3.30.100 \(\int \frac {1}{(1-2 x)^{5/2} (2+3 x)^{7/2} (3+5 x)^{5/2}} \, dx\) [3000]

Optimal. Leaf size=280 \[ \frac {4}{231 (1-2 x)^{3/2} (2+3 x)^{5/2} (3+5 x)^{3/2}}+\frac {632}{5929 \sqrt {1-2 x} (2+3 x)^{5/2} (3+5 x)^{3/2}}-\frac {3606 \sqrt {1-2 x}}{207515 (2+3 x)^{5/2} (3+5 x)^{3/2}}+\frac {649224 \sqrt {1-2 x}}{1452605 (2+3 x)^{3/2} (3+5 x)^{3/2}}+\frac {140700876 \sqrt {1-2 x}}{10168235 \sqrt {2+3 x} (3+5 x)^{3/2}}-\frac {6208896328 \sqrt {1-2 x} \sqrt {2+3 x}}{67110351 (3+5 x)^{3/2}}+\frac {412810345784 \sqrt {1-2 x} \sqrt {2+3 x}}{738213861 \sqrt {3+5 x}}-\frac {412810345784 E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{111850585 \sqrt {33}}-\frac {12417792656 F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{111850585 \sqrt {33}} \]

[Out]

4/231/(1-2*x)^(3/2)/(2+3*x)^(5/2)/(3+5*x)^(3/2)-412810345784/3691069305*EllipticE(1/7*21^(1/2)*(1-2*x)^(1/2),1
/33*1155^(1/2))*33^(1/2)-12417792656/3691069305*EllipticF(1/7*21^(1/2)*(1-2*x)^(1/2),1/33*1155^(1/2))*33^(1/2)
+632/5929/(2+3*x)^(5/2)/(3+5*x)^(3/2)/(1-2*x)^(1/2)-3606/207515*(1-2*x)^(1/2)/(2+3*x)^(5/2)/(3+5*x)^(3/2)+6492
24/1452605*(1-2*x)^(1/2)/(2+3*x)^(3/2)/(3+5*x)^(3/2)[phone]/10168235*(1-2*x)^(1/2)/(3+5*x)^(3/2)/(2+3*x)^(1
/2)-6208896328/67110351*(1-2*x)^(1/2)*(2+3*x)^(1/2)/(3+5*x)^(3/2)[phone]/738213861*(1-2*x)^(1/2)*(2+3*x)
^(1/2)/(3+5*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 = 5, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.179, Rules used = {106, 157, 164, 114, 120} \begin {gather*} -\frac {12417792656 F\left (\text {ArcSin}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{111850585 \sqrt {33}}-\frac {412810345784 E\left (\text {ArcSin}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{111850585 \sqrt {33}}+\frac {412810345784 \sqrt {1-2 x} \sqrt {3 x+2}}{738213861 \sqrt {5 x+3}}-\frac {6208896328 \sqrt {1-2 x} \sqrt {3 x+2}}{67110351 (5 x+3)^{3/2}}+\frac {140700876 \sqrt {1-2 x}}{10168235 \sqrt {3 x+2} (5 x+3)^{3/2}}+\frac {649224 \sqrt {1-2 x}}{1452605 (3 x+2)^{3/2} (5 x+3)^{3/2}}-\frac {3606 \sqrt {1-2 x}}{207515 (3 x+2)^{5/2} (5 x+3)^{3/2}}+\frac {632}{5929 \sqrt {1-2 x} (3 x+2)^{5/2} (5 x+3)^{3/2}}+\frac {4}{231 (1-2 x)^{3/2} (3 x+2)^{5/2} (5 x+3)^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/((1 - 2*x)^(5/2)*(2 + 3*x)^(7/2)*(3 + 5*x)^(5/2)),x]

[Out]

4/(231*(1 - 2*x)^(3/2)*(2 + 3*x)^(5/2)*(3 + 5*x)^(3/2)) + 632/(5929*Sqrt[1 - 2*x]*(2 + 3*x)^(5/2)*(3 + 5*x)^(3
/2)) - (3606*Sqrt[1 - 2*x])/(207515*(2 + 3*x)^(5/2)*(3 + 5*x)^(3/2)) + (649224*Sqrt[1 - 2*x])/(1452605*(2 + 3*
x)^(3/2)*(3 + 5*x)^(3/2)) + (140700876*Sqrt[1 - 2*x])/(10168235*Sqrt[2 + 3*x]*(3 + 5*x)^(3/2)) - (6208896328*S
qrt[1 - 2*x]*Sqrt[2 + 3*x])/(67110351*(3 + 5*x)^(3/2)) + (412810345784*Sqrt[1 - 2*x]*Sqrt[2 + 3*x])/(738213861
*Sqrt[3 + 5*x]) - (412810345784*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(111850585*Sqrt[33]) - (124
17792656*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(111850585*Sqrt[33])

