3.27.56 \(\int \frac {\sqrt {1-2 x} (3+5 x)^{3/2}}{\sqrt {2+3 x}} \, dx\) [2656]

Optimal. Leaf size=129 \[ -\frac {41}{135} \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}+\frac {2}{15} \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}-\frac {974}{675} \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )-\frac {41}{675} \sqrt {\frac {11}{3}} F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right ) \]

[Out]

-974/2025*EllipticE(1/7*21^(1/2)*(1-2*x)^(1/2),1/33*1155^(1/2))*33^(1/2)-41/2025*EllipticF(1/7*21^(1/2)*(1-2*x
)^(1/2),1/33*1155^(1/2))*33^(1/2)+2/15*(3+5*x)^(3/2)*(1-2*x)^(1/2)*(2+3*x)^(1/2)-41/135*(1-2*x)^(1/2)*(2+3*x)^
(1/2)*(3+5*x)^(1/2)

________________________________________________________________________________________

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 = {103, 159, 164, 114, 120} \begin {gather*} -\frac {41}{675} \sqrt {\frac {11}{3}} F\left (\text {ArcSin}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )-\frac {974}{675} \sqrt {\frac {11}{3}} E\left (\text {ArcSin}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )+\frac {2}{15} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{3/2}-\frac {41}{135} \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-41*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x])/135 + (2*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*(3 + 5*x)^(3/2))/15 - (97
4*Sqrt[11/3]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/675 - (41*Sqrt[11/3]*EllipticF[ArcSin[Sqrt[3/7
]*Sqrt[1 - 2*x]], 35/33])/675

Rule 103

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(a + b
*x)^m*(c + d*x)^n*((e + f*x)^(p + 1)/(f*(m + n + p + 1))), x] - Dist[1/(f*(m + n + p + 1)), Int[(a + b*x)^(m -
1)*(c + d*x)^(n - 1)*(e + f*x)^p*Simp[c*m*(b*e - a*f) + a*n*(d*e - c*f) + (d*m*(b*e - a*f) + b*n*(d*e - c*f))
*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && GtQ[m, 0] && GtQ[n, 0] && NeQ[m + n + p + 1, 0] && (Integ
ersQ[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 159

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)), x_Symb
ol] :> Simp[h*(a + b*x)^m*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(d*f*(m + n + p + 2))), x] + Dist[1/(d*f*(m + n
+ p + 2)), Int[(a + b*x)^(m - 1)*(c + d*x)^n*(e + f*x)^p*Simp[a*d*f*g*(m + n + p + 2) - h*(b*c*e*m + a*(d*e*(
n + 1) + c*f*(p + 1))) + (b*d*f*g*(m + n + p + 2) + h*(a*d*f*m - b*(d*e*(m + n + 1) + c*f*(m + p + 1))))*x, x]
, x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x] && GtQ[m, 0] && NeQ[m + n + p + 2, 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 {\sqrt {1-2 x} (3+5 x)^{3/2}}{\sqrt {2+3 x}} \, dx &=\frac {2}{15} \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}-\frac {2}{15} \int \frac {\left (-9-\frac {41 x}{2}\right ) \sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx\\ &=-\frac {41}{135} \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}+\frac {2}{15} \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}+\frac {2}{135} \int \frac {\frac {1259}{4}+487 x}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx\\ &=-\frac {41}{135} \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}+\frac {2}{15} \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}+\frac {451 \int \frac {1}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{1350}+\frac {974}{675} \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx\\ &=-\frac {41}{135} \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}+\frac {2}{15} \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}-\frac {974}{675} \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )-\frac {41}{675} \sqrt {\frac {11}{3}} F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]
time = 3.01, size = 92, normalized size = 0.71 \begin {gather*} \frac {15 \sqrt {2-4 x} \sqrt {2+3 x} \sqrt {3+5 x} (13+90 x)+1948 E\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )|-\frac {33}{2}\right )-595 F\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )|-\frac {33}{2}\right )}{2025 \sqrt {2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(15*Sqrt[2 - 4*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x]*(13 + 90*x) + 1948*EllipticE[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -
33/2] - 595*EllipticF[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2])/(2025*Sqrt[2])

________________________________________________________________________________________

Maple [A]
time = 0.10, size = 143, normalized size = 1.11

method	result	size
default	\(-\frac {\sqrt {3+5 x}\, \sqrt {1-2 x}\, \sqrt {2+3 x}\, \left (1353 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )-1948 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )-81000 x^{4}-73800 x^{3}+9930 x^{2}+18930 x +2340\right )}{4050 \left (30 x^{3}+23 x^{2}-7 x -6\right )}\)	\(143\)
elliptic	\(\frac {\sqrt {-\left (3+5 x \right ) \left (-1+2 x \right ) \left (2+3 x \right )}\, \left (\frac {2 x \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{3}+\frac {13 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{135}+\frac {1259 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{5670 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}+\frac {974 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \left (-\frac {\EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{15}-\frac {3 \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{5}\right )}{2835 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}\right )}{\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}\)	\(212\)
risch	\(-\frac {\left (13+90 x \right ) \sqrt {3+5 x}\, \left (-1+2 x \right ) \sqrt {2+3 x}\, \sqrt {\left (1-2 x \right ) \left (2+3 x \right ) \left (3+5 x \right )}}{135 \sqrt {-\left (3+5 x \right ) \left (-1+2 x \right ) \left (2+3 x \right )}\, \sqrt {1-2 x}}-\frac {\left (-\frac {1259 \sqrt {66+110 x}\, \sqrt {10+15 x}\, \sqrt {55-110 x}\, \EllipticF \left (\frac {\sqrt {66+110 x}}{11}, \frac {i \sqrt {66}}{2}\right )}{14850 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}-\frac {974 \sqrt {66+110 x}\, \sqrt {10+15 x}\, \sqrt {55-110 x}\, \left (\frac {\EllipticE \left (\frac {\sqrt {66+110 x}}{11}, \frac {i \sqrt {66}}{2}\right )}{15}-\frac {2 \EllipticF \left (\frac {\sqrt {66+110 x}}{11}, \frac {i \sqrt {66}}{2}\right )}{3}\right )}{7425 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}\right ) \sqrt {\left (1-2 x \right ) \left (2+3 x \right ) \left (3+5 x \right )}}{\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}\)	\(247\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4050*(3+5*x)^(1/2)*(1-2*x)^(1/2)*(2+3*x)^(1/2)*(1353*2^(1/2)*(2+3*x)^(1/2)*(-3-5*x)^(1/2)*(1-2*x)^(1/2)*Ell
ipticF(1/7*(28+42*x)^(1/2),1/2*70^(1/2))-1948*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))-81000*x^4-73800*x^3+9930*x^2+18930*x+2340)/(30*x^3+23*x^2-7*x-6)

________________________________________________________________________________________

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

[Out]

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

________________________________________________________________________________________

Fricas [A]
time = 0.14, size = 28, normalized size = 0.22 \begin {gather*} \frac {1}{135} \, {\left (90 \, x + 13\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/135*(90*x + 13)*sqrt(5*x + 3)*sqrt(3*x + 2)*sqrt(-2*x + 1)

________________________________________________________________________________________

Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {1 - 2 x} \left (5 x + 3\right )^{\frac {3}{2}}}{\sqrt {3 x + 2}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________