3.29.66 \(\int \frac {1}{\sqrt {1-2 x} (2+3 x)^{7/2} \sqrt {3+5 x}} \, dx\) [2866]

Optimal. Leaf size=158 \[ \frac {6 \sqrt {1-2 x} \sqrt {3+5 x}}{35 (2+3 x)^{5/2}}+\frac {296 \sqrt {1-2 x} \sqrt {3+5 x}}{245 (2+3 x)^{3/2}}+\frac {20644 \sqrt {1-2 x} \sqrt {3+5 x}}{1715 \sqrt {2+3 x}}-\frac {20644 \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{1715}-\frac {6856 F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{1715 \sqrt {33}} \]

[Out]

-20644/5145*EllipticE(1/7*21^(1/2)*(1-2*x)^(1/2),1/33*1155^(1/2))*33^(1/2)-6856/56595*EllipticF(1/7*21^(1/2)*(
1-2*x)^(1/2),1/33*1155^(1/2))*33^(1/2)+6/35*(1-2*x)^(1/2)*(3+5*x)^(1/2)/(2+3*x)^(5/2)+296/245*(1-2*x)^(1/2)*(3
+5*x)^(1/2)/(2+3*x)^(3/2)+20644/1715*(1-2*x)^(1/2)*(3+5*x)^(1/2)/(2+3*x)^(1/2)

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Rubi [A]
time = 0.03, antiderivative size = 158, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.179, Rules used = {106, 157, 164, 114, 120} \begin {gather*} -\frac {6856 F\left (\text {ArcSin}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{1715 \sqrt {33}}-\frac {20644 \sqrt {\frac {11}{3}} E\left (\text {ArcSin}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{1715}+\frac {20644 \sqrt {1-2 x} \sqrt {5 x+3}}{1715 \sqrt {3 x+2}}+\frac {296 \sqrt {1-2 x} \sqrt {5 x+3}}{245 (3 x+2)^{3/2}}+\frac {6 \sqrt {1-2 x} \sqrt {5 x+3}}{35 (3 x+2)^{5/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(6*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(35*(2 + 3*x)^(5/2)) + (296*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(245*(2 + 3*x)^(3/2))
+ (20644*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(1715*Sqrt[2 + 3*x]) - (20644*Sqrt[11/3]*EllipticE[ArcSin[Sqrt[3/7]*Sqr
t[1 - 2*x]], 35/33])/1715 - (6856*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(1715*Sqrt[33])

Rule 106

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[b*(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*(m + 1) - b*(d*e*(m + n + 2) +
c*f*(m + p + 2)) - b*d*f*(m + n + p + 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && LtQ[m, -1] &&
IntegersQ[2*m, 2*n, 2*p]

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 {1}{\sqrt {1-2 x} (2+3 x)^{7/2} \sqrt {3+5 x}} \, dx &=\frac {6 \sqrt {1-2 x} \sqrt {3+5 x}}{35 (2+3 x)^{5/2}}+\frac {2}{35} \int \frac {44-45 x}{\sqrt {1-2 x} (2+3 x)^{5/2} \sqrt {3+5 x}} \, dx\\ &=\frac {6 \sqrt {1-2 x} \sqrt {3+5 x}}{35 (2+3 x)^{5/2}}+\frac {296 \sqrt {1-2 x} \sqrt {3+5 x}}{245 (2+3 x)^{3/2}}+\frac {4}{735} \int \frac {\frac {3681}{2}-1110 x}{\sqrt {1-2 x} (2+3 x)^{3/2} \sqrt {3+5 x}} \, dx\\ &=\frac {6 \sqrt {1-2 x} \sqrt {3+5 x}}{35 (2+3 x)^{5/2}}+\frac {296 \sqrt {1-2 x} \sqrt {3+5 x}}{245 (2+3 x)^{3/2}}+\frac {20644 \sqrt {1-2 x} \sqrt {3+5 x}}{1715 \sqrt {2+3 x}}+\frac {8 \int \frac {24510+\frac {77415 x}{2}}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{5145}\\ &=\frac {6 \sqrt {1-2 x} \sqrt {3+5 x}}{35 (2+3 x)^{5/2}}+\frac {296 \sqrt {1-2 x} \sqrt {3+5 x}}{245 (2+3 x)^{3/2}}+\frac {20644 \sqrt {1-2 x} \sqrt {3+5 x}}{1715 \sqrt {2+3 x}}+\frac {3428 \int \frac {1}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{1715}+\frac {20644 \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx}{1715}\\ &=\frac {6 \sqrt {1-2 x} \sqrt {3+5 x}}{35 (2+3 x)^{5/2}}+\frac {296 \sqrt {1-2 x} \sqrt {3+5 x}}{245 (2+3 x)^{3/2}}+\frac {20644 \sqrt {1-2 x} \sqrt {3+5 x}}{1715 \sqrt {2+3 x}}-\frac {20644 \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{1715}-\frac {6856 F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{1715 \sqrt {33}}\\ \end {align*}

