3.29.13 \(\int \frac {(1-2 x)^{5/2}}{(2+3 x)^{7/2} (3+5 x)^{5/2}} \, dx\) [2813]

Optimal. Leaf size=222 \[ \frac {14 (1-2 x)^{3/2}}{15 (2+3 x)^{5/2} (3+5 x)^{3/2}}+\frac {1232 \sqrt {1-2 x}}{45 (2+3 x)^{3/2} (3+5 x)^{3/2}}+\frac {35948 \sqrt {1-2 x}}{45 \sqrt {2+3 x} (3+5 x)^{3/2}}-\frac {16016 \sqrt {1-2 x} \sqrt {2+3 x}}{3 (3+5 x)^{3/2}}+\frac {96808 \sqrt {1-2 x} \sqrt {2+3 x}}{3 \sqrt {3+5 x}}-\frac {96808}{5} \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )-\frac {2912}{5} \sqrt {\frac {11}{3}} F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right ) \]

[Out]

14/15*(1-2*x)^(3/2)/(2+3*x)^(5/2)/(3+5*x)^(3/2)-96808/15*EllipticE(1/7*21^(1/2)*(1-2*x)^(1/2),1/33*1155^(1/2))
*33^(1/2)-2912/15*EllipticF(1/7*21^(1/2)*(1-2*x)^(1/2),1/33*1155^(1/2))*33^(1/2)+1232/45*(1-2*x)^(1/2)/(2+3*x)
^(3/2)/(3+5*x)^(3/2)+35948/45*(1-2*x)^(1/2)/(3+5*x)^(3/2)/(2+3*x)^(1/2)-16016/3*(1-2*x)^(1/2)*(2+3*x)^(1/2)/(3
+5*x)^(3/2)+96808/3*(1-2*x)^(1/2)*(2+3*x)^(1/2)/(3+5*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 = {100, 155, 157, 164, 114, 120} \begin {gather*} -\frac {2912}{5} \sqrt {\frac {11}{3}} F\left (\text {ArcSin}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )-\frac {96808}{5} \sqrt {\frac {11}{3}} E\left (\text {ArcSin}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )+\frac {14 (1-2 x)^{3/2}}{15 (3 x+2)^{5/2} (5 x+3)^{3/2}}+\frac {96808 \sqrt {3 x+2} \sqrt {1-2 x}}{3 \sqrt {5 x+3}}-\frac {16016 \sqrt {3 x+2} \sqrt {1-2 x}}{3 (5 x+3)^{3/2}}+\frac {35948 \sqrt {1-2 x}}{45 \sqrt {3 x+2} (5 x+3)^{3/2}}+\frac {1232 \sqrt {1-2 x}}{45 (3 x+2)^{3/2} (5 x+3)^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(14*(1 - 2*x)^(3/2))/(15*(2 + 3*x)^(5/2)*(3 + 5*x)^(3/2)) + (1232*Sqrt[1 - 2*x])/(45*(2 + 3*x)^(3/2)*(3 + 5*x)
^(3/2)) + (35948*Sqrt[1 - 2*x])/(45*Sqrt[2 + 3*x]*(3 + 5*x)^(3/2)) - (16016*Sqrt[1 - 2*x]*Sqrt[2 + 3*x])/(3*(3
+ 5*x)^(3/2)) + (96808*Sqrt[1 - 2*x]*Sqrt[2 + 3*x])/(3*Sqrt[3 + 5*x]) - (96808*Sqrt[11/3]*EllipticE[ArcSin[Sq
rt[3/7]*Sqrt[1 - 2*x]], 35/33])/5 - (2912*Sqrt[11/3]*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/5

