3.30.51 \(\int \frac {\sqrt {3+5 x}}{(1-2 x)^{5/2} (2+3 x)^{3/2}} \, dx\) [2951]

Optimal. Leaf size=156 \[ \frac {2 \sqrt {3+5 x}}{21 (1-2 x)^{3/2} \sqrt {2+3 x}}+\frac {194 \sqrt {3+5 x}}{1617 \sqrt {1-2 x} \sqrt {2+3 x}}-\frac {458 \sqrt {1-2 x} \sqrt {3+5 x}}{3773 \sqrt {2+3 x}}+\frac {458 E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{343 \sqrt {33}}-\frac {178 F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{343 \sqrt {33}} \]

[Out]

458/11319*EllipticE(1/7*21^(1/2)*(1-2*x)^(1/2),1/33*1155^(1/2))*33^(1/2)-178/11319*EllipticF(1/7*21^(1/2)*(1-2
*x)^(1/2),1/33*1155^(1/2))*33^(1/2)+2/21*(3+5*x)^(1/2)/(1-2*x)^(3/2)/(2+3*x)^(1/2)+194/1617*(3+5*x)^(1/2)/(1-2
*x)^(1/2)/(2+3*x)^(1/2)-458/3773*(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 = 156, 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 = {101, 157, 164, 114, 120} \begin {gather*} -\frac {178 F\left (\text {ArcSin}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{343 \sqrt {33}}+\frac {458 E\left (\text {ArcSin}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{343 \sqrt {33}}-\frac {458 \sqrt {1-2 x} \sqrt {5 x+3}}{3773 \sqrt {3 x+2}}+\frac {194 \sqrt {5 x+3}}{1617 \sqrt {1-2 x} \sqrt {3 x+2}}+\frac {2 \sqrt {5 x+3}}{21 (1-2 x)^{3/2} \sqrt {3 x+2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*Sqrt[3 + 5*x])/(21*(1 - 2*x)^(3/2)*Sqrt[2 + 3*x]) + (194*Sqrt[3 + 5*x])/(1617*Sqrt[1 - 2*x]*Sqrt[2 + 3*x])
- (458*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(3773*Sqrt[2 + 3*x]) + (458*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/
33])/(343*Sqrt[33]) - (178*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(343*Sqrt[33])

Rule 101

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 + 1)/((m + 1)*(b*e - a*f))), x] - Dist[1/((m + 1)*(b*e - a*f)), Int[(a +
b*x)^(m + 1)*(c + d*x)^(n - 1)*(e + f*x)^p*Simp[d*e*n + c*f*(m + p + 2) + d*f*(m + n + p + 2)*x, x], x], x] /;
FreeQ[{a, b, c, d, e, f, p}, x] && LtQ[m, -1] && GtQ[n, 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 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 {\sqrt {3+5 x}}{(1-2 x)^{5/2} (2+3 x)^{3/2}} \, dx &=\frac {2 \sqrt {3+5 x}}{21 (1-2 x)^{3/2} \sqrt {2+3 x}}-\frac {2}{21} \int \frac {-13-\frac {45 x}{2}}{(1-2 x)^{3/2} (2+3 x)^{3/2} \sqrt {3+5 x}} \, dx\\ &=\frac {2 \sqrt {3+5 x}}{21 (1-2 x)^{3/2} \sqrt {2+3 x}}+\frac {194 \sqrt {3+5 x}}{1617 \sqrt {1-2 x} \sqrt {2+3 x}}+\frac {4 \int \frac {\frac {741}{4}+\frac {1455 x}{4}}{\sqrt {1-2 x} (2+3 x)^{3/2} \sqrt {3+5 x}} \, dx}{1617}\\ &=\frac {2 \sqrt {3+5 x}}{21 (1-2 x)^{3/2} \sqrt {2+3 x}}+\frac {194 \sqrt {3+5 x}}{1617 \sqrt {1-2 x} \sqrt {2+3 x}}-\frac {458 \sqrt {1-2 x} \sqrt {3+5 x}}{3773 \sqrt {2+3 x}}+\frac {8 \int \frac {-\frac {1185}{8}-\frac {3435 x}{4}}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{11319}\\ &=\frac {2 \sqrt {3+5 x}}{21 (1-2 x)^{3/2} \sqrt {2+3 x}}+\frac {194 \sqrt {3+5 x}}{1617 \sqrt {1-2 x} \sqrt {2+3 x}}-\frac {458 \sqrt {1-2 x} \sqrt {3+5 x}}{3773 \sqrt {2+3 x}}-\frac {458 \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx}{3773}+\frac {89}{343} \int \frac {1}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx\\ &=\frac {2 \sqrt {3+5 x}}{21 (1-2 x)^{3/2} \sqrt {2+3 x}}+\frac {194 \sqrt {3+5 x}}{1617 \sqrt {1-2 x} \sqrt {2+3 x}}-\frac {458 \sqrt {1-2 x} \sqrt {3+5 x}}{3773 \sqrt {2+3 x}}+\frac {458 E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{343 \sqrt {33}}-\frac {178 F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{343 \sqrt {33}}\\ \end {align*}

