3.27.45 \(\int \sqrt {1-2 x} (2+3 x)^{5/2} \sqrt {3+5 x} \, dx\) [2645]

Optimal. Leaf size=191 \[ -\frac {175111 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{236250}-\frac {1244 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}}{13125}-\frac {23 \sqrt {1-2 x} (2+3 x)^{3/2} (3+5 x)^{3/2}}{1575}+\frac {2}{45} \sqrt {1-2 x} (2+3 x)^{5/2} (3+5 x)^{3/2}-\frac {2911577 \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{590625}-\frac {175111 \sqrt {\frac {11}{3}} F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{1181250} \]

[Out]

-2911577/1771875*EllipticE(1/7*21^(1/2)*(1-2*x)^(1/2),1/33*1155^(1/2))*33^(1/2)-175111/3543750*EllipticF(1/7*2
1^(1/2)*(1-2*x)^(1/2),1/33*1155^(1/2))*33^(1/2)-23/1575*(2+3*x)^(3/2)*(3+5*x)^(3/2)*(1-2*x)^(1/2)+2/45*(2+3*x)
^(5/2)*(3+5*x)^(3/2)*(1-2*x)^(1/2)-1244/13125*(3+5*x)^(3/2)*(1-2*x)^(1/2)*(2+3*x)^(1/2)-175111/236250*(1-2*x)^
(1/2)*(2+3*x)^(1/2)*(3+5*x)^(1/2)

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Rubi [A]
time = 0.05, 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 = {103, 159, 164, 114, 120} \begin {gather*} -\frac {175111 \sqrt {\frac {11}{3}} F\left (\text {ArcSin}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{1181250}-\frac {2911577 \sqrt {\frac {11}{3}} E\left (\text {ArcSin}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{590625}+\frac {2}{45} \sqrt {1-2 x} (5 x+3)^{3/2} (3 x+2)^{5/2}-\frac {23 \sqrt {1-2 x} (5 x+3)^{3/2} (3 x+2)^{3/2}}{1575}-\frac {1244 \sqrt {1-2 x} (5 x+3)^{3/2} \sqrt {3 x+2}}{13125}-\frac {175111 \sqrt {1-2 x} \sqrt {5 x+3} \sqrt {3 x+2}}{236250} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-175111*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x])/236250 - (1244*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*(3 + 5*x)^(3/2)
)/13125 - (23*Sqrt[1 - 2*x]*(2 + 3*x)^(3/2)*(3 + 5*x)^(3/2))/1575 + (2*Sqrt[1 - 2*x]*(2 + 3*x)^(5/2)*(3 + 5*x)
^(3/2))/45 - (2911577*Sqrt[11/3]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/590625 - (175111*Sqrt[11/3
]*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/1181250

Rule 103

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(a + b
*x)^m*(c + d*x)^n*((e + f*x)^(p + 1)/(f*(m + n + p + 1))), x] - Dist[1/(f*(m + n + p + 1)), Int[(a + b*x)^(m -
 1)*(c + d*x)^(n - 1)*(e + f*x)^p*Simp[c*m*(b*e - a*f) + a*n*(d*e - c*f) + (d*m*(b*e - a*f) + b*n*(d*e - c*f))
*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && GtQ[m, 0] && GtQ[n, 0] && NeQ[m + n + p + 1, 0] && (Integ
ersQ[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 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 \sqrt {1-2 x} (2+3 x)^{5/2} \sqrt {3+5 x} \, dx &=\frac {2}{45} \sqrt {1-2 x} (2+3 x)^{5/2} (3+5 x)^{3/2}-\frac {2}{45} \int \frac {\left (-\frac {27}{2}-\frac {23 x}{2}\right ) (2+3 x)^{3/2} \sqrt {3+5 x}}{\sqrt {1-2 x}} \, dx\\ &=-\frac {23 \sqrt {1-2 x} (2+3 x)^{3/2} (3+5 x)^{3/2}}{1575}+\frac {2}{45} \sqrt {1-2 x} (2+3 x)^{5/2} (3+5 x)^{3/2}+\frac {2 \int \frac {\sqrt {2+3 x} \sqrt {3+5 x} \left (\frac {4815}{4}+1866 x\right )}{\sqrt {1-2 x}} \, dx}{1575}\\ &=-\frac {1244 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}}{13125}-\frac {23 \sqrt {1-2 x} (2+3 x)^{3/2} (3+5 x)^{3/2}}{1575}+\frac {2}{45} \sqrt {1-2 x} (2+3 x)^{5/2} (3+5 x)^{3/2}-\frac {2 \int \frac {\left (-\frac {170757}{2}-\frac {525333 x}{4}\right ) \sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx}{39375}\\ &=-\frac {175111 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{236250}-\frac {1244 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}}{13125}-\frac {23 \sqrt {1-2 x} (2+3 x)^{3/2} (3+5 x)^{3/2}}{1575}+\frac {2}{45} \sqrt {1-2 x} (2+3 x)^{5/2} (3+5 x)^{3/2}+\frac {2 \int \frac {\frac {22119087}{8}+\frac {8734731 x}{2}}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{354375}\\ &=-\frac {175111 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{236250}-\frac {1244 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}}{13125}-\frac {23 \sqrt {1-2 x} (2+3 x)^{3/2} (3+5 x)^{3/2}}{1575}+\frac {2}{45} \sqrt {1-2 x} (2+3 x)^{5/2} (3+5 x)^{3/2}+\frac {1926221 \int \frac {1}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{2362500}+\frac {2911577 \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx}{590625}\\ &=-\frac {175111 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{236250}-\frac {1244 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}}{13125}-\frac {23 \sqrt {1-2 x} (2+3 x)^{3/2} (3+5 x)^{3/2}}{1575}+\frac {2}{45} \sqrt {1-2 x} (2+3 x)^{5/2} (3+5 x)^{3/2}-\frac {2911577 \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{590625}-\frac {175111 \sqrt {\frac {11}{3}} F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{1181250}\\ \end {align*}

