937432816*2^(1/2)*EllipticE(1/7*(28+42*x)^(1/2),1/2*70^(1/2))*x^2*(2+3*x)^(1/2)*(-3-5*x)^(1/2)*(1-2
*x)^(1/2)[phone]*2^(1/2)*EllipticF(1/7*(28+42*x)^(1/2),1/2*70^(1/2))*x*(2+3*x)^(1/2)*(-3-5*x)^(1/2)*(1-2*x)
^(1/2)-1305525696*2^(1/2)*EllipticE(1/7*(28+42*x)^(1/2),1/2*70^(1/2))*x*(2+3*x)^(1/2)*(-3-5*x)^(1/2)*(1-2*x)^(
1/2)-33046119180*x^6+58458576*2^(1/2)*(2+3*x)^(1/2)*(-3-5*x)^(1/2)*(1-2*x)^(1/2)*EllipticF(1/7*(28+42*x)^(1/2)
,1/2*70^(1/2))-217587616*2^(1/2)*(2+3*x)^(1/2)*(-3-5*x)^(1/2)*(1-2*x)^(1/2)*EllipticE(1/7*(28+42*x)^(1/2),1/2*
70^(1/2))-93038777928*x^5-89171217657*x^4-21862930608*x^3+16533476400*x^2+11279323722*x+1956843891)*(1-2*x)^(1
/2)*(3+5*x)^(1/2)/(10*x^2+x-3)/(2+3*x)^(9/2)

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3+5*x)^(5/2)*(1-2*x)^(1/2)/(2+3*x)^(11/2),x, algorithm="maxima")

[Out]

integrate((5*x + 3)^(5/2)*sqrt(-2*x + 1)/(3*x + 2)^(11/2), x)

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Fricas [A]
time = 0.21, size = 70, normalized size = 0.32 \begin {gather*} \frac {2 \, {\left (1101537306 \, x^{4} + 2991138867 \, x^{3} + 3003721227 \, x^{2} + 1325733891 \, x + 217427099\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}}{20420505 \, {\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3+5*x)^(5/2)*(1-2*x)^(1/2)/(2+3*x)^(11/2),x, algorithm="fricas")

[Out]

2/20420505*(1101537306*x^4 + 2991138867*x^3 + 3003721227*x^2 + 1325733891*x + 217427099)*sqrt(5*x + 3)*sqrt(3*
x + 2)*sqrt(-2*x + 1)/(243*x^5 + 810*x^4 + 1080*x^3 + 720*x^2 + 240*x + 32)

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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3+5*x)**(5/2)*(1-2*x)**(1/2)/(2+3*x)**(11/2),x)

[Out]

Exception raised: SystemError >> excessive stack use: stack is 8856 deep

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3+5*x)^(5/2)*(1-2*x)^(1/2)/(2+3*x)^(11/2),x, algorithm="giac")

[Out]

integrate((5*x + 3)^(5/2)*sqrt(-2*x + 1)/(3*x + 2)^(11/2), x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {\sqrt {1-2\,x}\,{\left (5\,x+3\right )}^{5/2}}{{\left (3\,x+2\right )}^{11/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((1 - 2*x)^(1/2)*(5*x + 3)^(5/2))/(3*x + 2)^(11/2),x)

[Out]

int(((1 - 2*x)^(1/2)*(5*x + 3)^(5/2))/(3*x + 2)^(11/2), x)

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