3.30.72 \(\int \frac {(2+3 x)^{7/2}}{(1-2 x)^{5/2} \sqrt {3+5 x}} \, dx\) [2972]

Optimal. Leaf size=156 \[ -\frac {4517 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{1210}-\frac {679 (2+3 x)^{3/2} \sqrt {3+5 x}}{363 \sqrt {1-2 x}}+\frac {7 (2+3 x)^{5/2} \sqrt {3+5 x}}{33 (1-2 x)^{3/2}}-\frac {78472 E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{275 \sqrt {33}}-\frac {4721 F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{550 \sqrt {33}} \]

[Out]

-78472/9075*EllipticE(1/7*21^(1/2)*(1-2*x)^(1/2),1/33*1155^(1/2))*33^(1/2)-4721/18150*EllipticF(1/7*21^(1/2)*(
1-2*x)^(1/2),1/33*1155^(1/2))*33^(1/2)+7/33*(2+3*x)^(5/2)*(3+5*x)^(1/2)/(1-2*x)^(3/2)-679/363*(2+3*x)^(3/2)*(3
+5*x)^(1/2)/(1-2*x)^(1/2)-4517/1210*(1-2*x)^(1/2)*(2+3*x)^(1/2)*(3+5*x)^(1/2)

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Rubi [A]
time = 0.04, antiderivative size = 156, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {100, 155, 159, 164, 114, 120} \begin {gather*} -\frac {4721 F\left (\text {ArcSin}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{550 \sqrt {33}}-\frac {78472 E\left (\text {ArcSin}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{275 \sqrt {33}}+\frac {7 \sqrt {5 x+3} (3 x+2)^{5/2}}{33 (1-2 x)^{3/2}}-\frac {679 \sqrt {5 x+3} (3 x+2)^{3/2}}{363 \sqrt {1-2 x}}-\frac {4517 \sqrt {1-2 x} \sqrt {5 x+3} \sqrt {3 x+2}}{1210} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-4517*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x])/1210 - (679*(2 + 3*x)^(3/2)*Sqrt[3 + 5*x])/(363*Sqrt[1 - 2*x
]) + (7*(2 + 3*x)^(5/2)*Sqrt[3 + 5*x])/(33*(1 - 2*x)^(3/2)) - (78472*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]]
, 35/33])/(275*Sqrt[33]) - (4721*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(550*Sqrt[33])

Rule 100

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(b*c -
a*d)*(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*((e + f*x)^(p + 1)/(b*(b*e - a*f)*(m + 1))), x] + Dist[1/(b*(b*e - a*
f)*(m + 1)), Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 2)*(e + f*x)^p*Simp[a*d*(d*e*(n - 1) + c*f*(p + 1)) + b*c*(d
*e*(m - n + 2) - c*f*(m + p + 2)) + d*(a*d*f*(n + p) + b*(d*e*(m + 1) - c*f*(m + n + p + 1)))*x, x], x], x] /;
FreeQ[{a, b, c, d, e, f, p}, x] && LtQ[m, -1] && GtQ[n, 1] && (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 155

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*((e + f*x)^(p + 1)/(b*(b*e - a*f)*(m + 1))), x] - Dist[1
/(b*(b*e - a*f)*(m + 1)), Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*(e + f*x)^p*Simp[b*c*(f*g - e*h)*(m + 1) + (
b*g - a*h)*(d*e*n + c*f*(p + 1)) + d*(b*(f*g - e*h)*(m + 1) + f*(b*g - a*h)*(n + p + 1))*x, x], x], x] /; Free
Q[{a, b, c, d, e, f, g, h, p}, x] && LtQ[m, -1] && GtQ[n, 0] && IntegersQ[2*m, 2*n, 2*p]

