3.30.17 \(\int \frac {(3+5 x)^{5/2}}{(1-2 x)^{3/2} (2+3 x)^{11/2}} \, dx\) [2917]

Optimal. Leaf size=253 \[ \frac {295 \sqrt {1-2 x} \sqrt {3+5 x}}{1323 (2+3 x)^{9/2}}-\frac {67345 \sqrt {1-2 x} \sqrt {3+5 x}}{64827 (2+3 x)^{7/2}}-\frac {167228 \sqrt {1-2 x} \sqrt {3+5 x}}{453789 (2+3 x)^{5/2}}-\frac {392998 \sqrt {1-2 x} \sqrt {3+5 x}}{3176523 (2+3 x)^{3/2}}+\frac {6036028 \sqrt {1-2 x} \sqrt {3+5 x}}{22235661 \sqrt {2+3 x}}+\frac {11 (3+5 x)^{3/2}}{7 \sqrt {1-2 x} (2+3 x)^{9/2}}-\frac {6036028 \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{22235661}-\frac {1199452 \sqrt {\frac {11}{3}} F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{22235661} \]

[Out]

-6036028/66706983*EllipticE(1/7*21^(1/2)*(1-2*x)^(1/2),1/33*1155^(1/2))*33^(1/2)-1199452/66706983*EllipticF(1/
7*21^(1/2)*(1-2*x)^(1/2),1/33*1155^(1/2))*33^(1/2)+11/7*(3+5*x)^(3/2)/(2+3*x)^(9/2)/(1-2*x)^(1/2)+295/1323*(1-
2*x)^(1/2)*(3+5*x)^(1/2)/(2+3*x)^(9/2)-67345/64827*(1-2*x)^(1/2)*(3+5*x)^(1/2)/(2+3*x)^(7/2)-167228/453789*(1-
2*x)^(1/2)*(3+5*x)^(1/2)/(2+3*x)^(5/2)-392998/3176523*(1-2*x)^(1/2)*(3+5*x)^(1/2)/(2+3*x)^(3/2)+6036028/222356
61*(1-2*x)^(1/2)*(3+5*x)^(1/2)/(2+3*x)^(1/2)

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Rubi [A]
time = 0.07, antiderivative size = 253, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 6, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {100, 155, 157, 164, 114, 120} \begin {gather*} -\frac {1199452 \sqrt {\frac {11}{3}} F\left (\text {ArcSin}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{22235661}-\frac {6036028 \sqrt {\frac {11}{3}} E\left (\text {ArcSin}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{22235661}+\frac {11 (5 x+3)^{3/2}}{7 \sqrt {1-2 x} (3 x+2)^{9/2}}+\frac {6036028 \sqrt {1-2 x} \sqrt {5 x+3}}{22235661 \sqrt {3 x+2}}-\frac {392998 \sqrt {1-2 x} \sqrt {5 x+3}}{3176523 (3 x+2)^{3/2}}-\frac {167228 \sqrt {1-2 x} \sqrt {5 x+3}}{453789 (3 x+2)^{5/2}}-\frac {67345 \sqrt {1-2 x} \sqrt {5 x+3}}{64827 (3 x+2)^{7/2}}+\frac {295 \sqrt {1-2 x} \sqrt {5 x+3}}{1323 (3 x+2)^{9/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(295*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(1323*(2 + 3*x)^(9/2)) - (67345*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(64827*(2 + 3*x
)^(7/2)) - (167228*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(453789*(2 + 3*x)^(5/2)) - (392998*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]
)/(3176523*(2 + 3*x)^(3/2)) + (6036028*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(22235661*Sqrt[2 + 3*x]) + (11*(3 + 5*x)^(
3/2))/(7*Sqrt[1 - 2*x]*(2 + 3*x)^(9/2)) - (6036028*Sqrt[11/3]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33
])/22235661 - (1199452*Sqrt[11/3]*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/22235661

