3.30.70 \(\int \frac {(3+5 x)^{5/2}}{(1-2 x)^{5/2} (2+3 x)^{9/2}} \, dx\) [2970]

Optimal. Leaf size=253 \[ \frac {220 \sqrt {3+5 x}}{49 \sqrt {1-2 x} (2+3 x)^{7/2}}-\frac {4545 \sqrt {1-2 x} \sqrt {3+5 x}}{2401 (2+3 x)^{7/2}}-\frac {11433 \sqrt {1-2 x} \sqrt {3+5 x}}{16807 (2+3 x)^{5/2}}-\frac {33778 \sqrt {1-2 x} \sqrt {3+5 x}}{117649 (2+3 x)^{3/2}}-\frac {98642 \sqrt {1-2 x} \sqrt {3+5 x}}{823543 \sqrt {2+3 x}}+\frac {11 (3+5 x)^{3/2}}{21 (1-2 x)^{3/2} (2+3 x)^{7/2}}+\frac {98642 \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{823543}-\frac {65672 \sqrt {\frac {11}{3}} F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{823543} \]

[Out]

11/21*(3+5*x)^(3/2)/(1-2*x)^(3/2)/(2+3*x)^(7/2)+98642/2470629*EllipticE(1/7*21^(1/2)*(1-2*x)^(1/2),1/33*1155^(
1/2))*33^(1/2)-65672/2470629*EllipticF(1/7*21^(1/2)*(1-2*x)^(1/2),1/33*1155^(1/2))*33^(1/2)+220/49*(3+5*x)^(1/
2)/(2+3*x)^(7/2)/(1-2*x)^(1/2)-4545/2401*(1-2*x)^(1/2)*(3+5*x)^(1/2)/(2+3*x)^(7/2)-11433/16807*(1-2*x)^(1/2)*(
3+5*x)^(1/2)/(2+3*x)^(5/2)-33778/117649*(1-2*x)^(1/2)*(3+5*x)^(1/2)/(2+3*x)^(3/2)-98642/823543*(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 {65672 \sqrt {\frac {11}{3}} F\left (\text {ArcSin}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{823543}+\frac {98642 \sqrt {\frac {11}{3}} E\left (\text {ArcSin}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{823543}+\frac {11 (5 x+3)^{3/2}}{21 (1-2 x)^{3/2} (3 x+2)^{7/2}}-\frac {98642 \sqrt {1-2 x} \sqrt {5 x+3}}{823543 \sqrt {3 x+2}}-\frac {33778 \sqrt {1-2 x} \sqrt {5 x+3}}{117649 (3 x+2)^{3/2}}-\frac {11433 \sqrt {1-2 x} \sqrt {5 x+3}}{16807 (3 x+2)^{5/2}}-\frac {4545 \sqrt {1-2 x} \sqrt {5 x+3}}{2401 (3 x+2)^{7/2}}+\frac {220 \sqrt {5 x+3}}{49 \sqrt {1-2 x} (3 x+2)^{7/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(3 + 5*x)^(5/2)/((1 - 2*x)^(5/2)*(2 + 3*x)^(9/2)),x]

[Out]

(220*Sqrt[3 + 5*x])/(49*Sqrt[1 - 2*x]*(2 + 3*x)^(7/2)) - (4545*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(2401*(2 + 3*x)^(7
/2)) - (11433*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(16807*(2 + 3*x)^(5/2)) - (33778*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(1176
49*(2 + 3*x)^(3/2)) - (98642*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(823543*Sqrt[2 + 3*x]) + (11*(3 + 5*x)^(3/2))/(21*(1
- 2*x)^(3/2)*(2 + 3*x)^(7/2)) + (98642*Sqrt[11/3]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/823543 -
(65672*Sqrt[11/3]*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/823543