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Mathematica [A]
time = 3.55, size = 102, normalized size = 0.53 \begin {gather*} \frac {15 \sqrt {2-4 x} \sqrt {2+3 x} \sqrt {3+5 x} \left (-136987+410490 x+861750 x^2+472500 x^3\right )+11646308 E\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )|-\frac {33}{2}\right )-5867645 F\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )|-\frac {33}{2}\right )}{3543750 \sqrt {2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(15*Sqrt[2 - 4*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x]*(-136987 + 410490*x + 861750*x^2 + 472500*x^3) + 11646308*Ellipt
icE[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2] - 5867645*EllipticF[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2])/(35
43750*Sqrt[2])

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Maple [A]
time = 0.12, size = 153, normalized size = 0.80

method result size
default \(-\frac {\sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {3+5 x}\, \left (-425250000 x^{6}+5778663 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )-11646308 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )-1101600000 x^{5}-864823500 x^{4}+106067700 x^{3}+335838930 x^{2}+45120930 x -24657660\right )}{7087500 \left (30 x^{3}+23 x^{2}-7 x -6\right )}\) \(153\)
elliptic \(\frac {\sqrt {-\left (3+5 x \right ) \left (-1+2 x \right ) \left (2+3 x \right )}\, \left (2 x^{3} \sqrt {-30 x^{3}-23 x^{2}+7 x +6}+\frac {383 x^{2} \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{105}+\frac {4561 x \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{2625}-\frac {136987 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{236250}+\frac {7373029 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{9922500 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}+\frac {2911577 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \left (-\frac {\EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{15}-\frac {3 \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{5}\right )}{2480625 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}\right )}{\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}\) \(256\)
risch \(-\frac {\left (472500 x^{3}+861750 x^{2}+410490 x -136987\right ) \sqrt {3+5 x}\, \left (-1+2 x \right ) \sqrt {2+3 x}\, \sqrt {\left (1-2 x \right ) \left (2+3 x \right ) \left (3+5 x \right )}}{236250 \sqrt {-\left (3+5 x \right ) \left (-1+2 x \right ) \left (2+3 x \right )}\, \sqrt {1-2 x}}-\frac {\left (-\frac {7373029 \sqrt {66+110 x}\, \sqrt {10+15 x}\, \sqrt {55-110 x}\, \EllipticF \left (\frac {\sqrt {66+110 x}}{11}, \frac {i \sqrt {66}}{2}\right )}{25987500 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}-\frac {2911577 \sqrt {66+110 x}\, \sqrt {10+15 x}\, \sqrt {55-110 x}\, \left (\frac {\EllipticE \left (\frac {\sqrt {66+110 x}}{11}, \frac {i \sqrt {66}}{2}\right )}{15}-\frac {2 \EllipticF \left (\frac {\sqrt {66+110 x}}{11}, \frac {i \sqrt {66}}{2}\right )}{3}\right )}{6496875 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}\right ) \sqrt {\left (1-2 x \right ) \left (2+3 x \right ) \left (3+5 x \right )}}{\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}\) \(257\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/7087500*(2+3*x)^(1/2)*(1-2*x)^(1/2)*(3+5*x)^(1/2)*(-425250000*x^6+5778663*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))-11646308*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))-1101600000*x^5-864823500*x^4+106067700*x^3+335838930*x
^2+45120930*x-24657660)/(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((2+3*x)^(5/2)*(1-2*x)^(1/2)*(3+5*x)^(1/2),x, algorithm="maxima")

[Out]

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

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Fricas [A]
time = 0.25, size = 38, normalized size = 0.20 \begin {gather*} \frac {1}{236250} \, {\left (472500 \, x^{3} + 861750 \, x^{2} + 410490 \, x - 136987\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/236250*(472500*x^3 + 861750*x^2 + 410490*x - 136987)*sqrt(5*x + 3)*sqrt(3*x + 2)*sqrt(-2*x + 1)

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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