3.30.81 \(\int \frac {(2+3 x)^{9/2}}{(1-2 x)^{5/2} (3+5 x)^{3/2}} \, dx\) [2981]

Optimal. Leaf size=187 \[ \frac {3284 \sqrt {1-2 x} (2+3 x)^{3/2}}{19965 \sqrt {3+5 x}}-\frac {665 (2+3 x)^{5/2}}{363 \sqrt {1-2 x} \sqrt {3+5 x}}+\frac {7 (2+3 x)^{7/2}}{33 (1-2 x)^{3/2} \sqrt {3+5 x}}-\frac {153319 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{66550}-\frac {5327983 E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{30250 \sqrt {33}}-\frac {160297 F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{30250 \sqrt {33}} \]

[Out]

-5327983/998250*EllipticE(1/7*21^(1/2)*(1-2*x)^(1/2),1/33*1155^(1/2))*33^(1/2)-160297/998250*EllipticF(1/7*21^
(1/2)*(1-2*x)^(1/2),1/33*1155^(1/2))*33^(1/2)+7/33*(2+3*x)^(7/2)/(1-2*x)^(3/2)/(3+5*x)^(1/2)-665/363*(2+3*x)^(
5/2)/(1-2*x)^(1/2)/(3+5*x)^(1/2)+3284/19965*(2+3*x)^(3/2)*(1-2*x)^(1/2)/(3+5*x)^(1/2)-153319/66550*(1-2*x)^(1/
2)*(2+3*x)^(1/2)*(3+5*x)^(1/2)

________________________________________________________________________________________

Rubi [A]
time = 0.05, antiderivative size = 187, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {100, 155, 159, 164, 114, 120} \begin {gather*} -\frac {160297 F\left (\text {ArcSin}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{30250 \sqrt {33}}-\frac {5327983 E\left (\text {ArcSin}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{30250 \sqrt {33}}+\frac {7 (3 x+2)^{7/2}}{33 (1-2 x)^{3/2} \sqrt {5 x+3}}-\frac {665 (3 x+2)^{5/2}}{363 \sqrt {1-2 x} \sqrt {5 x+3}}+\frac {3284 \sqrt {1-2 x} (3 x+2)^{3/2}}{19965 \sqrt {5 x+3}}-\frac {153319 \sqrt {1-2 x} \sqrt {5 x+3} \sqrt {3 x+2}}{66550} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(2 + 3*x)^(9/2)/((1 - 2*x)^(5/2)*(3 + 5*x)^(3/2)),x]

[Out]

(3284*Sqrt[1 - 2*x]*(2 + 3*x)^(3/2))/(19965*Sqrt[3 + 5*x]) - (665*(2 + 3*x)^(5/2))/(363*Sqrt[1 - 2*x]*Sqrt[3 +
5*x]) + (7*(2 + 3*x)^(7/2))/(33*(1 - 2*x)^(3/2)*Sqrt[3 + 5*x]) - (153319*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 +
5*x])/66550 - (5327983*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(30250*Sqrt[33]) - (160297*Elliptic
F[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(30250*Sqrt[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 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 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 {(2+3 x)^{9/2}}{(1-2 x)^{5/2} (3+5 x)^{3/2}} \, dx &=\frac {7 (2+3 x)^{7/2}}{33 (1-2 x)^{3/2} \sqrt {3+5 x}}-\frac {1}{33} \int \frac {(2+3 x)^{5/2} \left (\frac {359}{2}+306 x\right )}{(1-2 x)^{3/2} (3+5 x)^{3/2}} \, dx\\ &=-\frac {665 (2+3 x)^{5/2}}{363 \sqrt {1-2 x} \sqrt {3+5 x}}+\frac {7 (2+3 x)^{7/2}}{33 (1-2 x)^{3/2} \sqrt {3+5 x}}-\frac {1}{363} \int \frac {\left (-\frac {23357}{2}-\frac {40023 x}{2}\right ) (2+3 x)^{3/2}}{\sqrt {1-2 x} (3+5 x)^{3/2}} \, dx\\ &=\frac {3284 \sqrt {1-2 x} (2+3 x)^{3/2}}{19965 \sqrt {3+5 x}}-\frac {665 (2+3 x)^{5/2}}{363 \sqrt {1-2 x} \sqrt {3+5 x}}+\frac {7 (2+3 x)^{7/2}}{33 (1-2 x)^{3/2} \sqrt {3+5 x}}-\frac {2 \int \frac {\left (-\frac {425475}{2}-\frac {1379871 x}{4}\right ) \sqrt {2+3 x}}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx}{19965}\\ &=\frac {3284 \sqrt {1-2 x} (2+3 x)^{3/2}}{19965 \sqrt {3+5 x}}-\frac {665 (2+3 x)^{5/2}}{363 \sqrt {1-2 x} \sqrt {3+5 x}}+\frac {7 (2+3 x)^{7/2}}{33 (1-2 x)^{3/2} \sqrt {3+5 x}}-\frac {153319 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{66550}+\frac {2 \int \frac {\frac {60716097}{8}+\frac {47951847 x}{4}}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{299475}\\ &=\frac {3284 \sqrt {1-2 x} (2+3 x)^{3/2}}{19965 \sqrt {3+5 x}}-\frac {665 (2+3 x)^{5/2}}{363 \sqrt {1-2 x} \sqrt {3+5 x}}+\frac {7 (2+3 x)^{7/2}}{33 (1-2 x)^{3/2} \sqrt {3+5 x}}-\frac {153319 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{66550}+\frac {160297 \int \frac {1}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{60500}+\frac {5327983 \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx}{332750}\\ &=\frac {3284 \sqrt {1-2 x} (2+3 x)^{3/2}}{19965 \sqrt {3+5 x}}-\frac {665 (2+3 x)^{5/2}}{363 \sqrt {1-2 x} \sqrt {3+5 x}}+\frac {7 (2+3 x)^{7/2}}{33 (1-2 x)^{3/2} \sqrt {3+5 x}}-\frac {153319 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{66550}-\frac {5327983 E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{30250 \sqrt {33}}-\frac {160297 F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{30250 \sqrt {33}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]
time = 8.73, size = 102, normalized size = 0.55 \begin {gather*} \frac {-\frac {5 \sqrt {4+6 x} \left (2438391-3117099 x-11321446 x^2+1078110 x^3\right )}{(1-2 x)^{3/2} \sqrt {3+5 x}}[phone] E\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )|-\frac {33}{2}\right )-5366165 F\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )|-\frac {33}{2}\right )}{998250 \sqrt {2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(2 + 3*x)^(9/2)/((1 - 2*x)^(5/2)*(3 + 5*x)^(3/2)),x]