Rule 100

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 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 {(3+5 x)^{5/2}}{(1-2 x)^{3/2} (2+3 x)^{11/2}} \, dx &=\frac {11 (3+5 x)^{3/2}}{7 \sqrt {1-2 x} (2+3 x)^{9/2}}-\frac {1}{7} \int \frac {\left (-\frac {555}{2}-490 x\right ) \sqrt {3+5 x}}{\sqrt {1-2 x} (2+3 x)^{11/2}} \, dx\\ &=\frac {295 \sqrt {1-2 x} \sqrt {3+5 x}}{1323 (2+3 x)^{9/2}}+\frac {11 (3+5 x)^{3/2}}{7 \sqrt {1-2 x} (2+3 x)^{9/2}}-\frac {2 \int \frac {-\frac {169585}{4}-\frac {144025 x}{2}}{\sqrt {1-2 x} (2+3 x)^{9/2} \sqrt {3+5 x}} \, dx}{1323}\\ &=\frac {295 \sqrt {1-2 x} \sqrt {3+5 x}}{1323 (2+3 x)^{9/2}}-\frac {67345 \sqrt {1-2 x} \sqrt {3+5 x}}{64827 (2+3 x)^{7/2}}+\frac {11 (3+5 x)^{3/2}}{7 \sqrt {1-2 x} (2+3 x)^{9/2}}-\frac {4 \int \frac {-245765-\frac {1683625 x}{4}}{\sqrt {1-2 x} (2+3 x)^{7/2} \sqrt {3+5 x}} \, dx}{64827}\\ &=\frac {295 \sqrt {1-2 x} \sqrt {3+5 x}}{1323 (2+3 x)^{9/2}}-\frac {67345 \sqrt {1-2 x} \sqrt {3+5 x}}{64827 (2+3 x)^{7/2}}-\frac {167228 \sqrt {1-2 x} \sqrt {3+5 x}}{453789 (2+3 x)^{5/2}}+\frac {11 (3+5 x)^{3/2}}{7 \sqrt {1-2 x} (2+3 x)^{9/2}}-\frac {8 \int \frac {-\frac {7378905}{8}-\frac {3135525 x}{2}}{\sqrt {1-2 x} (2+3 x)^{5/2} \sqrt {3+5 x}} \, dx}{2268945}\\ &=\frac {295 \sqrt {1-2 x} \sqrt {3+5 x}}{1323 (2+3 x)^{9/2}}-\frac {67345 \sqrt {1-2 x} \sqrt {3+5 x}}{64827 (2+3 x)^{7/2}}-\frac {167228 \sqrt {1-2 x} \sqrt {3+5 x}}{453789 (2+3 x)^{5/2}}-\frac {392998 \sqrt {1-2 x} \sqrt {3+5 x}}{3176523 (2+3 x)^{3/2}}+\frac {11 (3+5 x)^{3/2}}{7 \sqrt {1-2 x} (2+3 x)^{9/2}}-\frac {16 \int \frac {-\frac {17369985}{8}-\frac {14737425 x}{8}}{\sqrt {1-2 x} (2+3 x)^{3/2} \sqrt {3+5 x}} \, dx}{47647845}\\ &=\frac {295 \sqrt {1-2 x} \sqrt {3+5 x}}{1323 (2+3 x)^{9/2}}-\frac {67345 \sqrt {1-2 x} \sqrt {3+5 x}}{64827 (2+3 x)^{7/2}}-\frac {167228 \sqrt {1-2 x} \sqrt {3+5 x}}{453789 (2+3 x)^{5/2}}-\frac {392998 \sqrt {1-2 x} \sqrt {3+5 x}}{3176523 (2+3 x)^{3/2}}+\frac {6036028 \sqrt {1-2 x} \sqrt {3+5 x}}{22235661 \sqrt {2+3 x}}+\frac {11 (3+5 x)^{3/2}}{7 \sqrt {1-2 x} (2+3 x)^{9/2}}-\frac {32 \int \frac {-\frac {185288025}{16}-\frac {113175525 x}{8}}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{333534915}\\ &=\frac {295 \sqrt {1-2 x} \sqrt {3+5 x}}{1323 (2+3 x)^{9/2}}-\frac {67345 \sqrt {1-2 x} \sqrt {3+5 x}}{64827 (2+3 x)^{7/2}}-\frac {167228 \sqrt {1-2 x} \sqrt {3+5 x}}{453789 (2+3 x)^{5/2}}-\frac {392998 \sqrt {1-2 x} \sqrt {3+5 x}}{3176523 (2+3 x)^{3/2}}+\frac {6036028 \sqrt {1-2 x} \sqrt {3+5 x}}{22235661 \sqrt {2+3 x}}+\frac {11 (3+5 x)^{3/2}}{7 \sqrt {1-2 x} (2+3 x)^{9/2}}+\frac {6036028 \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx}{22235661}+\frac {6596986 \int \frac {1}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{22235661}\\ &=\frac {295 \sqrt {1-2 x} \sqrt {3+5 x}}{1323 (2+3 x)^{9/2}}-\frac {67345 \sqrt {1-2 x} \sqrt {3+5 x}}{64827 (2+3 x)^{7/2}}-\frac {167228 \sqrt {1-2 x} \sqrt {3+5 x}}{453789 (2+3 x)^{5/2}}-\frac {392998 \sqrt {1-2 x} \sqrt {3+5 x}}{3176523 (2+3 x)^{3/2}}+\frac {6036028 \sqrt {1-2 x} \sqrt {3+5 x}}{22235661 \sqrt {2+3 x}}+\frac {11 (3+5 x)^{3/2}}{7 \sqrt {1-2 x} (2+3 x)^{9/2}}-\frac {6036028 \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{22235661}-\frac {1199452 \sqrt {\frac {11}{3}} F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{22235661}\\ \end {align*}

