3.30.36 \(\int \frac {(2+3 x)^{9/2}}{(1-2 x)^{3/2} (3+5 x)^{5/2}} \, dx\) [2936]

Optimal. Leaf size=187 \[ -\frac {107 \sqrt {1-2 x} (2+3 x)^{5/2}}{1815 (3+5 x)^{3/2}}+\frac {7 (2+3 x)^{7/2}}{11 \sqrt {1-2 x} (3+5 x)^{3/2}}-\frac {4421 \sqrt {1-2 x} (2+3 x)^{3/2}}{99825 \sqrt {3+5 x}}+\frac {83093 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{166375}+\frac {5684677 E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{151250 \sqrt {33}}+\frac {84134 F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{75625 \sqrt {33}} \]

[Out]

5684677/4991250*EllipticE(1/7*21^(1/2)*(1-2*x)^(1/2),1/33*1155^(1/2))*33^(1/2)+84134/2495625*EllipticF(1/7*21^
(1/2)*(1-2*x)^(1/2),1/33*1155^(1/2))*33^(1/2)+7/11*(2+3*x)^(7/2)/(3+5*x)^(3/2)/(1-2*x)^(1/2)-107/1815*(2+3*x)^
(5/2)*(1-2*x)^(1/2)/(3+5*x)^(3/2)-4421/99825*(2+3*x)^(3/2)*(1-2*x)^(1/2)/(3+5*x)^(1/2)+83093/166375*(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 = 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 {84134 F\left (\text {ArcSin}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{75625 \sqrt {33}}+\frac {5684677 E\left (\text {ArcSin}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{151250 \sqrt {33}}+\frac {7 (3 x+2)^{7/2}}{11 \sqrt {1-2 x} (5 x+3)^{3/2}}-\frac {107 \sqrt {1-2 x} (3 x+2)^{5/2}}{1815 (5 x+3)^{3/2}}-\frac {4421 \sqrt {1-2 x} (3 x+2)^{3/2}}{99825 \sqrt {5 x+3}}+\frac {83093 \sqrt {1-2 x} \sqrt {5 x+3} \sqrt {3 x+2}}{166375} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-107*Sqrt[1 - 2*x]*(2 + 3*x)^(5/2))/(1815*(3 + 5*x)^(3/2)) + (7*(2 + 3*x)^(7/2))/(11*Sqrt[1 - 2*x]*(3 + 5*x)^
(3/2)) - (4421*Sqrt[1 - 2*x]*(2 + 3*x)^(3/2))/(99825*Sqrt[3 + 5*x]) + (83093*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[
3 + 5*x])/166375 + (5684677*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(151250*Sqrt[33]) + (84134*Elli
pticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(75625*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)^{3/2} (3+5 x)^{5/2}} \, dx &=\frac {7 (2+3 x)^{7/2}}{11 \sqrt {1-2 x} (3+5 x)^{3/2}}-\frac {1}{11} \int \frac {(2+3 x)^{5/2} \left (\frac {227}{2}+207 x\right )}{\sqrt {1-2 x} (3+5 x)^{5/2}} \, dx\\ &=-\frac {107 \sqrt {1-2 x} (2+3 x)^{5/2}}{1815 (3+5 x)^{3/2}}+\frac {7 (2+3 x)^{7/2}}{11 \sqrt {1-2 x} (3+5 x)^{3/2}}-\frac {2 \int \frac {(2+3 x)^{3/2} \left (\frac {24863}{4}+10728 x\right )}{\sqrt {1-2 x} (3+5 x)^{3/2}} \, dx}{1815}\\ &=-\frac {107 \sqrt {1-2 x} (2+3 x)^{5/2}}{1815 (3+5 x)^{3/2}}+\frac {7 (2+3 x)^{7/2}}{11 \sqrt {1-2 x} (3+5 x)^{3/2}}-\frac {4421 \sqrt {1-2 x} (2+3 x)^{3/2}}{99825 \sqrt {3+5 x}}-\frac {4 \int \frac {\sqrt {2+3 x} \left (\frac {904275}{8}+\frac {747837 x}{4}\right )}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx}{99825}\\ &=-\frac {107 \sqrt {1-2 x} (2+3 x)^{5/2}}{1815 (3+5 x)^{3/2}}+\frac {7 (2+3 x)^{7/2}}{11 \sqrt {1-2 x} (3+5 x)^{3/2}}-\frac {4421 \sqrt {1-2 x} (2+3 x)^{3/2}}{99825 \sqrt {3+5 x}}+\frac {83093 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{166375}+\frac {4 \int \frac {-\frac {32363109}{8}-\frac {51162093 x}{8}}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{1497375}\\ &=-\frac {107 \sqrt {1-2 x} (2+3 x)^{5/2}}{1815 (3+5 x)^{3/2}}+\frac {7 (2+3 x)^{7/2}}{11 \sqrt {1-2 x} (3+5 x)^{3/2}}-\frac {4421 \sqrt {1-2 x} (2+3 x)^{3/2}}{99825 \sqrt {3+5 x}}+\frac {83093 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{166375}-\frac {42067 \int \frac {1}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{75625}-\frac {5684677 \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx}{1663750}\\ &=-\frac {107 \sqrt {1-2 x} (2+3 x)^{5/2}}{1815 (3+5 x)^{3/2}}+\frac {7 (2+3 x)^{7/2}}{11 \sqrt {1-2 x} (3+5 x)^{3/2}}-\frac {4421 \sqrt {1-2 x} (2+3 x)^{3/2}}{99825 \sqrt {3+5 x}}+\frac {83093 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{166375}+\frac {5684677 E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{151250 \sqrt {33}}+\frac {84134 F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{75625 \sqrt {33}}\\ \end {align*}

