3.29.91 \(\int \frac {(2+3 x)^{7/2} \sqrt {3+5 x}}{(1-2 x)^{3/2}} \, dx\) [2891]

Optimal. Leaf size=186 \[ \frac {29293}{875} \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}+\frac {2517}{350} \sqrt {1-2 x} (2+3 x)^{3/2} \sqrt {3+5 x}+\frac {12}{7} \sqrt {1-2 x} (2+3 x)^{5/2} \sqrt {3+5 x}+\frac {(2+3 x)^{7/2} \sqrt {3+5 x}}{\sqrt {1-2 x}}+\frac {4071079 \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{17500}+\frac {673523 F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{8750 \sqrt {33}} \]

[Out]

4071079/52500*EllipticE(1/7*21^(1/2)*(1-2*x)^(1/2),1/33*1155^(1/2))*33^(1/2)+673523/288750*EllipticF(1/7*21^(1
/2)*(1-2*x)^(1/2),1/33*1155^(1/2))*33^(1/2)+(2+3*x)^(7/2)*(3+5*x)^(1/2)/(1-2*x)^(1/2)+2517/350*(2+3*x)^(3/2)*(
1-2*x)^(1/2)*(3+5*x)^(1/2)+12/7*(2+3*x)^(5/2)*(1-2*x)^(1/2)*(3+5*x)^(1/2)+29293/875*(1-2*x)^(1/2)*(2+3*x)^(1/2
)*(3+5*x)^(1/2)

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Rubi [A]
time = 0.04, antiderivative size = 186, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.179, Rules used = {99, 159, 164, 114, 120} \begin {gather*} \frac {673523 F\left (\text {ArcSin}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{8750 \sqrt {33}}+\frac {4071079 \sqrt {\frac {11}{3}} E\left (\text {ArcSin}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{17500}+\frac {\sqrt {5 x+3} (3 x+2)^{7/2}}{\sqrt {1-2 x}}+\frac {12}{7} \sqrt {1-2 x} \sqrt {5 x+3} (3 x+2)^{5/2}+\frac {2517}{350} \sqrt {1-2 x} \sqrt {5 x+3} (3 x+2)^{3/2}+\frac {29293}{875} \sqrt {1-2 x} \sqrt {5 x+3} \sqrt {3 x+2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((2 + 3*x)^(7/2)*Sqrt[3 + 5*x])/(1 - 2*x)^(3/2),x]

[Out]

(29293*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x])/875 + (2517*Sqrt[1 - 2*x]*(2 + 3*x)^(3/2)*Sqrt[3 + 5*x])/350
+ (12*Sqrt[1 - 2*x]*(2 + 3*x)^(5/2)*Sqrt[3 + 5*x])/7 + ((2 + 3*x)^(7/2)*Sqrt[3 + 5*x])/Sqrt[1 - 2*x] + (40710
79*Sqrt[11/3]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/17500 + (673523*EllipticF[ArcSin[Sqrt[3/7]*Sq
rt[1 - 2*x]], 35/33])/(8750*Sqrt[33])

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 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)^{7/2} \sqrt {3+5 x}}{(1-2 x)^{3/2}} \, dx &=\frac {(2+3 x)^{7/2} \sqrt {3+5 x}}{\sqrt {1-2 x}}-\int \frac {(2+3 x)^{5/2} \left (\frac {73}{2}+60 x\right )}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx\\ &=\frac {12}{7} \sqrt {1-2 x} (2+3 x)^{5/2} \sqrt {3+5 x}+\frac {(2+3 x)^{7/2} \sqrt {3+5 x}}{\sqrt {1-2 x}}+\frac {1}{35} \int \frac {\left (-3845-\frac {12585 x}{2}\right ) (2+3 x)^{3/2}}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx\\ &=\frac {2517}{350} \sqrt {1-2 x} (2+3 x)^{3/2} \sqrt {3+5 x}+\frac {12}{7} \sqrt {1-2 x} (2+3 x)^{5/2} \sqrt {3+5 x}+\frac {(2+3 x)^{7/2} \sqrt {3+5 x}}{\sqrt {1-2 x}}-\frac {1}{875} \int \frac {\sqrt {2+3 x} \left (\frac {1083625}{4}+439395 x\right )}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx\\ &=\frac {29293}{875} \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}+\frac {2517}{350} \sqrt {1-2 x} (2+3 x)^{3/2} \sqrt {3+5 x}+\frac {12}{7} \sqrt {1-2 x} (2+3 x)^{5/2} \sqrt {3+5 x}+\frac {(2+3 x)^{7/2} \sqrt {3+5 x}}{\sqrt {1-2 x}}+\frac {\int \frac {-9665070-\frac {61066185 x}{4}}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{13125}\\ &=\frac {29293}{875} \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}+\frac {2517}{350} \sqrt {1-2 x} (2+3 x)^{3/2} \sqrt {3+5 x}+\frac {12}{7} \sqrt {1-2 x} (2+3 x)^{5/2} \sqrt {3+5 x}+\frac {(2+3 x)^{7/2} \sqrt {3+5 x}}{\sqrt {1-2 x}}-\frac {673523 \int \frac {1}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{17500}-\frac {4071079 \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx}{17500}\\ &=\frac {29293}{875} \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}+\frac {2517}{350} \sqrt {1-2 x} (2+3 x)^{3/2} \sqrt {3+5 x}+\frac {12}{7} \sqrt {1-2 x} (2+3 x)^{5/2} \sqrt {3+5 x}+\frac {(2+3 x)^{7/2} \sqrt {3+5 x}}{\sqrt {1-2 x}}+\frac {4071079 \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{17500}+\frac {673523 F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{8750 \sqrt {33}}\\ \end {align*}

