3.30.22 \(\int \frac {1}{(1-2 x)^{3/2} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx\) [2922]

Optimal. Leaf size=81 \[ \frac {4 \sqrt {2+3 x} \sqrt {3+5 x}}{77 \sqrt {1-2 x}}+\frac {2 \sqrt {\frac {5}{7}} \sqrt {-3-5 x} E\left (\sin ^{-1}\left (\sqrt {5} \sqrt {2+3 x}\right )|\frac {2}{35}\right )}{11 \sqrt {3+5 x}} \]

[Out]

2/77*EllipticE(5^(1/2)*(2+3*x)^(1/2),1/35*70^(1/2))*35^(1/2)*(-3-5*x)^(1/2)/(3+5*x)^(1/2)+4/77*(2+3*x)^(1/2)*(
3+5*x)^(1/2)/(1-2*x)^(1/2)

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Rubi [A]
time = 0.02, antiderivative size = 81, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {106, 21, 115, 114} \begin {gather*} \frac {2 \sqrt {\frac {5}{7}} \sqrt {-5 x-3} E\left (\text {ArcSin}\left (\sqrt {5} \sqrt {3 x+2}\right )|\frac {2}{35}\right )}{11 \sqrt {5 x+3}}+\frac {4 \sqrt {3 x+2} \sqrt {5 x+3}}{77 \sqrt {1-2 x}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(4*Sqrt[2 + 3*x]*Sqrt[3 + 5*x])/(77*Sqrt[1 - 2*x]) + (2*Sqrt[5/7]*Sqrt[-3 - 5*x]*EllipticE[ArcSin[Sqrt[5]*Sqrt
[2 + 3*x]], 2/35])/(11*Sqrt[3 + 5*x])

Rule 21

Int[(u_.)*((a_) + (b_.)*(v_))^(m_.)*((c_) + (d_.)*(v_))^(n_.), x_Symbol] :> Dist[(b/d)^m, Int[u*(c + d*v)^(m +
n), x], x] /; FreeQ[{a, b, c, d, n}, x] && EqQ[b*c - a*d, 0] && IntegerQ[m] && ( !IntegerQ[n] || SimplerQ[c +
d*x, a + b*x])

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 115

Int[Sqrt[(e_.) + (f_.)*(x_)]/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]), x_Symbol] :> Dist[Sqrt[e + f*x
]*(Sqrt[b*((c + d*x)/(b*c - a*d))]/(Sqrt[c + d*x]*Sqrt[b*((e + f*x)/(b*e - a*f))])), Int[Sqrt[b*(e/(b*e - a*f)
) + b*f*(x/(b*e - a*f))]/(Sqrt[a + b*x]*Sqrt[b*(c/(b*c - a*d)) + b*d*(x/(b*c - a*d))]), x], 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]

Rubi steps

\begin {align*} \int \frac {1}{(1-2 x)^{3/2} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx &=\frac {4 \sqrt {2+3 x} \sqrt {3+5 x}}{77 \sqrt {1-2 x}}-\frac {2}{77} \int \frac {-\frac {15}{2}+15 x}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx\\ &=\frac {4 \sqrt {2+3 x} \sqrt {3+5 x}}{77 \sqrt {1-2 x}}+\frac {15}{77} \int \frac {\sqrt {1-2 x}}{\sqrt {2+3 x} \sqrt {3+5 x}} \, dx\\ &=\frac {4 \sqrt {2+3 x} \sqrt {3+5 x}}{77 \sqrt {1-2 x}}+\frac {\left (15 \sqrt {-3-5 x}\right ) \int \frac {\sqrt {\frac {3}{7}-\frac {6 x}{7}}}{\sqrt {-9-15 x} \sqrt {2+3 x}} \, dx}{11 \sqrt {7} \sqrt {3+5 x}}\\ &=\frac {4 \sqrt {2+3 x} \sqrt {3+5 x}}{77 \sqrt {1-2 x}}+\frac {2 \sqrt {\frac {5}{7}} \sqrt {-3-5 x} E\left (\sin ^{-1}\left (\sqrt {5} \sqrt {2+3 x}\right )|\frac {2}{35}\right )}{11 \sqrt {3+5 x}}\\ \end {align*}

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Mathematica [C] Result contains complex when optimal does not.
time = 2.69, size = 61, normalized size = 0.75 \begin {gather*} \frac {2}{77} \left (\frac {2 \sqrt {2+3 x} \sqrt {3+5 x}}{\sqrt {1-2 x}}-i \sqrt {33} E\left (i \sinh ^{-1}\left (\sqrt {9+15 x}\right )|-\frac {2}{33}\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*((2*Sqrt[2 + 3*x]*Sqrt[3 + 5*x])/Sqrt[1 - 2*x] - I*Sqrt[33]*EllipticE[I*ArcSinh[Sqrt[9 + 15*x]], -2/33]))/7
7

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(132\) vs. \(2(60)=120\).
time = 0.10, size = 133, normalized size = 1.64

method	result	size
default	\(-\frac {\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}\, \left (33 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )+2 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )+60 x^{2}+76 x +24\right )}{77 \left (30 x^{3}+23 x^{2}-7 x -6\right )}\)	\(133\)
elliptic	\(\frac {\sqrt {-\left (3+5 x \right ) \left (-1+2 x \right ) \left (2+3 x \right )}\, \left (-\frac {2 \left (-30 x^{2}-38 x -12\right )}{77 \sqrt {\left (-\frac {1}{2}+x \right ) \left (-30 x^{2}-38 x -12\right )}}+\frac {5 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{539 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}-\frac {10 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \left (-\frac {\EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{15}-\frac {3 \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{5}\right )}{539 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}\right )}{\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}\)	\(201\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/77*(1-2*x)^(1/2)*(2+3*x)^(1/2)*(3+5*x)^(1/2)*(33*2^(1/2)*(2+3*x)^(1/2)*(-3-5*x)^(1/2)*(1-2*x)^(1/2)*Ellipti
cF(1/7*(28+42*x)^(1/2),1/2*70^(1/2))+2*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))+60*x^2+76*x+24)/(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/(1-2*x)^(3/2)/(2+3*x)^(1/2)/(3+5*x)^(1/2),x, algorithm="maxima")

[Out]

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

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Fricas [A]
time = 0.16, size = 30, normalized size = 0.37 \begin {gather*} -\frac {4 \, \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}}{77 \, {\left (2 \, x - 1\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-4/77*sqrt(5*x + 3)*sqrt(3*x + 2)*sqrt(-2*x + 1)/(2*x - 1)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

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

[Out]

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

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