3.28.43 \(\int \frac {(1-2 x)^{3/2}}{(2+3 x)^{5/2} (3+5 x)^{3/2}} \, dx\) [2743]

Optimal. Leaf size=160 \[ \frac {14 \sqrt {1-2 x}}{9 (2+3 x)^{3/2} \sqrt {3+5 x}}+\frac {88 \sqrt {1-2 x}}{3 \sqrt {2+3 x} \sqrt {3+5 x}}-\frac {2660 \sqrt {1-2 x} \sqrt {2+3 x}}{9 \sqrt {3+5 x}}+\frac {532}{3} \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )+\frac {16}{3} \sqrt {\frac {11}{3}} F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right ) \]

[Out]

532/9*EllipticE(1/7*21^(1/2)*(1-2*x)^(1/2),1/33*1155^(1/2))*33^(1/2)+16/9*EllipticF(1/7*21^(1/2)*(1-2*x)^(1/2)
,1/33*1155^(1/2))*33^(1/2)+14/9*(1-2*x)^(1/2)/(2+3*x)^(3/2)/(3+5*x)^(1/2)+88/3*(1-2*x)^(1/2)/(2+3*x)^(1/2)/(3+
5*x)^(1/2)-2660/9*(1-2*x)^(1/2)*(2+3*x)^(1/2)/(3+5*x)^(1/2)

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Rubi [A]
time = 0.03, antiderivative size = 160, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.179, Rules used = {100, 157, 164, 114, 120} \begin {gather*} \frac {16}{3} \sqrt {\frac {11}{3}} F\left (\text {ArcSin}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )+\frac {532}{3} \sqrt {\frac {11}{3}} E\left (\text {ArcSin}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )-\frac {2660 \sqrt {1-2 x} \sqrt {3 x+2}}{9 \sqrt {5 x+3}}+\frac {88 \sqrt {1-2 x}}{3 \sqrt {3 x+2} \sqrt {5 x+3}}+\frac {14 \sqrt {1-2 x}}{9 (3 x+2)^{3/2} \sqrt {5 x+3}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(14*Sqrt[1 - 2*x])/(9*(2 + 3*x)^(3/2)*Sqrt[3 + 5*x]) + (88*Sqrt[1 - 2*x])/(3*Sqrt[2 + 3*x]*Sqrt[3 + 5*x]) - (2
660*Sqrt[1 - 2*x]*Sqrt[2 + 3*x])/(9*Sqrt[3 + 5*x]) + (532*Sqrt[11/3]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]]
, 35/33])/3 + (16*Sqrt[11/3]*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/3

