3.28.28 \(\int \frac {(1-2 x)^{3/2} (3+5 x)^{5/2}}{(2+3 x)^{15/2}} \, dx\) [2728]

Optimal. Leaf size=280 \[ -\frac {2174468 \sqrt {1-2 x} \sqrt {3+5 x}}{11918907 (2+3 x)^{7/2}}+\frac {73596464 \sqrt {1-2 x} \sqrt {3+5 x}}{417161745 (2+3 x)^{5/2}}+\frac {3523482724 \sqrt {1-2 x} \sqrt {3+5 x}}{2920132215 (2+3 x)^{3/2}}+\frac {245282464136 \sqrt {1-2 x} \sqrt {3+5 x}}{20440925505 \sqrt {2+3 x}}-\frac {20992 \sqrt {1-2 x} (3+5 x)^{3/2}}{81081 (2+3 x)^{9/2}}-\frac {2 (1-2 x)^{3/2} (3+5 x)^{5/2}}{39 (2+3 x)^{13/2}}+\frac {362 \sqrt {1-2 x} (3+5 x)^{5/2}}{1287 (2+3 x)^{11/2}}-\frac {245282464136 E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{1858265955 \sqrt {33}}-\frac {7391549624 F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{1858265955 \sqrt {33}} \]

[Out]

-2/39*(1-2*x)^(3/2)*(3+5*x)^(5/2)/(2+3*x)^(13/2)-245282464136/61322776515*EllipticE(1/7*21^(1/2)*(1-2*x)^(1/2)
,1/33*1155^(1/2))*33^(1/2)-7391549624/61322776515*EllipticF(1/7*21^(1/2)*(1-2*x)^(1/2),1/33*1155^(1/2))*33^(1/
2)-20992/81081*(3+5*x)^(3/2)*(1-2*x)^(1/2)/(2+3*x)^(9/2)+362/1287*(3+5*x)^(5/2)*(1-2*x)^(1/2)/(2+3*x)^(11/2)-2
174468/11918907*(1-2*x)^(1/2)*(3+5*x)^(1/2)/(2+3*x)^(7/2)[phone]/417161745*(1-2*x)^(1/2)*(3+5*x)^(1/2)/(2+3*
x)^(5/2)[phone]/2920132215*(1-2*x)^(1/2)*(3+5*x)^(1/2)/(2+3*x)^(3/2)[phone]/20440925505*(1-2*x)^(1/2
)*(3+5*x)^(1/2)/(2+3*x)^(1/2)

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Rubi [A]
time = 0.08, antiderivative size = 280, normalized size of antiderivative = 1.00, number of steps used = 10, 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 {7391549624 F\left (\text {ArcSin}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{1858265955 \sqrt {33}}-\frac {245282464136 E\left (\text {ArcSin}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{1858265955 \sqrt {33}}+\frac {362 \sqrt {1-2 x} (5 x+3)^{5/2}}{1287 (3 x+2)^{11/2}}-\frac {2 (1-2 x)^{3/2} (5 x+3)^{5/2}}{39 (3 x+2)^{13/2}}-\frac {20992 \sqrt {1-2 x} (5 x+3)^{3/2}}{81081 (3 x+2)^{9/2}}+\frac {245282464136 \sqrt {1-2 x} \sqrt {5 x+3}}{20440925505 \sqrt {3 x+2}}+\frac {3523482724 \sqrt {1-2 x} \sqrt {5 x+3}}{2920132215 (3 x+2)^{3/2}}+\frac {73596464 \sqrt {1-2 x} \sqrt {5 x+3}}{417161745 (3 x+2)^{5/2}}-\frac {2174468 \sqrt {1-2 x} \sqrt {5 x+3}}{11918907 (3 x+2)^{7/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((1 - 2*x)^(3/2)*(3 + 5*x)^(5/2))/(2 + 3*x)^(15/2),x]

[Out]

