3.28.4 \(\int \frac {(1-2 x)^{3/2} \sqrt {3+5 x}}{(2+3 x)^{5/2}} \, dx\) [2704]

Optimal. Leaf size=129 \[ -\frac {2 (1-2 x)^{3/2} \sqrt {3+5 x}}{9 (2+3 x)^{3/2}}+\frac {82 \sqrt {1-2 x} \sqrt {3+5 x}}{27 \sqrt {2+3 x}}-\frac {98}{27} \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )+\frac {16}{27} \sqrt {\frac {11}{3}} F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right ) \]

[Out]

-98/81*EllipticE(1/7*21^(1/2)*(1-2*x)^(1/2),1/33*1155^(1/2))*33^(1/2)+16/81*EllipticF(1/7*21^(1/2)*(1-2*x)^(1/
2),1/33*1155^(1/2))*33^(1/2)-2/9*(1-2*x)^(3/2)*(3+5*x)^(1/2)/(2+3*x)^(3/2)+82/27*(1-2*x)^(1/2)*(3+5*x)^(1/2)/(
2+3*x)^(1/2)

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Rubi [A]
time = 0.03, antiderivative size = 129, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.179, Rules used = {99, 155, 164, 114, 120} \begin {gather*} \frac {16}{27} \sqrt {\frac {11}{3}} F\left (\text {ArcSin}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )-\frac {98}{27} \sqrt {\frac {11}{3}} E\left (\text {ArcSin}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )-\frac {2 \sqrt {5 x+3} (1-2 x)^{3/2}}{9 (3 x+2)^{3/2}}+\frac {82 \sqrt {5 x+3} \sqrt {1-2 x}}{27 \sqrt {3 x+2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*(1 - 2*x)^(3/2)*Sqrt[3 + 5*x])/(9*(2 + 3*x)^(3/2)) + (82*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(27*Sqrt[2 + 3*x]) -
 (98*Sqrt[11/3]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/27 + (16*Sqrt[11/3]*EllipticF[ArcSin[Sqrt[3
/7]*Sqrt[1 - 2*x]], 35/33])/27

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 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} \sqrt {3+5 x}}{(2+3 x)^{5/2}} \, dx &=-\frac {2 (1-2 x)^{3/2} \sqrt {3+5 x}}{9 (2+3 x)^{3/2}}+\frac {2}{9} \int \frac {\left (-\frac {13}{2}-20 x\right ) \sqrt {1-2 x}}{(2+3 x)^{3/2} \sqrt {3+5 x}} \, dx\\ &=-\frac {2 (1-2 x)^{3/2} \sqrt {3+5 x}}{9 (2+3 x)^{3/2}}+\frac {82 \sqrt {1-2 x} \sqrt {3+5 x}}{27 \sqrt {2+3 x}}-\frac {4}{27} \int \frac {-\frac {103}{2}-\frac {245 x}{2}}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx\\ &=-\frac {2 (1-2 x)^{3/2} \sqrt {3+5 x}}{9 (2+3 x)^{3/2}}+\frac {82 \sqrt {1-2 x} \sqrt {3+5 x}}{27 \sqrt {2+3 x}}-\frac {88}{27} \int \frac {1}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx+\frac {98}{27} \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx\\ &=-\frac {2 (1-2 x)^{3/2} \sqrt {3+5 x}}{9 (2+3 x)^{3/2}}+\frac {82 \sqrt {1-2 x} \sqrt {3+5 x}}{27 \sqrt {2+3 x}}-\frac {98}{27} \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )+\frac {16}{27} \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 = 6.30, size = 97, normalized size = 0.75 \begin {gather*} \frac {2}{81} \left (\frac {3 \sqrt {1-2 x} \sqrt {3+5 x} (79+129 x)}{(2+3 x)^{3/2}}+49 \sqrt {2} E\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )|-\frac {33}{2}\right )-181 \sqrt {2} 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)*Sqrt[3 + 5*x])/(2 + 3*x)^(5/2),x]

[Out]

(2*((3*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(79 + 129*x))/(2 + 3*x)^(3/2) + 49*Sqrt[2]*EllipticE[ArcSin[Sqrt[2/11]*Sqrt
[3 + 5*x]], -33/2] - 181*Sqrt[2]*EllipticF[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2]))/81

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(214\) vs. \(2(93)=186\).
time = 0.10, size = 215, normalized size = 1.67

method result size
default \(\frac {2 \left (396 \sqrt {2}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}+147 \sqrt {2}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}+264 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )+98 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )+3870 x^{3}+2757 x^{2}-924 x -711\right ) \sqrt {1-2 x}\, \sqrt {3+5 x}}{81 \left (10 x^{2}+x -3\right ) \left (2+3 x \right )^{\frac {3}{2}}}\) \(215\)
elliptic \(\frac {\sqrt {-\left (3+5 x \right ) \left (-1+2 x \right ) \left (2+3 x \right )}\, \left (\frac {-\frac {860}{27} x^{2}-\frac {86}{27} x +\frac {86}{9}}{\sqrt {\left (\frac {2}{3}+x \right ) \left (-30 x^{2}-3 x +9\right )}}+\frac {206 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{567 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}+\frac {70 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \left (-\frac {\EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{15}-\frac {3 \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{5}\right )}{81 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}-\frac {14 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{243 \left (\frac {2}{3}+x \right )^{2}}\right )}{\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}\) \(225\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/81*(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)+147
*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))+98*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))+3870*x^3+2757*x^2-924*x-711)*(1-2*x
)^(1/2)*(3+5*x)^(1/2)/(10*x^2+x-3)/(2+3*x)^(3/2)

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

[Out]

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

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Fricas [A]
time = 0.15, size = 40, normalized size = 0.31 \begin {gather*} \frac {2 \, {\left (129 \, x + 79\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}}{27 \, {\left (9 \, x^{2} + 12 \, x + 4\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/27*(129*x + 79)*sqrt(5*x + 3)*sqrt(3*x + 2)*sqrt(-2*x + 1)/(9*x^2 + 12*x + 4)

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

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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