3.29.71 \(\int \frac {1}{\sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}} \, dx\) [2871]

Optimal. Leaf size=63 \[ -\frac {10 \sqrt {1-2 x} \sqrt {2+3 x}}{11 \sqrt {3+5 x}}+2 \sqrt {\frac {3}{11}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right ) \]

[Out]

2/11*EllipticE(1/7*21^(1/2)*(1-2*x)^(1/2),1/33*1155^(1/2))*33^(1/2)-10/11*(1-2*x)^(1/2)*(2+3*x)^(1/2)/(3+5*x)^
(1/2)

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Rubi [A]
time = 0.01, antiderivative size = 63, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.107, Rules used = {106, 21, 114} \begin {gather*} 2 \sqrt {\frac {3}{11}} E\left (\text {ArcSin}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )-\frac {10 \sqrt {1-2 x} \sqrt {3 x+2}}{11 \sqrt {5 x+3}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-10*Sqrt[1 - 2*x]*Sqrt[2 + 3*x])/(11*Sqrt[3 + 5*x]) + 2*Sqrt[3/11]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]],
35/33]

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])

Rubi steps

\begin {align*} \int \frac {1}{\sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}} \, dx &=-\frac {10 \sqrt {1-2 x} \sqrt {2+3 x}}{11 \sqrt {3+5 x}}-\frac {2}{11} \int \frac {9+15 x}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx\\ &=-\frac {10 \sqrt {1-2 x} \sqrt {2+3 x}}{11 \sqrt {3+5 x}}-\frac {6}{11} \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx\\ &=-\frac {10 \sqrt {1-2 x} \sqrt {2+3 x}}{11 \sqrt {3+5 x}}+2 \sqrt {\frac {3}{11}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )\\ \end {align*}

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Mathematica [A]
time = 2.63, size = 106, normalized size = 1.68 \begin {gather*} -\frac {2 \left (5 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}+\sqrt {2} (3+5 x) E\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )|-\frac {33}{2}\right )-\sqrt {2} (3+5 x) F\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )|-\frac {33}{2}\right )\right )}{33+55 x} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*(5*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x] + Sqrt[2]*(3 + 5*x)*EllipticE[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]
], -33/2] - Sqrt[2]*(3 + 5*x)*EllipticF[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2]))/(33 + 55*x)

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Maple [A]
time = 0.10, size = 91, normalized size = 1.44

method	result	size
default	\(-\frac {2 \sqrt {3+5 x}\, \sqrt {1-2 x}\, \sqrt {2+3 x}\, \left (\sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )+30 x^{2}+5 x -10\right )}{11 \left (30 x^{3}+23 x^{2}-7 x -6\right )}\)	\(91\)
elliptic	\(\frac {\sqrt {-\left (3+5 x \right ) \left (-1+2 x \right ) \left (2+3 x \right )}\, \left (-\frac {6 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{77 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}-\frac {10 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \left (-\frac {\EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{15}-\frac {3 \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{5}\right )}{77 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}-\frac {2 \left (-30 x^{2}-5 x +10\right )}{11 \sqrt {\left (x +\frac {3}{5}\right ) \left (-30 x^{2}-5 x +10\right )}}\right )}{\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}\)	\(201\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/11*(3+5*x)^(1/2)*(1-2*x)^(1/2)*(2+3*x)^(1/2)*(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))+30*x^2+5*x-10)/(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/(3+5*x)^(3/2)/(1-2*x)^(1/2)/(2+3*x)^(1/2),x, algorithm="maxima")

[Out]

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

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Fricas [A]
time = 0.23, size = 23, normalized size = 0.37 \begin {gather*} -\frac {10 \, \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}}{11 \, \sqrt {5 \, x + 3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-10/11*sqrt(3*x + 2)*sqrt(-2*x + 1)/sqrt(5*x + 3)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {1}{\sqrt {1-2\,x}\,\sqrt {3\,x+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/((1 - 2*x)^(1/2)*(3*x + 2)^(1/2)*(5*x + 3)^(3/2)),x)

[Out]

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

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