(-2174468*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(11918907*(2 + 3*x)^(7/2)) + (73596464*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(41
7161745*(2 + 3*x)^(5/2)) + (3523482724*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(2920132215*(2 + 3*x)^(3/2)) + (2452824641
36*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(20440925505*Sqrt[2 + 3*x]) - (20992*Sqrt[1 - 2*x]*(3 + 5*x)^(3/2))/(81081*(2
+ 3*x)^(9/2)) - (2*(1 - 2*x)^(3/2)*(3 + 5*x)^(5/2))/(39*(2 + 3*x)^(13/2)) + (362*Sqrt[1 - 2*x]*(3 + 5*x)^(5/2)
)/(1287*(2 + 3*x)^(11/2)) - (245282464136*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(1858265955*Sqrt[
33]) - (7391549624*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(1858265955*Sqrt[33])

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 {(1-2 x)^{3/2} (3+5 x)^{5/2}}{(2+3 x)^{15/2}} \, dx &=-\frac {2 (1-2 x)^{3/2} (3+5 x)^{5/2}}{39 (2+3 x)^{13/2}}+\frac {2}{39} \int \frac {\left (\frac {7}{2}-40 x\right ) \sqrt {1-2 x} (3+5 x)^{3/2}}{(2+3 x)^{13/2}} \, dx\\ &=-\frac {2 (1-2 x)^{3/2} (3+5 x)^{5/2}}{39 (2+3 x)^{13/2}}+\frac {362 \sqrt {1-2 x} (3+5 x)^{5/2}}{1287 (2+3 x)^{11/2}}-\frac {4 \int \frac {(3+5 x)^{3/2} \left (-1409+\frac {3645 x}{2}\right )}{\sqrt {1-2 x} (2+3 x)^{11/2}} \, dx}{1287}\\ &=-\frac {20992 \sqrt {1-2 x} (3+5 x)^{3/2}}{81081 (2+3 x)^{9/2}}-\frac {2 (1-2 x)^{3/2} (3+5 x)^{5/2}}{39 (2+3 x)^{13/2}}+\frac {362 \sqrt {1-2 x} (3+5 x)^{5/2}}{1287 (2+3 x)^{11/2}}-\frac {8 \int \frac {\sqrt {3+5 x} \left (-\frac {302967}{4}+\frac {360975 x}{4}\right )}{\sqrt {1-2 x} (2+3 x)^{9/2}} \, dx}{243243}\\ &=-\frac {2174468 \sqrt {1-2 x} \sqrt {3+5 x}}{11918907 (2+3 x)^{7/2}}-\frac {20992 \sqrt {1-2 x} (3+5 x)^{3/2}}{81081 (2+3 x)^{9/2}}-\frac {2 (1-2 x)^{3/2} (3+5 x)^{5/2}}{39 (2+3 x)^{13/2}}+\frac {362 \sqrt {1-2 x} (3+5 x)^{5/2}}{1287 (2+3 x)^{11/2}}-\frac {16 \int \frac {-\frac {6900783}{4}+\frac {6896325 x}{8}}{\sqrt {1-2 x} (2+3 x)^{7/2} \sqrt {3+5 x}} \, dx}{35756721}\\ &=-\frac {2174468 \sqrt {1-2 x} \sqrt {3+5 x}}{11918907 (2+3 x)^{7/2}}+\frac {73596464 \sqrt {1-2 x} \sqrt {3+5 x}}{417161745 (2+3 x)^{5/2}}-\frac {20992 \sqrt {1-2 x} (3+5 x)^{3/2}}{81081 (2+3 x)^{9/2}}-\frac {2 (1-2 x)^{3/2} (3+5 x)^{5/2}}{39 (2+3 x)^{13/2}}+\frac {362 \sqrt {1-2 x} (3+5 x)^{5/2}}{1287 (2+3 x)^{11/2}}-\frac {32 \int \frac {-\frac {1538665083}{16}+\frac {206990055 x}{2}}{\sqrt {1-2 x} (2+3 x)^{5/2} \sqrt {3+5 x}} \, dx}{1251485235}\\ &=-\frac {2174468 \sqrt {1-2 x} \sqrt {3+5 x}}{11918907 (2+3 x)^{7/2}}+\frac {73596464 \sqrt {1-2 x} \sqrt {3+5 x}}{417161745 (2+3 x)^{5/2}}+\frac {3523482724 \sqrt {1-2 x} \sqrt {3+5 x}}{2920132215 (2+3 x)^{3/2}}-\frac {20992 \sqrt {1-2 x} (3+5 x)^{3/2}}{81081 (2+3 x)^{9/2}}-\frac {2 (1-2 x)^{3/2} (3+5 x)^{5/2}}{39 (2+3 x)^{13/2}}+\frac {362 \sqrt {1-2 x} (3+5 x)^{5/2}}{1287 (2+3 x)^{11/2}}-\frac {64 \int \frac {-\frac {65554803621}{16}+\frac {39639180645 x}{16}}{\sqrt {1-2 x} (2+3 x)^{3/2} \sqrt {3+5 x}} \, dx}{26281189935}\\ &=-\frac {2174468 \sqrt {1-2 x} \sqrt {3+5 x}}{11918907 (2+3 x)^{7/2}}+\frac {73596464 \sqrt {1-2 x} \sqrt {3+5 x}}{417161745 (2+3 x)^{5/2}}+\frac {3523482724 \sqrt {1-2 x} \sqrt {3+5 x}}{2920132215 (2+3 x)^{3/2}}+\frac {245282464136 \sqrt {1-2 x} \sqrt {3+5 x}}{20440925505 \sqrt {2+3 x}}-\frac {20992 \sqrt {1-2 x} (3+5 x)^{3/2}}{81081 (2+3 x)^{9/2}}-\frac {2 (1-2 x)^{3/2} (3+5 x)^{5/2}}{39 (2+3 x)^{13/2}}+\frac {362 \sqrt {1-2 x} (3+5 x)^{5/2}}{1287 (2+3 x)^{11/2}}-\frac {128 \int \frac {-\frac {1747127059515}{32}-\frac {1379713860765 x}{16}}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{183968329545}\\ &=-\frac {2174468 \sqrt {1-2 x} \sqrt {3+5 x}}{11918907 (2+3 x)^{7/2}}+\frac {73596464 \sqrt {1-2 x} \sqrt {3+5 x}}{417161745 (2+3 x)^{5/2}}+\frac {3523482724 \sqrt {1-2 x} \sqrt {3+5 x}}{2920132215 (2+3 x)^{3/2}}+\frac {245282464136 \sqrt {1-2 x} \sqrt {3+5 x}}{20440925505 \sqrt {2+3 x}}-\frac {20992 \sqrt {1-2 x} (3+5 x)^{3/2}}{81081 (2+3 x)^{9/2}}-\frac {2 (1-2 x)^{3/2} (3+5 x)^{5/2}}{39 (2+3 x)^{13/2}}+\frac {362 \sqrt {1-2 x} (3+5 x)^{5/2}}{1287 (2+3 x)^{11/2}}+\frac {3695774812 \int \frac {1}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{1858265955}+\frac {245282464136 \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx}{20440925505}\\ &=-\frac {2174468 \sqrt {1-2 x} \sqrt {3+5 x}}{11918907 (2+3 x)^{7/2}}+\frac {73596464 \sqrt {1-2 x} \sqrt {3+5 x}}{417161745 (2+3 x)^{5/2}}+\frac {3523482724 \sqrt {1-2 x} \sqrt {3+5 x}}{2920132215 (2+3 x)^{3/2}}+\frac {245282464136 \sqrt {1-2 x} \sqrt {3+5 x}}{20440925505 \sqrt {2+3 x}}-\frac {20992 \sqrt {1-2 x} (3+5 x)^{3/2}}{81081 (2+3 x)^{9/2}}-\frac {2 (1-2 x)^{3/2} (3+5 x)^{5/2}}{39 (2+3 x)^{13/2}}+\frac {362 \sqrt {1-2 x} (3+5 x)^{5/2}}{1287 (2+3 x)^{11/2}}-\frac {245282464136 E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{1858265955 \sqrt {33}}-\frac {7391549624 F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{1858265955 \sqrt {33}}\\ \end {align*}