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 {(1-2 x)^{5/2}}{(2+3 x)^{7/2} (3+5 x)^{5/2}} \, dx &=\frac {14 (1-2 x)^{3/2}}{15 (2+3 x)^{5/2} (3+5 x)^{3/2}}+\frac {2}{15} \int \frac {(198-165 x) \sqrt {1-2 x}}{(2+3 x)^{5/2} (3+5 x)^{5/2}} \, dx\\ &=\frac {14 (1-2 x)^{3/2}}{15 (2+3 x)^{5/2} (3+5 x)^{3/2}}+\frac {1232 \sqrt {1-2 x}}{45 (2+3 x)^{3/2} (3+5 x)^{3/2}}-\frac {4}{135} \int \frac {-\frac {32769}{2}+22605 x}{\sqrt {1-2 x} (2+3 x)^{3/2} (3+5 x)^{5/2}} \, dx\\ &=\frac {14 (1-2 x)^{3/2}}{15 (2+3 x)^{5/2} (3+5 x)^{3/2}}+\frac {1232 \sqrt {1-2 x}}{45 (2+3 x)^{3/2} (3+5 x)^{3/2}}+\frac {35948 \sqrt {1-2 x}}{45 \sqrt {2+3 x} (3+5 x)^{3/2}}-\frac {8}{945} \int \frac {-\frac {2463615}{2}+\frac {2830905 x}{2}}{\sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{5/2}} \, dx\\ &=\frac {14 (1-2 x)^{3/2}}{15 (2+3 x)^{5/2} (3+5 x)^{3/2}}+\frac {1232 \sqrt {1-2 x}}{45 (2+3 x)^{3/2} (3+5 x)^{3/2}}+\frac {35948 \sqrt {1-2 x}}{45 \sqrt {2+3 x} (3+5 x)^{3/2}}-\frac {16016 \sqrt {1-2 x} \sqrt {2+3 x}}{3 (3+5 x)^{3/2}}+\frac {16 \int \frac {-\frac {201818925}{4}[phone] x}{\sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}} \, dx}{31185}\\ &=\frac {14 (1-2 x)^{3/2}}{15 (2+3 x)^{5/2} (3+5 x)^{3/2}}+\frac {1232 \sqrt {1-2 x}}{45 (2+3 x)^{3/2} (3+5 x)^{3/2}}+\frac {35948 \sqrt {1-2 x}}{45 \sqrt {2+3 x} (3+5 x)^{3/2}}-\frac {16016 \sqrt {1-2 x} \sqrt {2+3 x}}{3 (3+5 x)^{3/2}}+\frac {96808 \sqrt {1-2 x} \sqrt {2+3 x}}{3 \sqrt {3+5 x}}-\frac {32 \int \frac {-\frac {2627991135}{4}-\frac {4151066535 x}{4}}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{343035}\\ &=\frac {14 (1-2 x)^{3/2}}{15 (2+3 x)^{5/2} (3+5 x)^{3/2}}+\frac {1232 \sqrt {1-2 x}}{45 (2+3 x)^{3/2} (3+5 x)^{3/2}}+\frac {35948 \sqrt {1-2 x}}{45 \sqrt {2+3 x} (3+5 x)^{3/2}}-\frac {16016 \sqrt {1-2 x} \sqrt {2+3 x}}{3 (3+5 x)^{3/2}}+\frac {96808 \sqrt {1-2 x} \sqrt {2+3 x}}{3 \sqrt {3+5 x}}+\frac {16016}{5} \int \frac {1}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx+\frac {96808}{5} \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx\\ &=\frac {14 (1-2 x)^{3/2}}{15 (2+3 x)^{5/2} (3+5 x)^{3/2}}+\frac {1232 \sqrt {1-2 x}}{45 (2+3 x)^{3/2} (3+5 x)^{3/2}}+\frac {35948 \sqrt {1-2 x}}{45 \sqrt {2+3 x} (3+5 x)^{3/2}}-\frac {16016 \sqrt {1-2 x} \sqrt {2+3 x}}{3 (3+5 x)^{3/2}}+\frac {96808 \sqrt {1-2 x} \sqrt {2+3 x}}{3 \sqrt {3+5 x}}-\frac {96808}{5} \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )-\frac {2912}{5} \sqrt {\frac {11}{3}} F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )\\ \end {align*}