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Mathematica [A]
time = 8.25, size = 115, normalized size = 0.45 \begin {gather*} \frac {-\frac {24 \sqrt {3+5 x} \left (-52688263-243200677 x-227945505 x^2+466728543 x^3+985046292 x^4+488918268 x^5\right )}{\sqrt {1-2 x} (2+3 x)^{9/2}}+8 \sqrt {2} \left (3018014 E\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )|-\frac {33}{2}\right )+6877465 F\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )|-\frac {33}{2}\right )\right )}{266827932} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((-24*Sqrt[3 + 5*x]*(-52688263 - 243200677*x - 227945505*x^2 + 466728543*x^3 + 985046292*x^4 + 488918268*x^5))
/(Sqrt[1 - 2*x]*(2 + 3*x)^(9/2)) + 8*Sqrt[2]*(3018014*EllipticE[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2] + 687
7465*EllipticF[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2]))/266827932

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(493\) vs. \(2(185)=370\).
time = 0.11, size = 494, normalized size = 1.95

method	result	size
elliptic	\(\frac {\sqrt {-\left (3+5 x \right ) \left (-1+2 x \right ) \left (2+3 x \right )}\, \left (-\frac {2 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{321489 \left (\frac {2}{3}+x \right )^{5}}-\frac {968 \left (-30 x^{2}-38 x -12\right )}{117649 \sqrt {\left (-\frac {1}{2}+x \right ) \left (-30 x^{2}-38 x -12\right )}}-\frac {30014 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{12252303 \left (\frac {2}{3}+x \right )^{3}}+\frac {1262 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{5250987 \left (\frac {2}{3}+x \right )^{4}}+\frac {-\frac {65848840}{22235661} x^{2}-\frac {6584884}{22235661} x +\frac {6584884}{7411887}}{\sqrt {\left (\frac {2}{3}+x \right ) \left (-30 x^{2}-3 x +9\right )}}-\frac {118570 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{28588707 \left (\frac {2}{3}+x \right )^{2}}+\frac {24705070 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{466948881 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}+\frac {30180140 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \left (-\frac {\EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{15}-\frac {3 \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{5}\right )}{466948881 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}\right )}{\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}\)	\(325\)
default	\(-\frac {2 \sqrt {3+5 x}\, \sqrt {1-2 x}\, \left (801533799 \sqrt {2}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x^{4} \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}-244459134 \sqrt {2}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x^{4} \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}[phone] \sqrt {2}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x^{3} \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}-651891024 \sqrt {2}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x^{3} \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}[phone] \sqrt {2}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x^{2} \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}-651891024 \sqrt {2}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x^{2} \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}[phone] \sqrt {2}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}-289729344 \sqrt {2}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}-7333774020 x^{6}[phone] \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )-48288224 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )-19175958792 x^{5}-15866344773 x^{4}-781374312 x^{3}[phone] x^{2}[phone] x [phone]\right )}{66706983 \left (2+3 x \right )^{\frac {9}{2}} \left (10 x^{2}+x -3\right )}\)	\(494\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/66706983*(3+5*x)^(1/2)*(1-2*x)^(1/2)*(801533799*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)-244459134*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)-651891024*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)-651891024*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)-289729344*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)-7333774020*x^6+158327664*2^(1/2)*(2+3*x)^(1/2)*(-3-5*x)^(1/2)*(1-2*x)^(1/2)*Ell
ipticF(1/7*(28+42*x)^(1/2),1/2*70^(1/2))-48288224*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))-19175958792*x^5-15866344773*x^4-781374312*x^3+5699519700*x^2+2979130038*x+4
74194367)/(2+3*x)^(9/2)/(10*x^2+x-3)

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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)^(3/2)/(2+3*x)^(11/2),x, algorithm="maxima")

[Out]

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

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Fricas [A]
time = 0.21, size = 80, normalized size = 0.32 \begin {gather*} \frac {2 \, {\left (488918268 \, x^{5} + 985046292 \, x^{4} + 466728543 \, x^{3} - 227945505 \, x^{2} - 243200677 \, x - 52688263\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}}{22235661 \, {\left (486 \, x^{6} + 1377 \, x^{5} + 1350 \, x^{4} + 360 \, x^{3} - 240 \, x^{2} - 176 \, 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)^(3/2)/(2+3*x)^(11/2),x, algorithm="fricas")

[Out]

2/22235661*(488918268*x^5 + 985046292*x^4 + 466728543*x^3 - 227945505*x^2 - 243200677*x - 52688263)*sqrt(5*x +
3)*sqrt(3*x + 2)*sqrt(-2*x + 1)/(486*x^6 + 1377*x^5 + 1350*x^4 + 360*x^3 - 240*x^2 - 176*x - 32)

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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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)^(3/2)/(2+3*x)^(11/2),x, algorithm="giac")

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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