[Out]

((-5*Sqrt[4 + 6*x]*(2438391 - 3117099*x - 11321446*x^2 + 1078110*x^3))/((1 - 2*x)^(3/2)*Sqrt[3 + 5*x]) + 10655
966*EllipticE[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2] - 5366165*EllipticF[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -
33/2])/(998250*Sqrt[2])

________________________________________________________________________________________

Maple [A]
time = 0.11, size = 229, normalized size = 1.22

method	result	size
default	\(-\frac {\sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {3+5 x}\, \left (10579602 \sqrt {2}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}-21311932 \sqrt {2}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}-5289801 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )[phone] \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )[phone] x^{4}-318081180 x^{3}-319941890 x^{2}[phone] x [phone]\right )}{1996500 \left (15 x^{2}+19 x +6\right ) \left (-1+2 x \right )^{2}}\)	\(229\)
elliptic	\(\frac {\sqrt {-\left (3+5 x \right ) \left (-1+2 x \right ) \left (2+3 x \right )}\, \left (\frac {-\frac {217805}{2662} x^{2}-\frac {827659}{7986} x -\frac {43561}{1331}}{\sqrt {\left (-\frac {1}{2}+x \right ) \left (-30 x^{2}-38 x -12\right )}}+\frac {6746233 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{2795100 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}+\frac {5327983 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \left (-\frac {\EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{15}-\frac {3 \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{5}\right )}{1397550 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}-\frac {27 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{100}+\frac {2401 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{5808 \left (-\frac {1}{2}+x \right )^{2}}-\frac {2 \left (-30 x^{2}-5 x +10\right )}{166375 \sqrt {\left (x +\frac {3}{5}\right ) \left (-30 x^{2}-5 x +10\right )}}\right )}{\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}\)	\(272\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2+3*x)^(9/2)/(1-2*x)^(5/2)/(3+5*x)^(3/2),x,method=_RETURNVERBOSE)

[Out]

-1/1996500*(2+3*x)^(1/2)*(1-2*x)^(1/2)*(3+5*x)^(1/2)*(10579602*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)-21311932*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)-5289801*2^(1/2)*(2+3*x)^(1/2)*(-3-5*x)^(1/2)*(1-2*x)^(1/2)*Ellip
ticF(1/7*(28+42*x)^(1/2),1/2*70^(1/2))[phone]*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))[phone]*x^4-318081180*x^3-319941890*x^2+10809750*x+48767820)/(15*x^2+19*x+6)
/(-1+2*x)^2

________________________________________________________________________________________

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((2+3*x)^(9/2)/(1-2*x)^(5/2)/(3+5*x)^(3/2),x, algorithm="maxima")

[Out]

integrate((3*x + 2)^(9/2)/((5*x + 3)^(3/2)*(-2*x + 1)^(5/2)), x)

________________________________________________________________________________________

Fricas [A]
time = 0.18, size = 55, normalized size = 0.29 \begin {gather*} -\frac {{\left (1078110 \, x^{3} - 11321446 \, x^{2} - 3117099 \, x + 2438391\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}}{199650 \, {\left (20 \, x^{3} - 8 \, x^{2} - 7 \, x + 3\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2+3*x)^(9/2)/(1-2*x)^(5/2)/(3+5*x)^(3/2),x, algorithm="fricas")

[Out]

-1/199650*(1078110*x^3 - 11321446*x^2 - 3117099*x + 2438391)*sqrt(5*x + 3)*sqrt(3*x + 2)*sqrt(-2*x + 1)/(20*x^
3 - 8*x^2 - 7*x + 3)

________________________________________________________________________________________

Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2+3*x)**(9/2)/(1-2*x)**(5/2)/(3+5*x)**(3/2),x)

[Out]

Exception raised: SystemError >> excessive stack use: stack is 7317 deep

________________________________________________________________________________________

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((2+3*x)^(9/2)/(1-2*x)^(5/2)/(3+5*x)^(3/2),x, algorithm="giac")

[Out]

integrate((3*x + 2)^(9/2)/((5*x + 3)^(3/2)*(-2*x + 1)^(5/2)), x)

________________________________________________________________________________________

Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\left (3\,x+2\right )}^{9/2}}{{\left (1-2\,x\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((3*x + 2)^(9/2)/((1 - 2*x)^(5/2)*(5*x + 3)^(3/2)),x)

[Out]

int((3*x + 2)^(9/2)/((1 - 2*x)^(5/2)*(5*x + 3)^(3/2)), x)

________________________________________________________________________________________