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)^{5/2} (2+3 x)^{9/2}} \, dx &=\frac {11 (3+5 x)^{3/2}}{21 (1-2 x)^{3/2} (2+3 x)^{7/2}}-\frac {1}{21} \int \frac {\left (-\frac {345}{2}-315 x\right ) \sqrt {3+5 x}}{(1-2 x)^{3/2} (2+3 x)^{9/2}} \, dx\\ &=\frac {220 \sqrt {3+5 x}}{49 \sqrt {1-2 x} (2+3 x)^{7/2}}+\frac {11 (3+5 x)^{3/2}}{21 (1-2 x)^{3/2} (2+3 x)^{7/2}}-\frac {1}{147} \int \frac {-\frac {34305}{2}-\frac {58275 x}{2}}{\sqrt {1-2 x} (2+3 x)^{9/2} \sqrt {3+5 x}} \, dx\\ &=\frac {220 \sqrt {3+5 x}}{49 \sqrt {1-2 x} (2+3 x)^{7/2}}-\frac {4545 \sqrt {1-2 x} \sqrt {3+5 x}}{2401 (2+3 x)^{7/2}}+\frac {11 (3+5 x)^{3/2}}{21 (1-2 x)^{3/2} (2+3 x)^{7/2}}-\frac {2 \int \frac {-\frac {397335}{4}-\frac {340875 x}{2}}{\sqrt {1-2 x} (2+3 x)^{7/2} \sqrt {3+5 x}} \, dx}{7203}\\ &=\frac {220 \sqrt {3+5 x}}{49 \sqrt {1-2 x} (2+3 x)^{7/2}}-\frac {4545 \sqrt {1-2 x} \sqrt {3+5 x}}{2401 (2+3 x)^{7/2}}-\frac {11433 \sqrt {1-2 x} \sqrt {3+5 x}}{16807 (2+3 x)^{5/2}}+\frac {11 (3+5 x)^{3/2}}{21 (1-2 x)^{3/2} (2+3 x)^{7/2}}-\frac {4 \int \frac {-\frac {1461615}{4}-\frac {2572425 x}{4}}{\sqrt {1-2 x} (2+3 x)^{5/2} \sqrt {3+5 x}} \, dx}{252105}\\ &=\frac {220 \sqrt {3+5 x}}{49 \sqrt {1-2 x} (2+3 x)^{7/2}}-\frac {4545 \sqrt {1-2 x} \sqrt {3+5 x}}{2401 (2+3 x)^{7/2}}-\frac {11433 \sqrt {1-2 x} \sqrt {3+5 x}}{16807 (2+3 x)^{5/2}}-\frac {33778 \sqrt {1-2 x} \sqrt {3+5 x}}{117649 (2+3 x)^{3/2}}+\frac {11 (3+5 x)^{3/2}}{21 (1-2 x)^{3/2} (2+3 x)^{7/2}}-\frac {8 \int \frac {-\frac {4326885}{8}-\frac {3800025 x}{4}}{\sqrt {1-2 x} (2+3 x)^{3/2} \sqrt {3+5 x}} \, dx}{5294205}\\ &=\frac {220 \sqrt {3+5 x}}{49 \sqrt {1-2 x} (2+3 x)^{7/2}}-\frac {4545 \sqrt {1-2 x} \sqrt {3+5 x}}{2401 (2+3 x)^{7/2}}-\frac {11433 \sqrt {1-2 x} \sqrt {3+5 x}}{16807 (2+3 x)^{5/2}}-\frac {33778 \sqrt {1-2 x} \sqrt {3+5 x}}{117649 (2+3 x)^{3/2}}-\frac {98642 \sqrt {1-2 x} \sqrt {3+5 x}}{823543 \sqrt {2+3 x}}+\frac {11 (3+5 x)^{3/2}}{21 (1-2 x)^{3/2} (2+3 x)^{7/2}}-\frac {16 \int \frac {-\frac {1468575}{8}+\frac {11097225 x}{8}}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{37059435}\\ &=\frac {220 \sqrt {3+5 x}}{49 \sqrt {1-2 x} (2+3 x)^{7/2}}-\frac {4545 \sqrt {1-2 x} \sqrt {3+5 x}}{2401 (2+3 x)^{7/2}}-\frac {11433 \sqrt {1-2 x} \sqrt {3+5 x}}{16807 (2+3 x)^{5/2}}-\frac {33778 \sqrt {1-2 x} \sqrt {3+5 x}}{117649 (2+3 x)^{3/2}}-\frac {98642 \sqrt {1-2 x} \sqrt {3+5 x}}{823543 \sqrt {2+3 x}}+\frac {11 (3+5 x)^{3/2}}{21 (1-2 x)^{3/2} (2+3 x)^{7/2}}-\frac {98642 \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx}{823543}+\frac {361196 \int \frac {1}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{823543}\\ &=\frac {220 \sqrt {3+5 x}}{49 \sqrt {1-2 x} (2+3 x)^{7/2}}-\frac {4545 \sqrt {1-2 x} \sqrt {3+5 x}}{2401 (2+3 x)^{7/2}}-\frac {11433 \sqrt {1-2 x} \sqrt {3+5 x}}{16807 (2+3 x)^{5/2}}-\frac {33778 \sqrt {1-2 x} \sqrt {3+5 x}}{117649 (2+3 x)^{3/2}}-\frac {98642 \sqrt {1-2 x} \sqrt {3+5 x}}{823543 \sqrt {2+3 x}}+\frac {11 (3+5 x)^{3/2}}{21 (1-2 x)^{3/2} (2+3 x)^{7/2}}+\frac {98642 \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{823543}-\frac {65672 \sqrt {\frac {11}{3}} F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{823543}\\ \end {align*}

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Mathematica [A]
time = 9.04, size = 113, normalized size = 0.45 \begin {gather*} \frac {2 \left (\frac {\sqrt {3+5 x} \left (866085+6524789 x+10746933 x^2-7681599 x^3-28748088 x^4-15980004 x^5\right )}{(1-2 x)^{3/2} (2+3 x)^{7/2}}+\sqrt {2} \left (-49321 E\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )|-\frac {33}{2}\right )+591115 F\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )|-\frac {33}{2}\right )\right )\right )}{2470629} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(3 + 5*x)^(5/2)/((1 - 2*x)^(5/2)*(2 + 3*x)^(9/2)),x]