Rule 106

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[b*(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*(m + 1) - b*(d*e*(m + n + 2) +
c*f*(m + p + 2)) - b*d*f*(m + n + p + 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && LtQ[m, -1] &&
IntegersQ[2*m, 2*n, 2*p]

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 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}{(1-2 x)^{5/2} (2+3 x)^{7/2} (3+5 x)^{5/2}} \, dx &=\frac {4}{231 (1-2 x)^{3/2} (2+3 x)^{5/2} (3+5 x)^{3/2}}-\frac {2}{231} \int \frac {-\frac {309}{2}-165 x}{(1-2 x)^{3/2} (2+3 x)^{7/2} (3+5 x)^{5/2}} \, dx\\ &=\frac {4}{231 (1-2 x)^{3/2} (2+3 x)^{5/2} (3+5 x)^{3/2}}+\frac {632}{5929 \sqrt {1-2 x} (2+3 x)^{5/2} (3+5 x)^{3/2}}+\frac {4 \int \frac {\frac {83517}{4}+31995 x}{\sqrt {1-2 x} (2+3 x)^{7/2} (3+5 x)^{5/2}} \, dx}{17787}\\ &=\frac {4}{231 (1-2 x)^{3/2} (2+3 x)^{5/2} (3+5 x)^{3/2}}+\frac {632}{5929 \sqrt {1-2 x} (2+3 x)^{5/2} (3+5 x)^{3/2}}-\frac {3606 \sqrt {1-2 x}}{207515 (2+3 x)^{5/2} (3+5 x)^{3/2}}+\frac {8 \int \frac {153282+\frac {189315 x}{4}}{\sqrt {1-2 x} (2+3 x)^{5/2} (3+5 x)^{5/2}} \, dx}{622545}\\ &=\frac {4}{231 (1-2 x)^{3/2} (2+3 x)^{5/2} (3+5 x)^{3/2}}+\frac {632}{5929 \sqrt {1-2 x} (2+3 x)^{5/2} (3+5 x)^{3/2}}-\frac {3606 \sqrt {1-2 x}}{207515 (2+3 x)^{5/2} (3+5 x)^{3/2}}+\frac {649224 \sqrt {1-2 x}}{1452605 (2+3 x)^{3/2} (3+5 x)^{3/2}}+\frac {16 \int \frac {\frac {56833857}{8}-\frac {18259425 x}{2}}{\sqrt {1-2 x} (2+3 x)^{3/2} (3+5 x)^{5/2}} \, dx}{13073445}\\ &=\frac {4}{231 (1-2 x)^{3/2} (2+3 x)^{5/2} (3+5 x)^{3/2}}+\frac {632}{5929 \sqrt {1-2 x} (2+3 x)^{5/2} (3+5 x)^{3/2}}-\frac {3606 \sqrt {1-2 x}}{207515 (2+3 x)^{5/2} (3+5 x)^{3/2}}+\frac {649224 \sqrt {1-2 x}}{1452605 (2+3 x)^{3/2} (3+5 x)^{3/2}}+\frac {140700876 \sqrt {1-2 x}}{10168235 \sqrt {2+3 x} (3+5 x)^{3/2}}+\frac {32 \int \frac {\frac {2067907815}{4}-\frac {4748654565 x}{8}}{\sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{5/2}} \, dx}{91514115}\\ &=\frac {4}{231 (1-2 x)^{3/2} (2+3 x)^{5/2} (3+5 x)^{3/2}}+\frac {632}{5929 \sqrt {1-2 x} (2+3 x)^{5/2} (3+5 x)^{3/2}}-\frac {3606 \sqrt {1-2 x}}{207515 (2+3 x)^{5/2} (3+5 x)^{3/2}}+\frac {649224 \sqrt {1-2 x}}{1452605 (2+3 x)^{3/2} (3+5 x)^{3/2}}+\frac {140700876 \sqrt {1-2 x}}{10168235 \sqrt {2+3 x} (3+5 x)^{3/2}}-\frac {6208896328 \sqrt {1-2 x} \sqrt {2+3 x}}{67110351 (3+5 x)^{3/2}}-\frac {64 \int \frac {\frac {338681488365}{16}-\frac {104775125535 x}{8}}{\sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}} \, dx}{3019965795}\\ &=\frac {4}{231 (1-2 x)^{3/2} (2+3 x)^{5/2} (3+5 x)^{3/2}}+\frac {632}{5929 \sqrt {1-2 x} (2+3 x)^{5/2} (3+5 x)^{3/2}}-\frac {3606 \sqrt {1-2 x}}{207515 (2+3 x)^{5/2} (3+5 x)^{3/2}}+\frac {649224 \sqrt {1-2 x}}{1452605 (2+3 x)^{3/2} (3+5 x)^{3/2}}+\frac {140700876 \sqrt {1-2 x}}{10168235 \sqrt {2+3 x} (3+5 x)^{3/2}}-\frac {6208896328 \sqrt {1-2 x} \sqrt {2+3 x}}{67110351 (3+5 x)^{3/2}}+\frac {412810345784 \sqrt {1-2 x} \sqrt {2+3 x}}{738213861 \sqrt {3+5 x}}+\frac {128 \int \frac {\frac {551276253405}{2}+\frac {6966174585105 x}{16}}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{33219623745}\\ &=\frac {4}{231 (1-2 x)^{3/2} (2+3 x)^{5/2} (3+5 x)^{3/2}}+\frac {632}{5929 \sqrt {1-2 x} (2+3 x)^{5/2} (3+5 x)^{3/2}}-\frac {3606 \sqrt {1-2 x}}{207515 (2+3 x)^{5/2} (3+5 x)^{3/2}}+\frac {649224 \sqrt {1-2 x}}{1452605 (2+3 x)^{3/2} (3+5 x)^{3/2}}+\frac {140700876 \sqrt {1-2 x}}{10168235 \sqrt {2+3 x} (3+5 x)^{3/2}}-\frac {6208896328 \sqrt {1-2 x} \sqrt {2+3 x}}{67110351 (3+5 x)^{3/2}}+\frac {412810345784 \sqrt {1-2 x} \sqrt {2+3 x}}{738213861 \sqrt {3+5 x}}+\frac {6208896328 \int \frac {1}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{111850585}+\frac {412810345784 \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx}{1230356435}\\ &=\frac {4}{231 (1-2 x)^{3/2} (2+3 x)^{5/2} (3+5 x)^{3/2}}+\frac {632}{5929 \sqrt {1-2 x} (2+3 x)^{5/2} (3+5 x)^{3/2}}-\frac {3606 \sqrt {1-2 x}}{207515 (2+3 x)^{5/2} (3+5 x)^{3/2}}+\frac {649224 \sqrt {1-2 x}}{1452605 (2+3 x)^{3/2} (3+5 x)^{3/2}}+\frac {140700876 \sqrt {1-2 x}}{10168235 \sqrt {2+3 x} (3+5 x)^{3/2}}-\frac {6208896328 \sqrt {1-2 x} \sqrt {2+3 x}}{67110351 (3+5 x)^{3/2}}+\frac {412810345784 \sqrt {1-2 x} \sqrt {2+3 x}}{738213861 \sqrt {3+5 x}}-\frac {412810345784 E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{111850585 \sqrt {33}}-\frac {12417792656 F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{111850585 \sqrt {33}}\\ \end {align*}

