3.27.52 \(\int \frac {\sqrt {1-2 x} \sqrt {3+5 x}}{(2+3 x)^{9/2}} \, dx\) [2652]

Optimal. Leaf size=191 \[ -\frac {2 \sqrt {1-2 x} \sqrt {3+5 x}}{21 (2+3 x)^{7/2}}+\frac {74 \sqrt {1-2 x} \sqrt {3+5 x}}{735 (2+3 x)^{5/2}}+\frac {3184 \sqrt {1-2 x} \sqrt {3+5 x}}{5145 (2+3 x)^{3/2}}+\frac {220076 \sqrt {1-2 x} \sqrt {3+5 x}}{36015 \sqrt {2+3 x}}-\frac {220076 \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{36015}-\frac {6584 \sqrt {\frac {11}{3}} F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{36015} \]

[Out]

-220076/108045*EllipticE(1/7*21^(1/2)*(1-2*x)^(1/2),1/33*1155^(1/2))*33^(1/2)-6584/108045*EllipticF(1/7*21^(1/
2)*(1-2*x)^(1/2),1/33*1155^(1/2))*33^(1/2)-2/21*(1-2*x)^(1/2)*(3+5*x)^(1/2)/(2+3*x)^(7/2)+74/735*(1-2*x)^(1/2)
*(3+5*x)^(1/2)/(2+3*x)^(5/2)+3184/5145*(1-2*x)^(1/2)*(3+5*x)^(1/2)/(2+3*x)^(3/2)+220076/36015*(1-2*x)^(1/2)*(3
+5*x)^(1/2)/(2+3*x)^(1/2)

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Rubi [A]
time = 0.04, antiderivative size = 191, 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, 157, 164, 114, 120} \begin {gather*} -\frac {6584 \sqrt {\frac {11}{3}} F\left (\text {ArcSin}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{36015}-\frac {220076 \sqrt {\frac {11}{3}} E\left (\text {ArcSin}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{36015}+\frac {220076 \sqrt {1-2 x} \sqrt {5 x+3}}{36015 \sqrt {3 x+2}}+\frac {3184 \sqrt {1-2 x} \sqrt {5 x+3}}{5145 (3 x+2)^{3/2}}+\frac {74 \sqrt {1-2 x} \sqrt {5 x+3}}{735 (3 x+2)^{5/2}}-\frac {2 \sqrt {1-2 x} \sqrt {5 x+3}}{21 (3 x+2)^{7/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(21*(2 + 3*x)^(7/2)) + (74*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(735*(2 + 3*x)^(5/2))
+ (3184*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(5145*(2 + 3*x)^(3/2)) + (220076*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(36015*Sqr
t[2 + 3*x]) - (220076*Sqrt[11/3]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/36015 - (6584*Sqrt[11/3]*E
llipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/36015

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 157

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 + 1)*((e + f*x)^(p + 1)/((m + 1)*(b*c - a*d)*(b*e - a*f
))), x] + Dist[1/((m + 1)*(b*c - a*d)*(b*e - a*f)), Int[(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*x)^p*Simp[(a*d*f*
g - b*(d*e + c*f)*g + b*c*e*h)*(m + 1) - (b*g - a*h)*(d*e*(n + 1) + c*f*(p + 1)) - d*f*(b*g - a*h)*(m + n + p
+ 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x] && LtQ[m, -1] && 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} \sqrt {3+5 x}}{(2+3 x)^{9/2}} \, dx &=-\frac {2 \sqrt {1-2 x} \sqrt {3+5 x}}{21 (2+3 x)^{7/2}}+\frac {2}{21} \int \frac {-\frac {1}{2}-10 x}{\sqrt {1-2 x} (2+3 x)^{7/2} \sqrt {3+5 x}} \, dx\\ &=-\frac {2 \sqrt {1-2 x} \sqrt {3+5 x}}{21 (2+3 x)^{7/2}}+\frac {74 \sqrt {1-2 x} \sqrt {3+5 x}}{735 (2+3 x)^{5/2}}+\frac {4}{735} \int \frac {213-\frac {555 x}{2}}{\sqrt {1-2 x} (2+3 x)^{5/2} \sqrt {3+5 x}} \, dx\\ &=-\frac {2 \sqrt {1-2 x} \sqrt {3+5 x}}{21 (2+3 x)^{7/2}}+\frac {74 \sqrt {1-2 x} \sqrt {3+5 x}}{735 (2+3 x)^{5/2}}+\frac {3184 \sqrt {1-2 x} \sqrt {3+5 x}}{5145 (2+3 x)^{3/2}}+\frac {8 \int \frac {\frac {39099}{4}-5970 x}{\sqrt {1-2 x} (2+3 x)^{3/2} \sqrt {3+5 x}} \, dx}{15435}\\ &=-\frac {2 \sqrt {1-2 x} \sqrt {3+5 x}}{21 (2+3 x)^{7/2}}+\frac {74 \sqrt {1-2 x} \sqrt {3+5 x}}{735 (2+3 x)^{5/2}}+\frac {3184 \sqrt {1-2 x} \sqrt {3+5 x}}{5145 (2+3 x)^{3/2}}+\frac {220076 \sqrt {1-2 x} \sqrt {3+5 x}}{36015 \sqrt {2+3 x}}+\frac {16 \int \frac {\frac {261165}{2}+\frac {825285 x}{4}}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{108045}\\ &=-\frac {2 \sqrt {1-2 x} \sqrt {3+5 x}}{21 (2+3 x)^{7/2}}+\frac {74 \sqrt {1-2 x} \sqrt {3+5 x}}{735 (2+3 x)^{5/2}}+\frac {3184 \sqrt {1-2 x} \sqrt {3+5 x}}{5145 (2+3 x)^{3/2}}+\frac {220076 \sqrt {1-2 x} \sqrt {3+5 x}}{36015 \sqrt {2+3 x}}+\frac {36212 \int \frac {1}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{36015}+\frac {220076 \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx}{36015}\\ &=-\frac {2 \sqrt {1-2 x} \sqrt {3+5 x}}{21 (2+3 x)^{7/2}}+\frac {74 \sqrt {1-2 x} \sqrt {3+5 x}}{735 (2+3 x)^{5/2}}+\frac {3184 \sqrt {1-2 x} \sqrt {3+5 x}}{5145 (2+3 x)^{3/2}}+\frac {220076 \sqrt {1-2 x} \sqrt {3+5 x}}{36015 \sqrt {2+3 x}}-\frac {220076 \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{36015}-\frac {6584 \sqrt {\frac {11}{3}} F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{36015}\\ \end {align*}

