3.28.52 \(\int \frac {(1-2 x)^{3/2}}{(2+3 x)^{5/2} (3+5 x)^{5/2}} \, dx\) [2752]

Optimal. Leaf size=185 \[ \frac {14 \sqrt {1-2 x}}{9 (2+3 x)^{3/2} (3+5 x)^{3/2}}+\frac {404 \sqrt {1-2 x}}{9 \sqrt {2+3 x} (3+5 x)^{3/2}}-\frac {300 \sqrt {1-2 x} \sqrt {2+3 x}}{(3+5 x)^{3/2}}+\frac {5440 \sqrt {1-2 x} \sqrt {2+3 x}}{3 \sqrt {3+5 x}}-1088 \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )-120 \sqrt {\frac {3}{11}} F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right ) \]

[Out]

-120/11*EllipticF(1/7*21^(1/2)*(1-2*x)^(1/2),1/33*1155^(1/2))*33^(1/2)-1088/3*EllipticE(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)^(3/2)+404/9*(1-2*x)^(1/2)/(3+5*x)^(3/2
)/(2+3*x)^(1/2)-300*(1-2*x)^(1/2)*(2+3*x)^(1/2)/(3+5*x)^(3/2)+5440/3*(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 = 185, normalized size of antiderivative = 1.00, number of steps used = 7, 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*} -120 \sqrt {\frac {3}{11}} F\left (\text {ArcSin}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )-1088 \sqrt {\frac {11}{3}} E\left (\text {ArcSin}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )+\frac {5440 \sqrt {1-2 x} \sqrt {3 x+2}}{3 \sqrt {5 x+3}}-\frac {300 \sqrt {1-2 x} \sqrt {3 x+2}}{(5 x+3)^{3/2}}+\frac {404 \sqrt {1-2 x}}{9 \sqrt {3 x+2} (5 x+3)^{3/2}}+\frac {14 \sqrt {1-2 x}}{9 (3 x+2)^{3/2} (5 x+3)^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(14*Sqrt[1 - 2*x])/(9*(2 + 3*x)^(3/2)*(3 + 5*x)^(3/2)) + (404*Sqrt[1 - 2*x])/(9*Sqrt[2 + 3*x]*(3 + 5*x)^(3/2))
- (300*Sqrt[1 - 2*x]*Sqrt[2 + 3*x])/(3 + 5*x)^(3/2) + (5440*Sqrt[1 - 2*x]*Sqrt[2 + 3*x])/(3*Sqrt[3 + 5*x]) -
1088*Sqrt[11/3]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33] - 120*Sqrt[3/11]*EllipticF[ArcSin[Sqrt[3/7]*
Sqrt[1 - 2*x]], 35/33]

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)^{5/2}} \, dx &=\frac {14 \sqrt {1-2 x}}{9 (2+3 x)^{3/2} (3+5 x)^{3/2}}+\frac {2}{9} \int \frac {123-169 x}{\sqrt {1-2 x} (2+3 x)^{3/2} (3+5 x)^{5/2}} \, dx\\ &=\frac {14 \sqrt {1-2 x}}{9 (2+3 x)^{3/2} (3+5 x)^{3/2}}+\frac {404 \sqrt {1-2 x}}{9 \sqrt {2+3 x} (3+5 x)^{3/2}}+\frac {4}{63} \int \frac {\frac {18459}{2}-10605 x}{\sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{5/2}} \, dx\\ &=\frac {14 \sqrt {1-2 x}}{9 (2+3 x)^{3/2} (3+5 x)^{3/2}}+\frac {404 \sqrt {1-2 x}}{9 \sqrt {2+3 x} (3+5 x)^{3/2}}-\frac {300 \sqrt {1-2 x} \sqrt {2+3 x}}{(3+5 x)^{3/2}}-\frac {8 \int \frac {\frac {756063}{2}-\frac {467775 x}{2}}{\sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}} \, dx}{2079}\\ &=\frac {14 \sqrt {1-2 x}}{9 (2+3 x)^{3/2} (3+5 x)^{3/2}}+\frac {404 \sqrt {1-2 x}}{9 \sqrt {2+3 x} (3+5 x)^{3/2}}-\frac {300 \sqrt {1-2 x} \sqrt {2+3 x}}{(3+5 x)^{3/2}}+\frac {5440 \sqrt {1-2 x} \sqrt {2+3 x}}{3 \sqrt {3+5 x}}+\frac {16 \int \frac {\frac {19690209}{4}+7775460 x}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{22869}\\ &=\frac {14 \sqrt {1-2 x}}{9 (2+3 x)^{3/2} (3+5 x)^{3/2}}+\frac {404 \sqrt {1-2 x}}{9 \sqrt {2+3 x} (3+5 x)^{3/2}}-\frac {300 \sqrt {1-2 x} \sqrt {2+3 x}}{(3+5 x)^{3/2}}+\frac {5440 \sqrt {1-2 x} \sqrt {2+3 x}}{3 \sqrt {3+5 x}}+180 \int \frac {1}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx+1088 \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} (3+5 x)^{3/2}}+\frac {404 \sqrt {1-2 x}}{9 \sqrt {2+3 x} (3+5 x)^{3/2}}-\frac {300 \sqrt {1-2 x} \sqrt {2+3 x}}{(3+5 x)^{3/2}}+\frac {5440 \sqrt {1-2 x} \sqrt {2+3 x}}{3 \sqrt {3+5 x}}-1088 \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )-120 \sqrt {\frac {3}{11}} 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 = 8.13, size = 104, normalized size = 0.56 \begin {gather*} \frac {2}{3} \left (\frac {\sqrt {1-2 x} \left (30977+147122 x+232590 x^2+122400 x^3\right )}{(2+3 x)^{3/2} (3+5 x)^{3/2}}+2 \sqrt {2} \left (272 E\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )|-\frac {33}{2}\right )-137 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)^(3/2)/((2 + 3*x)^(5/2)*(3 + 5*x)^(5/2)),x]

