3.28.37 \(\int \frac {(1-2 x)^{3/2} (2+3 x)^{7/2}}{(3+5 x)^{3/2}} \, dx\) [2737]

Optimal. Leaf size=222 \[ -\frac {2 (1-2 x)^{3/2} (2+3 x)^{7/2}}{5 \sqrt {3+5 x}}-\frac {12601 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{196875}+\frac {5153 \sqrt {1-2 x} (2+3 x)^{3/2} \sqrt {3+5 x}}{39375}+\frac {958 \sqrt {1-2 x} (2+3 x)^{5/2} \sqrt {3+5 x}}{1575}-\frac {8}{45} \sqrt {1-2 x} (2+3 x)^{7/2} \sqrt {3+5 x}-\frac {1473539 \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{1968750}-\frac {31288 \sqrt {\frac {11}{3}} F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{984375} \]

[Out]

-1473539/5906250*EllipticE(1/7*21^(1/2)*(1-2*x)^(1/2),1/33*1155^(1/2))*33^(1/2)-31288/2953125*EllipticF(1/7*21
^(1/2)*(1-2*x)^(1/2),1/33*1155^(1/2))*33^(1/2)-2/5*(1-2*x)^(3/2)*(2+3*x)^(7/2)/(3+5*x)^(1/2)+5153/39375*(2+3*x
)^(3/2)*(1-2*x)^(1/2)*(3+5*x)^(1/2)+958/1575*(2+3*x)^(5/2)*(1-2*x)^(1/2)*(3+5*x)^(1/2)-8/45*(2+3*x)^(7/2)*(1-2
*x)^(1/2)*(3+5*x)^(1/2)-12601/196875*(1-2*x)^(1/2)*(2+3*x)^(1/2)*(3+5*x)^(1/2)

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Rubi [A]
time = 0.06, antiderivative size = 222, normalized size of antiderivative = 1.00, number of steps used = 8, 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 {31288 \sqrt {\frac {11}{3}} F\left (\text {ArcSin}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{984375}-\frac {1473539 \sqrt {\frac {11}{3}} E\left (\text {ArcSin}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{1968750}-\frac {8}{45} \sqrt {1-2 x} \sqrt {5 x+3} (3 x+2)^{7/2}-\frac {2 (1-2 x)^{3/2} (3 x+2)^{7/2}}{5 \sqrt {5 x+3}}+\frac {958 \sqrt {1-2 x} \sqrt {5 x+3} (3 x+2)^{5/2}}{1575}+\frac {5153 \sqrt {1-2 x} \sqrt {5 x+3} (3 x+2)^{3/2}}{39375}-\frac {12601 \sqrt {1-2 x} \sqrt {5 x+3} \sqrt {3 x+2}}{196875} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*(1 - 2*x)^(3/2)*(2 + 3*x)^(7/2))/(5*Sqrt[3 + 5*x]) - (12601*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x])/196
875 + (5153*Sqrt[1 - 2*x]*(2 + 3*x)^(3/2)*Sqrt[3 + 5*x])/39375 + (958*Sqrt[1 - 2*x]*(2 + 3*x)^(5/2)*Sqrt[3 + 5
*x])/1575 - (8*Sqrt[1 - 2*x]*(2 + 3*x)^(7/2)*Sqrt[3 + 5*x])/45 - (1473539*Sqrt[11/3]*EllipticE[ArcSin[Sqrt[3/7
]*Sqrt[1 - 2*x]], 35/33])/1968750 - (31288*Sqrt[11/3]*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/98437
5