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Mathematica [A]
time = 3.53, size = 101, normalized size = 0.64 \begin {gather*} \frac {4 \left (\frac {3 \sqrt {1-2 x} \sqrt {3+5 x} \left (43507+126972 x+92898 x^2\right )}{2 (2+3 x)^{5/2}}+\sqrt {2} \left (5161 E\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )|-\frac {33}{2}\right )-2590 F\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )|-\frac {33}{2}\right )\right )\right )}{5145} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(4*((3*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(43507 + 126972*x + 92898*x^2))/(2*(2 + 3*x)^(5/2)) + Sqrt[2]*(5161*Ellipti
cE[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2] - 2590*EllipticF[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2])))/5145

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(307\) vs. \(2(116)=232\).
time = 0.10, size = 308, normalized size = 1.95

method	result	size
elliptic	\(\frac {\sqrt {-\left (3+5 x \right ) \left (-1+2 x \right ) \left (2+3 x \right )}\, \left (\frac {2 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{315 \left (\frac {2}{3}+x \right )^{3}}+\frac {296 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{2205 \left (\frac {2}{3}+x \right )^{2}}+\frac {-\frac {41288}{343} x^{2}-\frac {20644}{1715} x +\frac {61932}{1715}}{\sqrt {\left (\frac {2}{3}+x \right ) \left (-30 x^{2}-3 x +9\right )}}+\frac {13072 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{7203 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}+\frac {20644 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \left (-\frac {\EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{15}-\frac {3 \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{5}\right )}{7203 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}\right )}{\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}\)	\(249\)
default	\(-\frac {2 \left (46278 \sqrt {2}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x^{2} \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}-92898 \sqrt {2}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x^{2} \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}+61704 \sqrt {2}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}-123864 \sqrt {2}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}+20568 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )-41288 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )-2786940 x^{4}-4087854 x^{3}-850044 x^{2}+1012227 x +391563\right ) \sqrt {3+5 x}\, \sqrt {1-2 x}}{5145 \left (10 x^{2}+x -3\right ) \left (2+3 x \right )^{\frac {5}{2}}}\)	\(308\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/5145*(46278*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)-92898*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)+6
1704*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)-123864*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)+20568*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))-41288*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))-2786940*x^4-4087854*x^3-850044*x
^2+1012227*x+391563)*(3+5*x)^(1/2)*(1-2*x)^(1/2)/(10*x^2+x-3)/(2+3*x)^(5/2)

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

[Out]

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

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Fricas [A]
time = 0.24, size = 50, normalized size = 0.32 \begin {gather*} \frac {2 \, {\left (92898 \, x^{2} + 126972 \, x + 43507\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}}{1715 \, {\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/1715*(92898*x^2 + 126972*x + 43507)*sqrt(5*x + 3)*sqrt(3*x + 2)*sqrt(-2*x + 1)/(27*x^3 + 54*x^2 + 36*x + 8)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\sqrt {1 - 2 x} \left (3 x + 2\right )^{\frac {7}{2}} \sqrt {5 x + 3}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {1}{\sqrt {1-2\,x}\,{\left (3\,x+2\right )}^{7/2}\,\sqrt {5\,x+3}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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