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Mathematica [A]
time = 8.38, size = 117, normalized size = 0.42 \begin {gather*} \frac {\frac {48 \sqrt {2-4 x} \sqrt {3+5 x} \left (8272877174903+73802680969881 x+274263621177573 x^2+543590753927373 x^3+606171513555828 x^4+360618554767050 x^5+89405458177572 x^6\right )}{(2+3 x)^{13/2}}[phone] E\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )|-\frac {33}{2}\right )-1973150325440 F\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )|-\frac {33}{2}\right )}{490582212120 \sqrt {2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((1 - 2*x)^(3/2)*(3 + 5*x)^(5/2))/(2 + 3*x)^(15/2),x]

[Out]

((48*Sqrt[2 - 4*x]*Sqrt[3 + 5*x]*(8272877174903 + 73802680969881*x + 274263621177573*x^2 + 543590753927373*x^3
+ 606171513555828*x^4 + 360618554767050*x^5 + 89405458177572*x^6))/(2 + 3*x)^(13/2) + 3924519426176*EllipticE
[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2] - 1973150325440*EllipticF[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2])/
(490582212120*Sqrt[2])

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(679\) vs. \(2(208)=416\).
time = 0.11, size = 680, normalized size = 2.43

method	result	size
elliptic	\(-\frac {\sqrt {1-2 x}\, \sqrt {3+5 x}\, \sqrt {-\left (3+5 x \right ) \left (-1+2 x \right ) \left (2+3 x \right )}\, \left (-\frac {145118 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{59108049 \left (\frac {2}{3}+x \right )^{5}}+\frac {1946 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{8444007 \left (\frac {2}{3}+x \right )^{6}}-\frac {14 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{2302911 \left (\frac {2}{3}+x \right )^{7}}+\frac {73596464 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{11263367115 \left (\frac {2}{3}+x \right )^{3}}+\frac {3126842 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{965431467 \left (\frac {2}{3}+x \right )^{4}}+\frac {-\frac {490564928272}{4088185101} x^{2}-\frac {245282464136}{20440925505} x +\frac {245282464136}{6813641835}}{\sqrt {\left (\frac {2}{3}+x \right ) \left (-30 x^{2}-3 x +9\right )}}+\frac {3523482724 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{26281189935 \left (\frac {2}{3}+x \right )^{2}}+\frac {155300183068 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{85851887121 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}+\frac {245282464136 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \left (-\frac {\EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{15}-\frac {3 \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{5}\right )}{85851887121 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}\right )}{\left (10 x^{2}+x -3\right ) \sqrt {2+3 x}}\)	\(356\)
default	\(\frac {2 \left (89405458177572 \sqrt {2}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x^{6} \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}-44454627326142 \sqrt {2}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x^{6} \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}[phone] \sqrt {2}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x^{5} \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}-177818509304568 \sqrt {2}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x^{5} \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}[phone] \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}-296364182174280 \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}[phone] \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}-263434828599360 \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}+2682163745327160 x^{8}[phone] \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}-131717414299680 \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}+11086773017544216 x^{7}[phone] \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}-35124643813248 \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}+18462351947377842 x^{6}[phone] \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 )-3902738201472 \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 )+14880670165585224 x^{5}+4403137275106857 x^{4}-1855445492717208 x^{3}-1998778232441424 x^{2}-639405497204220 x -74455894574127\right ) \sqrt {3+5 x}\, \sqrt {1-2 x}}{61322776515 \left (10 x^{2}+x -3\right ) \left (2+3 x \right )^{\frac {13}{2}}}\)	\(680\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1-2*x)^(3/2)*(3+5*x)^(5/2)/(2+3*x)^(15/2),x,method=_RETURNVERBOSE)