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 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}}{(2+3 x)^{5/2} (3+5 x)^{3/2}} \, dx &=\frac {14 \sqrt {1-2 x}}{9 (2+3 x)^{3/2} \sqrt {3+5 x}}+\frac {2}{9} \int \frac {88-99 x}{\sqrt {1-2 x} (2+3 x)^{3/2} (3+5 x)^{3/2}} \, dx\\ &=\frac {14 \sqrt {1-2 x}}{9 (2+3 x)^{3/2} \sqrt {3+5 x}}+\frac {88 \sqrt {1-2 x}}{3 \sqrt {2+3 x} \sqrt {3+5 x}}+\frac {4}{63} \int \frac {\frac {7469}{2}-2310 x}{\sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}} \, dx\\ &=\frac {14 \sqrt {1-2 x}}{9 (2+3 x)^{3/2} \sqrt {3+5 x}}+\frac {88 \sqrt {1-2 x}}{3 \sqrt {2+3 x} \sqrt {3+5 x}}-\frac {2660 \sqrt {1-2 x} \sqrt {2+3 x}}{9 \sqrt {3+5 x}}-\frac {8}{693} \int \frac {\frac {97251}{2}+\frac {153615 x}{2}}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx\\ &=\frac {14 \sqrt {1-2 x}}{9 (2+3 x)^{3/2} \sqrt {3+5 x}}+\frac {88 \sqrt {1-2 x}}{3 \sqrt {2+3 x} \sqrt {3+5 x}}-\frac {2660 \sqrt {1-2 x} \sqrt {2+3 x}}{9 \sqrt {3+5 x}}-\frac {88}{3} \int \frac {1}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx-\frac {532}{3} \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx\\ &=\frac {14 \sqrt {1-2 x}}{9 (2+3 x)^{3/2} \sqrt {3+5 x}}+\frac {88 \sqrt {1-2 x}}{3 \sqrt {2+3 x} \sqrt {3+5 x}}-\frac {2660 \sqrt {1-2 x} \sqrt {2+3 x}}{9 \sqrt {3+5 x}}+\frac {532}{3} \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )+\frac {16}{3} \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 = 6.92, size = 100, normalized size = 0.62 \begin {gather*} -\frac {2 \sqrt {1-2 x} \left (1683+5188 x+3990 x^2\right )}{3 (2+3 x)^{3/2} \sqrt {3+5 x}}-\frac {4}{9} \sqrt {2} \left (133 E\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )|-\frac {33}{2}\right )-67 F\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )|-\frac {33}{2}\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*Sqrt[1 - 2*x]*(1683 + 5188*x + 3990*x^2))/(3*(2 + 3*x)^(3/2)*Sqrt[3 + 5*x]) - (4*Sqrt[2]*(133*EllipticE[Ar
cSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2] - 67*EllipticF[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2]))/9

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Maple [A]
time = 0.10, size = 215, normalized size = 1.34

method	result	size
default	\(\frac {2 \sqrt {1-2 x}\, \sqrt {3+5 x}\, \left (396 \sqrt {2}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}-798 \sqrt {2}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}+264 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )-532 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )-23940 x^{3}-19158 x^{2}+5466 x +5049\right )}{9 \left (2+3 x \right )^{\frac {3}{2}} \left (10 x^{2}+x -3\right )}\)	\(215\)
elliptic	\(\frac {\sqrt {-\left (3+5 x \right ) \left (-1+2 x \right ) \left (2+3 x \right )}\, \left (-\frac {334 \left (-30 x^{2}-3 x +9\right )}{9 \sqrt {\left (\frac {2}{3}+x \right ) \left (-30 x^{2}-3 x +9\right )}}-\frac {1684 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{63 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}-\frac {380 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \left (-\frac {\EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{15}-\frac {3 \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{5}\right )}{9 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}-\frac {14 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{27 \left (\frac {2}{3}+x \right )^{2}}-\frac {22 \left (-30 x^{2}-5 x +10\right )}{\sqrt {\left (x +\frac {3}{5}\right ) \left (-30 x^{2}-5 x +10\right )}}\right )}{\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}\)	\(253\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/9*(1-2*x)^(1/2)*(3+5*x)^(1/2)*(396*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)-798*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)+264*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))-532*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))-23940*x
^3-19158*x^2+5466*x+5049)/(2+3*x)^(3/2)/(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((1-2*x)^(3/2)/(2+3*x)^(5/2)/(3+5*x)^(3/2),x, algorithm="maxima")

[Out]

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

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Fricas [A]
time = 0.32, size = 50, normalized size = 0.31 \begin {gather*} -\frac {2 \, {\left (3990 \, x^{2} + 5188 \, x + 1683\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}}{3 \, {\left (45 \, x^{3} + 87 \, x^{2} + 56 \, x + 12\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/3*(3990*x^2 + 5188*x + 1683)*sqrt(5*x + 3)*sqrt(3*x + 2)*sqrt(-2*x + 1)/(45*x^3 + 87*x^2 + 56*x + 12)

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

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\left (1-2\,x\right )}^{3/2}}{{\left (3\,x+2\right )}^{5/2}\,{\left (5\,x+3\right )}^{3/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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