[Out]

(2*((Sqrt[1 - 2*x]*(30977 + 147122*x + 232590*x^2 + 122400*x^3))/((2 + 3*x)^(3/2)*(3 + 5*x)^(3/2)) + 2*Sqrt[2]
*(272*EllipticE[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2] - 137*EllipticF[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33
/2])))/3

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(304\) vs. \(2(139)=278\).
time = 0.10, size = 305, normalized size = 1.65

method	result	size
elliptic	\(\frac {\sqrt {-\left (3+5 x \right ) \left (-1+2 x \right ) \left (2+3 x \right )}\, \left (\frac {\left (-\frac {86}{675}-\frac {136 x}{675}\right ) \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{\left (x^{2}+\frac {19}{15} x +\frac {2}{5}\right )^{2}}-\frac {2 \left (15-30 x \right ) \left (-\frac {1034}{9}-\frac {544 x}{3}\right )}{\sqrt {\left (x^{2}+\frac {19}{15} x +\frac {2}{5}\right ) \left (15-30 x \right )}}+\frac {164 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{\sqrt {-30 x^{3}-23 x^{2}+7 x +6}}+\frac {5440 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \left (-\frac {\EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{15}-\frac {3 \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{5}\right )}{21 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}\right )}{\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}\)	\(234\)
default	\(-\frac {2 \sqrt {1-2 x}\, \left (4050 \sqrt {2}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x^{2} \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}-8160 \sqrt {2}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x^{2} \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}+5130 \sqrt {2}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}-10336 \sqrt {2}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}+1620 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )-3264 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )-244800 x^{4}-342780 x^{3}-61654 x^{2}+85168 x +30977\right )}{3 \left (2+3 x \right )^{\frac {3}{2}} \left (3+5 x \right )^{\frac {3}{2}} \left (-1+2 x \right )}\)	\(305\)

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)^(5/2),x,method=_RETURNVERBOSE)

[Out]

-2/3*(1-2*x)^(1/2)*(4050*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)-8160*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)+5130*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)-
10336*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)+1620*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))-3264*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))-244800*x^4-342780*x^3-61654
*x^2+85168*x+30977)/(2+3*x)^(3/2)/(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)^(3/2)/(2+3*x)^(5/2)/(3+5*x)^(5/2),x, algorithm="maxima")

[Out]

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

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Fricas [A]
time = 0.21, size = 60, normalized size = 0.32 \begin {gather*} \frac {2 \, {\left (122400 \, x^{3} + 232590 \, x^{2} + 147122 \, x + 30977\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}}{3 \, {\left (225 \, x^{4} + 570 \, x^{3} + 541 \, x^{2} + 228 \, x + 36\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)^(5/2),x, algorithm="fricas")

[Out]

2/3*(122400*x^3 + 232590*x^2 + 147122*x + 30977)*sqrt(5*x + 3)*sqrt(3*x + 2)*sqrt(-2*x + 1)/(225*x^4 + 570*x^3
+ 541*x^2 + 228*x + 36)

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

[Out]

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

[Out]

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

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