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Mathematica [A]
time = 8.74, size = 109, normalized size = 0.49 \begin {gather*} \frac {2}{15} \left (\frac {\sqrt {1-2 x} \left (5512543+34450018 x+80662602 x^2+83867940 x^3+32672700 x^4\right )}{(2+3 x)^{5/2} (3+5 x)^{3/2}}+4 \sqrt {2} \left (12101 E\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )|-\frac {33}{2}\right )-6095 F\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )|-\frac {33}{2}\right )\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*((Sqrt[1 - 2*x]*(5512543 + 34450018*x + 80662602*x^2 + 83867940*x^3 + 32672700*x^4))/((2 + 3*x)^(5/2)*(3 +
5*x)^(3/2)) + 4*Sqrt[2]*(12101*EllipticE[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2] - 6095*EllipticF[ArcSin[Sqrt
[2/11]*Sqrt[3 + 5*x]], -33/2])))/15

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(397\) vs. \(2(162)=324\).
time = 0.11, size = 398, normalized size = 1.79

method	result	size
elliptic	\(\frac {\sqrt {-\left (3+5 x \right ) \left (-1+2 x \right ) \left (2+3 x \right )}\, \left (-\frac {242 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{15 \left (x +\frac {3}{5}\right )^{2}}+\frac {-82940 x^{2}-\frac {41470}{3} x +\frac {82940}{3}}{\sqrt {\left (x +\frac {3}{5}\right ) \left (-30 x^{2}-5 x +10\right )}}+\frac {98 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{135 \left (\frac {2}{3}+x \right )^{3}}+\frac {-110676 x^{2}-\frac {55338}{5} x +\frac {166014}{5}}{\sqrt {\left (\frac {2}{3}+x \right ) \left (-30 x^{2}-3 x +9\right )}}+\frac {728 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{15 \left (\frac {2}{3}+x \right )^{2}}+\frac {61288 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{21 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}+\frac {96808 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \left (-\frac {\EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{15}-\frac {3 \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{5}\right )}{21 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}\right )}{\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}\)	\(301\)
default	\(-\frac {2 \sqrt {1-2 x}\, \left (1081080 \sqrt {2}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x^{3} \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}-2178180 \sqrt {2}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x^{3} \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}+2090088 \sqrt {2}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x^{2} \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}-4211148 \sqrt {2}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x^{2} \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}+1345344 \sqrt {2}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}-2710624 \sqrt {2}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}+288288 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )-580848 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )-65345400 x^{5}-135063180 x^{4}-77457264 x^{3}[phone] x^{2}[phone] x +5512543\right )}{15 \left (2+3 x \right )^{\frac {5}{2}} \left (3+5 x \right )^{\frac {3}{2}} \left (-1+2 x \right )}\)	\(398\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/15*(1-2*x)^(1/2)*(1081080*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)-2178180*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)+2090088*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)-4211148*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)+1345344*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)-2710624*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)+2
88288*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))-580848*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))-65345400*x^5-1350
63180*x^4-77457264*x^3+11762566*x^2+23424932*x+5512543)/(2+3*x)^(5/2)/(3+5*x)^(3/2)/(-1+2*x)

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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)^(5/2)/(2+3*x)^(7/2)/(3+5*x)^(5/2),x, algorithm="maxima")

[Out]

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

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Fricas [A]
time = 0.25, size = 70, normalized size = 0.32 \begin {gather*} \frac {2 \, {\left (32672700 \, x^{4} + 83867940 \, x^{3} + 80662602 \, x^{2} + 34450018 \, x + 5512543\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}}{15 \, {\left (675 \, x^{5} + 2160 \, x^{4} + 2763 \, x^{3} + 1766 \, x^{2} + 564 \, x + 72\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/15*(32672700*x^4 + 83867940*x^3 + 80662602*x^2 + 34450018*x + 5512543)*sqrt(5*x + 3)*sqrt(3*x + 2)*sqrt(-2*x
+ 1)/(675*x^5 + 2160*x^4 + 2763*x^3 + 1766*x^2 + 564*x + 72)

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

[Out]

Exception raised: SystemError >> excessive stack use: stack is 5987 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((1-2*x)^(5/2)/(2+3*x)^(7/2)/(3+5*x)^(5/2),x, algorithm="giac")

[Out]

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

[Out]

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

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