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 {(2+3 x)^{7/2}}{(1-2 x)^{5/2} \sqrt {3+5 x}} \, dx &=\frac {7 (2+3 x)^{5/2} \sqrt {3+5 x}}{33 (1-2 x)^{3/2}}-\frac {1}{33} \int \frac {(2+3 x)^{3/2} \left (\frac {373}{2}+306 x\right )}{(1-2 x)^{3/2} \sqrt {3+5 x}} \, dx\\ &=-\frac {679 (2+3 x)^{3/2} \sqrt {3+5 x}}{363 \sqrt {1-2 x}}+\frac {7 (2+3 x)^{5/2} \sqrt {3+5 x}}{33 (1-2 x)^{3/2}}-\frac {1}{363} \int \frac {\left (-\frac {25065}{2}-\frac {40653 x}{2}\right ) \sqrt {2+3 x}}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx\\ &=-\frac {4517 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{1210}-\frac {679 (2+3 x)^{3/2} \sqrt {3+5 x}}{363 \sqrt {1-2 x}}+\frac {7 (2+3 x)^{5/2} \sqrt {3+5 x}}{33 (1-2 x)^{3/2}}+\frac {\int \frac {\frac {1788471}{4}+706248 x}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{5445}\\ &=-\frac {4517 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{1210}-\frac {679 (2+3 x)^{3/2} \sqrt {3+5 x}}{363 \sqrt {1-2 x}}+\frac {7 (2+3 x)^{5/2} \sqrt {3+5 x}}{33 (1-2 x)^{3/2}}+\frac {4721 \int \frac {1}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{1100}+\frac {78472 \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx}{3025}\\ &=-\frac {4517 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{1210}-\frac {679 (2+3 x)^{3/2} \sqrt {3+5 x}}{363 \sqrt {1-2 x}}+\frac {7 (2+3 x)^{5/2} \sqrt {3+5 x}}{33 (1-2 x)^{3/2}}-\frac {78472 E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{275 \sqrt {33}}-\frac {4721 F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{550 \sqrt {33}}\\ \end {align*}

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Mathematica [A]
time = 8.13, size = 120, normalized size = 0.77 \begin {gather*} -\frac {10 \sqrt {2+3 x} \sqrt {3+5 x} \left (24051-70234 x+6534 x^2\right )+313888 \sqrt {2-4 x} (-1+2 x) E\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )|-\frac {33}{2}\right )-158095 \sqrt {2-4 x} (-1+2 x) F\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )|-\frac {33}{2}\right )}{36300 (1-2 x)^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/36300*(10*Sqrt[2 + 3*x]*Sqrt[3 + 5*x]*(24051 - 70234*x + 6534*x^2) + 313888*Sqrt[2 - 4*x]*(-1 + 2*x)*Ellipt
icE[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2] - 158095*Sqrt[2 - 4*x]*(-1 + 2*x)*EllipticF[ArcSin[Sqrt[2/11]*Sqr
t[3 + 5*x]], -33/2])/(1 - 2*x)^(3/2)

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Maple [A]
time = 0.10, size = 229, normalized size = 1.47

method	result	size
default	\(-\frac {\left (311586 \sqrt {2}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}-627776 \sqrt {2}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}-155793 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )+313888 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )+980100 x^{4}-9293640 x^{3}-9344770 x^{2}+355650 x +1443060\right ) \sqrt {3+5 x}\, \sqrt {1-2 x}\, \sqrt {2+3 x}}{36300 \left (-1+2 x \right )^{2} \left (15 x^{2}+19 x +6\right )}\)	\(229\)
elliptic	\(\frac {\sqrt {-\left (3+5 x \right ) \left (-1+2 x \right ) \left (2+3 x \right )}\, \left (\frac {-\frac {15925}{121} x^{2}-\frac {60515}{363} x -\frac {6370}{121}}{\sqrt {\left (-\frac {1}{2}+x \right ) \left (-30 x^{2}-38 x -12\right )}}+\frac {198719 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{50820 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}+\frac {78472 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \left (-\frac {\EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{15}-\frac {3 \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{5}\right )}{12705 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}-\frac {9 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{20}+\frac {343 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{528 \left (-\frac {1}{2}+x \right )^{2}}\right )}{\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}\)	\(244\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/36300*(311586*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)-627776*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)-15
5793*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))+313888*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))+980100*x^4-9293640
*x^3-9344770*x^2+355650*x+1443060)*(3+5*x)^(1/2)*(1-2*x)^(1/2)*(2+3*x)^(1/2)/(-1+2*x)^2/(15*x^2+19*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((2+3*x)^(7/2)/(1-2*x)^(5/2)/(3+5*x)^(1/2),x, algorithm="maxima")

[Out]

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

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Fricas [A]
time = 0.28, size = 45, normalized size = 0.29 \begin {gather*} -\frac {{\left (6534 \, x^{2} - 70234 \, x + 24051\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}}{3630 \, {\left (4 \, x^{2} - 4 \, x + 1\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3630*(6534*x^2 - 70234*x + 24051)*sqrt(5*x + 3)*sqrt(3*x + 2)*sqrt(-2*x + 1)/(4*x^2 - 4*x + 1)

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

[Out]

Exception raised: SystemError >> excessive stack use: stack is 3062 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((2+3*x)^(7/2)/(1-2*x)^(5/2)/(3+5*x)^(1/2),x, algorithm="giac")

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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