[Out]

2/61322776515*(89405458177572*2^(1/2)*EllipticE(1/7*(28+42*x)^(1/2),1/2*70^(1/2))*x^6*(2+3*x)^(1/2)*(-3-5*x)^(
1/2)*(1-2*x)^(1/2)-44454627326142*2^(1/2)*EllipticF(1/7*(28+42*x)^(1/2),1/2*70^(1/2))*x^6*(2+3*x)^(1/2)*(-3-5*
x)^(1/2)*(1-2*x)^(1/2)[phone]*2^(1/2)*EllipticE(1/7*(28+42*x)^(1/2),1/2*70^(1/2))*x^5*(2+3*x)^(1/2)*(
-3-5*x)^(1/2)*(1-2*x)^(1/2)-177818509304568*2^(1/2)*EllipticF(1/7*(28+42*x)^(1/2),1/2*70^(1/2))*x^5*(2+3*x)^(1
/2)*(-3-5*x)^(1/2)*(1-2*x)^(1/2)[phone]*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)-296364182174280*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)[phone]*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)-263434828599360*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)+2682163745327160*x^8+264905061266880*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)-131717414299680*2^(1/2)*Ellipti
cF(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)+11086773017544216*x^7+7064
1349671168*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)-35
124643813248*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)+
18462351947377842*x^6+7849038852352*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))-3902738201472*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))+14880670165585224*x^5+4403137275106857*x^4-1855445492717208*x^3-1998778232441424*x^2-63940
5497204220*x-74455894574127)*(3+5*x)^(1/2)*(1-2*x)^(1/2)/(10*x^2+x-3)/(2+3*x)^(13/2)

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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((1-2*x)^(3/2)*(3+5*x)^(5/2)/(2+3*x)^(15/2),x, algorithm="maxima")

[Out]

integrate((5*x + 3)^(5/2)*(-2*x + 1)^(3/2)/(3*x + 2)^(15/2), x)

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Fricas [A]
time = 0.27, size = 90, normalized size = 0.32 \begin {gather*} \frac {2 \, {\left (89405458177572 \, x^{6} + 360618554767050 \, x^{5} + 606171513555828 \, x^{4} + 543590753927373 \, x^{3} + 274263621177573 \, x^{2} + 73802680969881 \, x + 8272877174903\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}}{20440925505 \, {\left (2187 \, x^{7} + 10206 \, x^{6} + 20412 \, x^{5} + 22680 \, x^{4} + 15120 \, x^{3} + 6048 \, x^{2} + 1344 \, x + 128\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-2*x)^(3/2)*(3+5*x)^(5/2)/(2+3*x)^(15/2),x, algorithm="fricas")

[Out]

2/20440925505*(89405458177572*x^6 + 360618554767050*x^5 + 606171513555828*x^4 + 543590753927373*x^3 + 27426362
1177573*x^2 + 73802680969881*x + 8272877174903)*sqrt(5*x + 3)*sqrt(3*x + 2)*sqrt(-2*x + 1)/(2187*x^7 + 10206*x
^6 + 20412*x^5 + 22680*x^4 + 15120*x^3 + 6048*x^2 + 1344*x + 128)

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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((1-2*x)**(3/2)*(3+5*x)**(5/2)/(2+3*x)**(15/2),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((1-2*x)^(3/2)*(3+5*x)^(5/2)/(2+3*x)^(15/2),x, algorithm="giac")

[Out]

integrate((5*x + 3)^(5/2)*(-2*x + 1)^(3/2)/(3*x + 2)^(15/2), x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\left (1-2\,x\right )}^{3/2}\,{\left (5\,x+3\right )}^{5/2}}{{\left (3\,x+2\right )}^{15/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((1 - 2*x)^(3/2)*(5*x + 3)^(5/2))/(3*x + 2)^(15/2),x)

[Out]

int(((1 - 2*x)^(3/2)*(5*x + 3)^(5/2))/(3*x + 2)^(15/2), x)

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