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Mathematica [A]
time = 9.08, size = 119, normalized size = 0.42 \begin {gather*} \frac {2 \left (\frac {23506658680609+52875828155808 x-149619576926754 x^2-430611138612568 x^3+84649478011164 x^4+873229924799280 x^5+557293966808400 x^6}{(1-2 x)^{3/2} (2+3 x)^{5/2} (3+5 x)^{3/2}}+4 \sqrt {2} \left (51601293223 E\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )|-\frac {33}{2}\right )-25989595870 F\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )|-\frac {33}{2}\right )\right )\right )}{3691069305} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/((1 - 2*x)^(5/2)*(2 + 3*x)^(7/2)*(3 + 5*x)^(5/2)),x]

[Out]

(2*((23506658680609 + 52875828155808*x - 149619576926754*x^2 - 430611138612568*x^3 + 84649478011164*x^4 + 8732
29924799280*x^5 + 557293966808400*x^6)/((1 - 2*x)^(3/2)*(2 + 3*x)^(5/2)*(3 + 5*x)^(3/2)) + 4*Sqrt[2]*(51601293
223*EllipticE[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2] - 25989595870*EllipticF[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]
], -33/2])))/3691069305

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(490\) vs. \(2(208)=416\).
time = 0.11, size = 491, normalized size = 1.75

method	result	size
elliptic	\(\frac {\sqrt {-\left (3+5 x \right ) \left (-1+2 x \right ) \left (2+3 x \right )}\, \left (\frac {\left (-\frac {7503029}{43578150}+\frac {1500641 x}{4357815}\right ) \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{\left (x^{2}+\frac {1}{10} x -\frac {3}{10}\right )^{2}}-\frac {2 \left (-20-30 x \right ) \left (\frac {92281045511}{7382138610}-\frac {18225070049 x}{738213861}\right )}{\sqrt {\left (x^{2}+\frac {1}{10} x -\frac {3}{10}\right ) \left (-20-30 x \right )}}+\frac {18 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{1715 \left (\frac {2}{3}+x \right )^{3}}+\frac {1332 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{1715 \left (\frac {2}{3}+x \right )^{2}}+\frac {-\frac {31495068}{16807} x^{2}-\frac {15747534}{84035} x +\frac {47242602}{84035}}{\sqrt {\left (\frac {2}{3}+x \right ) \left (-30 x^{2}-3 x +9\right )}}+\frac {261345779392 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{5167497027 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}+\frac {412810345784 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \left (-\frac {\EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{15}-\frac {3 \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{5}\right )}{5167497027 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}\right )}{\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}\)	\(310\)
default	\(-\frac {2 \sqrt {1-2 x}\, \left (9220211047080 \sqrt {2}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x^{4} \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}-18576465560280 \sqrt {2}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x^{4} \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}[phone] \sqrt {2}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x^{3} \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}-26626267303068 \sqrt {2}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x^{3} \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}[phone] \sqrt {2}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x^{2} \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}-5160129322300 \sqrt {2}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x^{2} \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}-3278297261184 \sqrt {2}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}[phone] \sqrt {2}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}-557293966808400 x^{6}-1229361472944 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )[phone] \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )-873229924799280 x^{5}-84649478011164 x^{4}[phone] x^{3}[phone] x^{2}-52875828155808 x -23506658680609\right )}{3691069305 \left (2+3 x \right )^{\frac {5}{2}} \left (3+5 x \right )^{\frac {3}{2}} \left (-1+2 x \right )^{2}}\)	\(491\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(1-2*x)^(5/2)/(2+3*x)^(7/2)/(3+5*x)^(5/2),x,method=_RETURNVERBOSE)

[Out]

-2/3691069305*(1-2*x)^(1/2)*(9220211047080*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)-18576465560280*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)-26626267303068*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)-5160129322300*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)-3278297261184*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)[phone]*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)-557293966808400*x^6-1229361472944*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))[phone]*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))-873229924799280*x^5-84649478011164*x^4+4
30611138612568*x^3+149619576926754*x^2-52875828155808*x-23506658680609)/(2+3*x)^(5/2)/(3+5*x)^(3/2)/(-1+2*x)^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/(1-2*x)^(5/2)/(2+3*x)^(7/2)/(3+5*x)^(5/2),x, algorithm="maxima")

[Out]

integrate(1/((5*x + 3)^(5/2)*(3*x + 2)^(7/2)*(-2*x + 1)^(5/2)), x)

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Fricas [A]
time = 0.20, size = 90, normalized size = 0.32 \begin {gather*} \frac {2 \, {\left (557293966808400 \, x^{6} + 873229924799280 \, x^{5} + 84649478011164 \, x^{4} - 430611138612568 \, x^{3} - 149619576926754 \, x^{2} + 52875828155808 \, x + 23506658680609\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}}{3691069305 \, {\left (2700 \, x^{7} + 5940 \, x^{6} + 3087 \, x^{5} - 1828 \, x^{4} - 2045 \, x^{3} - 202 \, x^{2} + 276 \, x + 72\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(1-2*x)^(5/2)/(2+3*x)^(7/2)/(3+5*x)^(5/2),x, algorithm="fricas")

[Out]

2/3691069305*(557293966808400*x^6 + 873229924799280*x^5 + 84649478011164*x^4 - 430611138612568*x^3 - 149619576
926754*x^2 + 52875828155808*x + 23506658680609)*sqrt(5*x + 3)*sqrt(3*x + 2)*sqrt(-2*x + 1)/(2700*x^7 + 5940*x^
6 + 3087*x^5 - 1828*x^4 - 2045*x^3 - 202*x^2 + 276*x + 72)

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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/(1-2*x)**(5/2)/(2+3*x)**(7/2)/(3+5*x)**(5/2),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/(1-2*x)^(5/2)/(2+3*x)^(7/2)/(3+5*x)^(5/2),x, algorithm="giac")

[Out]

integrate(1/((5*x + 3)^(5/2)*(3*x + 2)^(7/2)*(-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 {1}{{\left (1-2\,x\right )}^{5/2}\,{\left (3\,x+2\right )}^{7/2}\,{\left (5\,x+3\right )}^{5/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/((1 - 2*x)^(5/2)*(3*x + 2)^(7/2)*(5*x + 3)^(5/2)),x)

[Out]

int(1/((1 - 2*x)^(5/2)*(3*x + 2)^(7/2)*(5*x + 3)^(5/2)), x)

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