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Mathematica [A]
time = 7.63, size = 136, normalized size = 0.73 \begin {gather*} \frac {10 \sqrt {2+3 x} \sqrt {3+5 x} \left (4534181+14153413 x+9376775 x^2-2695275 x^3\right )-5684677 \sqrt {2-4 x} (3+5 x)^2 E\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )|-\frac {33}{2}\right )+2908255 \sqrt {2-4 x} (3+5 x)^2 F\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )|-\frac {33}{2}\right )}{4991250 \sqrt {1-2 x} (3+5 x)^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(10*Sqrt[2 + 3*x]*Sqrt[3 + 5*x]*(4534181 + 14153413*x + 9376775*x^2 - 2695275*x^3) - 5684677*Sqrt[2 - 4*x]*(3
+ 5*x)^2*EllipticE[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2] + 2908255*Sqrt[2 - 4*x]*(3 + 5*x)^2*EllipticF[ArcS
in[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2])/(4991250*Sqrt[1 - 2*x]*(3 + 5*x)^2)

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Maple [A]
time = 0.10, size = 220, normalized size = 1.18

method	result	size
default	\(-\frac {\sqrt {2+3 x}\, \sqrt {1-2 x}\, \left (28423385 \sqrt {2}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}-13882110 \sqrt {2}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}[phone] \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )-8329266 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )-80858250 x^{4}[phone] x^{3}[phone] x^{2}[phone] x [phone]\right )}{4991250 \left (3+5 x \right )^{\frac {3}{2}} \left (6 x^{2}+x -2\right )}\)	\(220\)
elliptic	\(\frac {\sqrt {-\left (3+5 x \right ) \left (-1+2 x \right ) \left (2+3 x \right )}\, \left (-\frac {2401 \left (-30 x^{2}-38 x -12\right )}{5324 \sqrt {\left (-\frac {1}{2}+x \right ) \left (-30 x^{2}-38 x -12\right )}}-\frac {3595901 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{6987750 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}-\frac {5684677 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \left (-\frac {\EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{15}-\frac {3 \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{5}\right )}{6987750 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}+\frac {27 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{250}-\frac {2 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{1134375 \left (x +\frac {3}{5}\right )^{2}}-\frac {878 \left (-30 x^{2}-5 x +10\right )}{2495625 \sqrt {\left (x +\frac {3}{5}\right ) \left (-30 x^{2}-5 x +10\right )}}\right )}{\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}\)	\(272\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4991250*(2+3*x)^(1/2)*(1-2*x)^(1/2)*(28423385*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)-13882110*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)[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))-8329266*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))-80858250*x^4+227397750*x^3+612137890*x^2+419093690*x+90683620)/(3+5*x)^(3/2)/(6*x^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((2+3*x)^(9/2)/(1-2*x)^(3/2)/(3+5*x)^(5/2),x, algorithm="maxima")

[Out]

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

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Fricas [A]
time = 0.16, size = 55, normalized size = 0.29 \begin {gather*} \frac {{\left (2695275 \, x^{3} - 9376775 \, x^{2} - 14153413 \, x - 4534181\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}}{499125 \, {\left (50 \, x^{3} + 35 \, x^{2} - 12 \, x - 9\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/499125*(2695275*x^3 - 9376775*x^2 - 14153413*x - 4534181)*sqrt(5*x + 3)*sqrt(3*x + 2)*sqrt(-2*x + 1)/(50*x^3
+ 35*x^2 - 12*x - 9)

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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)**(3/2)/(3+5*x)**(5/2),x)

[Out]

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

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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