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Mathematica [A]
time = 6.57, size = 115, normalized size = 0.62 \begin {gather*} \frac {-30 \sqrt {2+3 x} \sqrt {3+5 x} \left (-109756+54757 x+26010 x^2+6750 x^3\right )-4071079 \sqrt {2-4 x} E\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )|-\frac {33}{2}\right )+2050510 \sqrt {2-4 x} F\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )|-\frac {33}{2}\right )}{52500 \sqrt {1-2 x}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((2 + 3*x)^(7/2)*Sqrt[3 + 5*x])/(1 - 2*x)^(3/2),x]

[Out]

(-30*Sqrt[2 + 3*x]*Sqrt[3 + 5*x]*(-109756 + 54757*x + 26010*x^2 + 6750*x^3) - 4071079*Sqrt[2 - 4*x]*EllipticE[
ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2] + 2050510*Sqrt[2 - 4*x]*EllipticF[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -
33/2])/(52500*Sqrt[1 - 2*x])

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Maple [A]
time = 0.10, size = 148, normalized size = 0.80

method	result	size
default	\(-\frac {\sqrt {2+3 x}\, \sqrt {3+5 x}\, \sqrt {1-2 x}\, \left (4071079 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )-2020569 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )-3037500 x^{5}-15552000 x^{4}-40681350 x^{3}[phone] x^{2}[phone] x [phone]\right )}{52500 \left (30 x^{3}+23 x^{2}-7 x -6\right )}\)	\(148\)
elliptic	\(\frac {\sqrt {-\left (3+5 x \right ) \left (-1+2 x \right ) \left (2+3 x \right )}\, \left (\frac {27 x^{2} \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{14}+\frac {5877 x \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{700}+\frac {138899 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{7000}-\frac {644338 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{18375 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}-\frac {4071079 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \left (-\frac {\EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{15}-\frac {3 \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{5}\right )}{73500 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}-\frac {343 \left (-30 x^{2}-38 x -12\right )}{16 \sqrt {\left (-\frac {1}{2}+x \right ) \left (-30 x^{2}-38 x -12\right )}}\right )}{\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}\)	\(262\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2+3*x)^(7/2)*(3+5*x)^(1/2)/(1-2*x)^(3/2),x,method=_RETURNVERBOSE)

[Out]

-1/52500*(2+3*x)^(1/2)*(3+5*x)^(1/2)*(1-2*x)^(1/2)*(4071079*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))-2020569*2^(1/2)*(2+3*x)^(1/2)*(-3-5*x)^(1/2)*(1-2*x)^(1/2)*Ellipt
icF(1/7*(28+42*x)^(1/2),1/2*70^(1/2))-3037500*x^5-15552000*x^4-40681350*x^3+13496910*x^2+52704660*x+19756080)/
(30*x^3+23*x^2-7*x-6)

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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)^(7/2)*(3+5*x)^(1/2)/(1-2*x)^(3/2),x, algorithm="maxima")

[Out]

integrate(sqrt(5*x + 3)*(3*x + 2)^(7/2)/(-2*x + 1)^(3/2), x)

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Fricas [A]
time = 0.28, size = 45, normalized size = 0.24 \begin {gather*} \frac {{\left (6750 \, x^{3} + 26010 \, x^{2} + 54757 \, x - 109756\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}}{1750 \, {\left (2 \, x - 1\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2+3*x)^(7/2)*(3+5*x)^(1/2)/(1-2*x)^(3/2),x, algorithm="fricas")

[Out]

1/1750*(6750*x^3 + 26010*x^2 + 54757*x - 109756)*sqrt(5*x + 3)*sqrt(3*x + 2)*sqrt(-2*x + 1)/(2*x - 1)

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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)**(7/2)*(3+5*x)**(1/2)/(1-2*x)**(3/2),x)

[Out]

Exception raised: SystemError >> excessive stack use: stack is 3877 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)^(7/2)*(3+5*x)^(1/2)/(1-2*x)^(3/2),x, algorithm="giac")

[Out]

integrate(sqrt(5*x + 3)*(3*x + 2)^(7/2)/(-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 )}^{7/2}\,\sqrt {5\,x+3}}{{\left (1-2\,x\right )}^{3/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((3*x + 2)^(7/2)*(5*x + 3)^(1/2))/(1 - 2*x)^(3/2),x)

[Out]

int(((3*x + 2)^(7/2)*(5*x + 3)^(1/2))/(1 - 2*x)^(3/2), x)

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