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 {(1-2 x)^{3/2} (2+3 x)^{7/2}}{(3+5 x)^{3/2}} \, dx &=-\frac {2 (1-2 x)^{3/2} (2+3 x)^{7/2}}{5 \sqrt {3+5 x}}+\frac {2}{5} \int \frac {\left (\frac {9}{2}-30 x\right ) \sqrt {1-2 x} (2+3 x)^{5/2}}{\sqrt {3+5 x}} \, dx\\ &=-\frac {2 (1-2 x)^{3/2} (2+3 x)^{7/2}}{5 \sqrt {3+5 x}}-\frac {8}{45} \sqrt {1-2 x} (2+3 x)^{7/2} \sqrt {3+5 x}+\frac {4}{675} \int \frac {\left (\frac {4875}{4}-\frac {7185 x}{2}\right ) (2+3 x)^{5/2}}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx\\ &=-\frac {2 (1-2 x)^{3/2} (2+3 x)^{7/2}}{5 \sqrt {3+5 x}}+\frac {958 \sqrt {1-2 x} (2+3 x)^{5/2} \sqrt {3+5 x}}{1575}-\frac {8}{45} \sqrt {1-2 x} (2+3 x)^{7/2} \sqrt {3+5 x}-\frac {4 \int \frac {(2+3 x)^{3/2} \left (-\frac {32295}{4}+\frac {77295 x}{4}\right )}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx}{23625}\\ &=-\frac {2 (1-2 x)^{3/2} (2+3 x)^{7/2}}{5 \sqrt {3+5 x}}+\frac {5153 \sqrt {1-2 x} (2+3 x)^{3/2} \sqrt {3+5 x}}{39375}+\frac {958 \sqrt {1-2 x} (2+3 x)^{5/2} \sqrt {3+5 x}}{1575}-\frac {8}{45} \sqrt {1-2 x} (2+3 x)^{7/2} \sqrt {3+5 x}+\frac {4 \int \frac {\sqrt {2+3 x} \left (\frac {1297125}{8}+\frac {567045 x}{4}\right )}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx}{590625}\\ &=-\frac {2 (1-2 x)^{3/2} (2+3 x)^{7/2}}{5 \sqrt {3+5 x}}-\frac {12601 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{196875}+\frac {5153 \sqrt {1-2 x} (2+3 x)^{3/2} \sqrt {3+5 x}}{39375}+\frac {958 \sqrt {1-2 x} (2+3 x)^{5/2} \sqrt {3+5 x}}{1575}-\frac {8}{45} \sqrt {1-2 x} (2+3 x)^{7/2} \sqrt {3+5 x}-\frac {4 \int \frac {-\frac {42883065}{8}-\frac {66309255 x}{8}}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{8859375}\\ &=-\frac {2 (1-2 x)^{3/2} (2+3 x)^{7/2}}{5 \sqrt {3+5 x}}-\frac {12601 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{196875}+\frac {5153 \sqrt {1-2 x} (2+3 x)^{3/2} \sqrt {3+5 x}}{39375}+\frac {958 \sqrt {1-2 x} (2+3 x)^{5/2} \sqrt {3+5 x}}{1575}-\frac {8}{45} \sqrt {1-2 x} (2+3 x)^{7/2} \sqrt {3+5 x}+\frac {172084 \int \frac {1}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{984375}+\frac {1473539 \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx}{1968750}\\ &=-\frac {2 (1-2 x)^{3/2} (2+3 x)^{7/2}}{5 \sqrt {3+5 x}}-\frac {12601 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{196875}+\frac {5153 \sqrt {1-2 x} (2+3 x)^{3/2} \sqrt {3+5 x}}{39375}+\frac {958 \sqrt {1-2 x} (2+3 x)^{5/2} \sqrt {3+5 x}}{1575}-\frac {8}{45} \sqrt {1-2 x} (2+3 x)^{7/2} \sqrt {3+5 x}-\frac {1473539 \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{1968750}-\frac {31288 \sqrt {\frac {11}{3}} F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{984375}\\ \end {align*}

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Mathematica [A]
time = 7.61, size = 132, normalized size = 0.59 \begin {gather*} \frac {1473539 \sqrt {2} (3+5 x) E\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )|-\frac {33}{2}\right )-5 \left (6 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x} \left (-83787-377530 x-252225 x^2+517500 x^3+472500 x^4\right )+88207 \sqrt {2} (3+5 x) F\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )|-\frac {33}{2}\right )\right )}{5906250 (3+5 x)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(1473539*Sqrt[2]*(3 + 5*x)*EllipticE[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2] - 5*(6*Sqrt[1 - 2*x]*Sqrt[2 + 3*
x]*Sqrt[3 + 5*x]*(-83787 - 377530*x - 252225*x^2 + 517500*x^3 + 472500*x^4) + 88207*Sqrt[2]*(3 + 5*x)*Elliptic
F[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2]))/(5906250*(3 + 5*x))

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Maple [A]
time = 0.10, size = 153, normalized size = 0.69

method	result	size
default	\(-\frac {\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}\, \left (85050000 x^{6}+1032504 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )-1473539 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )[phone] x^{5}-58225500 x^{4}-106572150 x^{3}-11274060 x^{2}[phone] x +5027220\right )}{5906250 \left (30 x^{3}+23 x^{2}-7 x -6\right )}\)	\(153\)
elliptic	\(-\frac {\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {-\left (3+5 x \right ) \left (-1+2 x \right ) \left (2+3 x \right )}\, \left (-\frac {12 x^{3} \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{25}-\frac {208 x^{2} \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{875}+\frac {349 x \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{875}+\frac {28391 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{196875}+\frac {952957 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{8268750 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}+\frac {1473539 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \left (-\frac {\EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{15}-\frac {3 \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{5}\right )}{8268750 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}-\frac {22 \left (-30 x^{2}-5 x +10\right )}{15625 \sqrt {\left (x +\frac {3}{5}\right ) \left (-30 x^{2}-5 x +10\right )}}\right )}{\left (6 x^{2}+x -2\right ) \sqrt {3+5 x}}\)	\(295\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/5906250*(1-2*x)^(1/2)*(2+3*x)^(1/2)*(3+5*x)^(1/2)*(85050000*x^6+1032504*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))-1473539*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^5-58225500*x^4-106572150*x^3-11274060*x^2+201
38190*x+5027220)/(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((1-2*x)^(3/2)*(2+3*x)^(7/2)/(3+5*x)^(3/2),x, algorithm="maxima")

[Out]

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

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Fricas [A]
time = 0.20, size = 43, normalized size = 0.19 \begin {gather*} -\frac {{\left (472500 \, x^{4} + 517500 \, x^{3} - 252225 \, x^{2} - 377530 \, x - 83787\right )} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}}{196875 \, \sqrt {5 \, x + 3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/196875*(472500*x^4 + 517500*x^3 - 252225*x^2 - 377530*x - 83787)*sqrt(3*x + 2)*sqrt(-2*x + 1)/sqrt(5*x + 3)

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

[Out]

Exception raised: SystemError >> excessive stack use: stack is 5986 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((1-2*x)^(3/2)*(2+3*x)^(7/2)/(3+5*x)^(3/2),x, algorithm="giac")

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\left (1-2\,x\right )}^{3/2}\,{\left (3\,x+2\right )}^{7/2}}{{\left (5\,x+3\right )}^{3/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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