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Mathematica [A]
time = 6.35, size = 99, normalized size = 0.63 \begin {gather*} \frac {-\frac {2 \sqrt {3+5 x} \left (-531-1390 x+2748 x^2\right )}{(1-2 x)^{3/2} \sqrt {2+3 x}}+\sqrt {2} \left (-458 E\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )|-\frac {33}{2}\right )+3395 F\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )|-\frac {33}{2}\right )\right )}{11319} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((-2*Sqrt[3 + 5*x]*(-531 - 1390*x + 2748*x^2))/((1 - 2*x)^(3/2)*Sqrt[2 + 3*x]) + Sqrt[2]*(-458*EllipticE[ArcSi
n[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2] + 3395*EllipticF[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2]))/11319

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Maple [A]
time = 0.10, size = 224, normalized size = 1.44

method	result	size
default	\(-\frac {\left (5874 \sqrt {2}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}+916 \sqrt {2}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}-2937 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )-458 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )+27480 x^{3}+2588 x^{2}-13650 x -3186\right ) \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {3+5 x}}{11319 \left (15 x^{2}+19 x +6\right ) \left (-1+2 x \right )^{2}}\)	\(224\)
elliptic	\(\frac {\sqrt {-\left (3+5 x \right ) \left (-1+2 x \right ) \left (2+3 x \right )}\, \left (-\frac {260 \left (-30 x^{2}-38 x -12\right )}{11319 \sqrt {\left (-\frac {1}{2}+x \right ) \left (-30 x^{2}-38 x -12\right )}}-\frac {395 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{79233 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}-\frac {2290 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \left (-\frac {\EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{15}-\frac {3 \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{5}\right )}{79233 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}-\frac {6 \left (-30 x^{2}-3 x +9\right )}{343 \sqrt {\left (\frac {2}{3}+x \right ) \left (-30 x^{2}-3 x +9\right )}}+\frac {\sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{147 \left (-\frac {1}{2}+x \right )^{2}}\right )}{\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}\)	\(253\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/11319*(5874*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
)+916*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)-2937*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))-458*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))+27480*x^3+2588*x^2-13650*x-3
186)*(2+3*x)^(1/2)*(1-2*x)^(1/2)*(3+5*x)^(1/2)/(15*x^2+19*x+6)/(-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((3+5*x)^(1/2)/(1-2*x)^(5/2)/(2+3*x)^(3/2),x, algorithm="maxima")

[Out]

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

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Fricas [A]
time = 0.24, size = 50, normalized size = 0.32 \begin {gather*} -\frac {2 \, {\left (2748 \, x^{2} - 1390 \, x - 531\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}}{11319 \, {\left (12 \, x^{3} - 4 \, x^{2} - 5 \, x + 2\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/11319*(2748*x^2 - 1390*x - 531)*sqrt(5*x + 3)*sqrt(3*x + 2)*sqrt(-2*x + 1)/(12*x^3 - 4*x^2 - 5*x + 2)

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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