[Out]

(2*((Sqrt[3 + 5*x]*(866085 + 6524789*x + 10746933*x^2 - 7681599*x^3 - 28748088*x^4 - 15980004*x^5))/((1 - 2*x)
^(3/2)*(2 + 3*x)^(7/2)) + Sqrt[2]*(-49321*EllipticE[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2] + 591115*Elliptic
F[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2])))/2470629

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(490\) vs. \(2(185)=370\).
time = 0.11, size = 491, normalized size = 1.94

method	result	size
elliptic	\(\frac {\sqrt {-\left (3+5 x \right ) \left (-1+2 x \right ) \left (2+3 x \right )}\, \left (\frac {242 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{50421 \left (-\frac {1}{2}+x \right )^{2}}-\frac {1364 \left (-30 x^{2}-38 x -12\right )}{50421 \sqrt {\left (-\frac {1}{2}+x \right ) \left (-30 x^{2}-38 x -12\right )}}+\frac {194 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{453789 \left (\frac {2}{3}+x \right )^{3}}-\frac {2 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{194481 \left (\frac {2}{3}+x \right )^{4}}-\frac {10602 \left (-30 x^{2}-3 x +9\right )}{823543 \sqrt {\left (\frac {2}{3}+x \right ) \left (-30 x^{2}-3 x +9\right )}}-\frac {1814 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{352947 \left (\frac {2}{3}+x \right )^{2}}+\frac {65270 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{17294403 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}-\frac {493210 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \left (-\frac {\EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{15}-\frac {3 \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{5}\right )}{17294403 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}\right )}{\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}\)	\(325\)
default	\(-\frac {2 \sqrt {1-2 x}\, \left (2663334 \sqrt {2}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x^{4} \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}[phone] \sqrt {2}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x^{4} \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}+3995001 \sqrt {2}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x^{3} \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}[phone] \sqrt {2}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x^{3} \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}+887778 \sqrt {2}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x^{2} \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}+9752292 \sqrt {2}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x^{2} \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}-986420 \sqrt {2}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}-10835880 \sqrt {2}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}[phone] x^{6}-394568 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )-4334352 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )[phone] x^{5}[phone] x^{4}-30689868 x^{3}-64864744 x^{2}-23904792 x -2598255\right )}{2470629 \left (2+3 x \right )^{\frac {7}{2}} \left (-1+2 x \right )^{2} \sqrt {3+5 x}}\)	\(491\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((3+5*x)^(5/2)/(1-2*x)^(5/2)/(2+3*x)^(9/2),x,method=_RETURNVERBOSE)

[Out]

-2/2470629*(1-2*x)^(1/2)*(2663334*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^4*(2+3*x)^(1/2)*(-3-5*x)
^(1/2)*(1-2*x)^(1/2)+3995001*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^3*(2+3*x)^(1/2)*(-3-5*x)^(1/2
)*(1-2*x)^(1/2)+887778*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)+9752292*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)-986420*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)-10835880*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)+7
9900020*x^6-394568*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))-4334352*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))+191
680452*x^5+124652259*x^4-30689868*x^3-64864744*x^2-23904792*x-2598255)/(2+3*x)^(7/2)/(-1+2*x)^2/(3+5*x)^(1/2)

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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)^(5/2)/(2+3*x)^(9/2),x, algorithm="maxima")

[Out]

integrate((5*x + 3)^(5/2)/((3*x + 2)^(9/2)*(-2*x + 1)^(5/2)), x)

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Fricas [A]
time = 0.25, size = 80, normalized size = 0.32 \begin {gather*} -\frac {2 \, {\left (15980004 \, x^{5} + 28748088 \, x^{4} + 7681599 \, x^{3} - 10746933 \, x^{2} - 6524789 \, x - 866085\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}}{2470629 \, {\left (324 \, x^{6} + 540 \, x^{5} + 81 \, x^{4} - 264 \, x^{3} - 104 \, x^{2} + 32 \, x + 16\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3+5*x)^(5/2)/(1-2*x)^(5/2)/(2+3*x)^(9/2),x, algorithm="fricas")

[Out]

-2/2470629*(15980004*x^5 + 28748088*x^4 + 7681599*x^3 - 10746933*x^2 - 6524789*x - 866085)*sqrt(5*x + 3)*sqrt(
3*x + 2)*sqrt(-2*x + 1)/(324*x^6 + 540*x^5 + 81*x^4 - 264*x^3 - 104*x^2 + 32*x + 16)

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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)**(5/2)/(2+3*x)**(9/2),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)^(5/2)/(2+3*x)^(9/2),x, algorithm="giac")

[Out]

integrate((5*x + 3)^(5/2)/((3*x + 2)^(9/2)*(-2*x + 1)^(5/2)), 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 )}^{5/2}\,{\left (3\,x+2\right )}^{9/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((5*x + 3)^(5/2)/((1 - 2*x)^(5/2)*(3*x + 2)^(9/2)),x)

[Out]

int((5*x + 3)^(5/2)/((1 - 2*x)^(5/2)*(3*x + 2)^(9/2)), x)

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