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Mathematica [A]
time = 3.71, size = 106, normalized size = 0.55 \begin {gather*} \frac {4 \left (\frac {3 \sqrt {1-2 x} \sqrt {3+5 x} \left (926791+4100535 x+6042348 x^2+2971026 x^3\right )}{2 (2+3 x)^{7/2}}+\sqrt {2} \left (55019 E\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )|-\frac {33}{2}\right )-27860 F\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )|-\frac {33}{2}\right )\right )\right )}{108045} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(4*((3*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(926791 + 4100535*x + 6042348*x^2 + 2971026*x^3))/(2*(2 + 3*x)^(7/2)) + Sqr
t[2]*(55019*EllipticE[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2] - 27860*EllipticF[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*
x]], -33/2])))/108045

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(400\) vs. \(2(139)=278\).
time = 0.11, size = 401, normalized size = 2.10

method	result	size
elliptic	\(\frac {\sqrt {-\left (3+5 x \right ) \left (-1+2 x \right ) \left (2+3 x \right )}\, \left (-\frac {2 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{1701 \left (\frac {2}{3}+x \right )^{4}}+\frac {74 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{19845 \left (\frac {2}{3}+x \right )^{3}}+\frac {3184 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{46305 \left (\frac {2}{3}+x \right )^{2}}+\frac {-\frac {440152}{7203} x^{2}-\frac {220076}{36015} x +\frac {220076}{12005}}{\sqrt {\left (\frac {2}{3}+x \right ) \left (-30 x^{2}-3 x +9\right )}}+\frac {139288 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{151263 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}+\frac {220076 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \left (-\frac {\EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{15}-\frac {3 \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{5}\right )}{151263 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}\right )}{\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}\)	\(273\)
default	\(-\frac {2 \left (1466586 \sqrt {2}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x^{3} \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}-2971026 \sqrt {2}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x^{3} \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}+2933172 \sqrt {2}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x^{2} \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}-5942052 \sqrt {2}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x^{2} \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}+1955448 \sqrt {2}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}-3961368 \sqrt {2}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}+434544 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )-880304 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )-89130780 x^{5}-190183518 x^{4}-114403860 x^{3}[phone] x^{2}[phone] x +8341119\right ) \sqrt {1-2 x}\, \sqrt {3+5 x}}{108045 \left (10 x^{2}+x -3\right ) \left (2+3 x \right )^{\frac {7}{2}}}\)	\(401\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/108045*(1466586*2^(1/2)*EllipticF(1/7*(28+42*x)^(1/2),1/2*70^(1/2))*x^3*(2+3*x)^(1/2)*(-3-5*x)^(1/2)*(1-2*x
)^(1/2)-2971026*2^(1/2)*EllipticE(1/7*(28+42*x)^(1/2),1/2*70^(1/2))*x^3*(2+3*x)^(1/2)*(-3-5*x)^(1/2)*(1-2*x)^(
1/2)+2933172*2^(1/2)*EllipticF(1/7*(28+42*x)^(1/2),1/2*70^(1/2))*x^2*(2+3*x)^(1/2)*(-3-5*x)^(1/2)*(1-2*x)^(1/2
)-5942052*2^(1/2)*EllipticE(1/7*(28+42*x)^(1/2),1/2*70^(1/2))*x^2*(2+3*x)^(1/2)*(-3-5*x)^(1/2)*(1-2*x)^(1/2)+1
955448*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)-396136
8*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)+434544*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))-880304*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))-89130780*x^5-190183518*x^4-
114403860*x^3+14275797*x^2+34124442*x+8341119)*(1-2*x)^(1/2)*(3+5*x)^(1/2)/(10*x^2+x-3)/(2+3*x)^(7/2)

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

[Out]

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

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Fricas [A]
time = 0.28, size = 60, normalized size = 0.31 \begin {gather*} \frac {2 \, {\left (2971026 \, x^{3} + 6042348 \, x^{2} + 4100535 \, x + 926791\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}}{36015 \, {\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/36015*(2971026*x^3 + 6042348*x^2 + 4100535*x + 926791)*sqrt(5*x + 3)*sqrt(3*x + 2)*sqrt(-2*x + 1)/(81*x^4 +
216*x^3 + 216*x^2 + 96*x + 16)

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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