3.27.91 \(\int \frac {\sqrt {1-2 x} (2+3 x)^{7/2}}{(3+5 x)^{5/2}} \, dx\) [2691]

Optimal. Leaf size=189 \[ -\frac {2 \sqrt {1-2 x} (2+3 x)^{7/2}}{15 (3+5 x)^{3/2}}-\frac {458 \sqrt {1-2 x} (2+3 x)^{5/2}}{825 \sqrt {3+5 x}}+\frac {2719 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{34375}+\frac {2818 \sqrt {1-2 x} (2+3 x)^{3/2} \sqrt {3+5 x}}{6875}-\frac {47342 E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{15625 \sqrt {33}}-\frac {523 \sqrt {\frac {11}{3}} F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{15625} \]

[Out]

-523/46875*EllipticF(1/7*21^(1/2)*(1-2*x)^(1/2),1/33*1155^(1/2))*33^(1/2)-47342/515625*EllipticE(1/7*21^(1/2)*
(1-2*x)^(1/2),1/33*1155^(1/2))*33^(1/2)-2/15*(2+3*x)^(7/2)*(1-2*x)^(1/2)/(3+5*x)^(3/2)-458/825*(2+3*x)^(5/2)*(
1-2*x)^(1/2)/(3+5*x)^(1/2)+2818/6875*(2+3*x)^(3/2)*(1-2*x)^(1/2)*(3+5*x)^(1/2)+2719/34375*(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 = 189, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {99, 155, 159, 164, 114, 120} \begin {gather*} -\frac {523 \sqrt {\frac {11}{3}} F\left (\text {ArcSin}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{15625}-\frac {47342 E\left (\text {ArcSin}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{15625 \sqrt {33}}-\frac {2 \sqrt {1-2 x} (3 x+2)^{7/2}}{15 (5 x+3)^{3/2}}-\frac {458 \sqrt {1-2 x} (3 x+2)^{5/2}}{825 \sqrt {5 x+3}}+\frac {2818 \sqrt {1-2 x} \sqrt {5 x+3} (3 x+2)^{3/2}}{6875}+\frac {2719 \sqrt {1-2 x} \sqrt {5 x+3} \sqrt {3 x+2}}{34375} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*Sqrt[1 - 2*x]*(2 + 3*x)^(7/2))/(15*(3 + 5*x)^(3/2)) - (458*Sqrt[1 - 2*x]*(2 + 3*x)^(5/2))/(825*Sqrt[3 + 5*
x]) + (2719*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x])/34375 + (2818*Sqrt[1 - 2*x]*(2 + 3*x)^(3/2)*Sqrt[3 + 5*
x])/6875 - (47342*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(15625*Sqrt[33]) - (523*Sqrt[11/3]*Ellipt
icF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/15625

Rule 99

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(a + b*
x)^(m + 1)*(c + d*x)^n*((e + f*x)^p/(b*(m + 1))), x] - Dist[1/(b*(m + 1)), Int[(a + b*x)^(m + 1)*(c + d*x)^(n
- 1)*(e + f*x)^(p - 1)*Simp[d*e*n + c*f*p + d*f*(n + p)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && LtQ[m
, -1] && GtQ[n, 0] && GtQ[p, 0] && (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 {\sqrt {1-2 x} (2+3 x)^{7/2}}{(3+5 x)^{5/2}} \, dx &=-\frac {2 \sqrt {1-2 x} (2+3 x)^{7/2}}{15 (3+5 x)^{3/2}}+\frac {2}{15} \int \frac {\left (\frac {17}{2}-24 x\right ) (2+3 x)^{5/2}}{\sqrt {1-2 x} (3+5 x)^{3/2}} \, dx\\ &=-\frac {2 \sqrt {1-2 x} (2+3 x)^{7/2}}{15 (3+5 x)^{3/2}}-\frac {458 \sqrt {1-2 x} (2+3 x)^{5/2}}{825 \sqrt {3+5 x}}+\frac {4}{825} \int \frac {\left (\frac {2379}{4}-\frac {4227 x}{2}\right ) (2+3 x)^{3/2}}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx\\ &=-\frac {2 \sqrt {1-2 x} (2+3 x)^{7/2}}{15 (3+5 x)^{3/2}}-\frac {458 \sqrt {1-2 x} (2+3 x)^{5/2}}{825 \sqrt {3+5 x}}+\frac {2818 \sqrt {1-2 x} (2+3 x)^{3/2} \sqrt {3+5 x}}{6875}-\frac {4 \int \frac {\sqrt {2+3 x} \left (-\frac {13275}{4}+\frac {24471 x}{4}\right )}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx}{20625}\\ &=-\frac {2 \sqrt {1-2 x} (2+3 x)^{7/2}}{15 (3+5 x)^{3/2}}-\frac {458 \sqrt {1-2 x} (2+3 x)^{5/2}}{825 \sqrt {3+5 x}}+\frac {2719 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{34375}+\frac {2818 \sqrt {1-2 x} (2+3 x)^{3/2} \sqrt {3+5 x}}{6875}+\frac {4 \int \frac {\frac {625203}{8}+\frac {213039 x}{2}}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{309375}\\ &=-\frac {2 \sqrt {1-2 x} (2+3 x)^{7/2}}{15 (3+5 x)^{3/2}}-\frac {458 \sqrt {1-2 x} (2+3 x)^{5/2}}{825 \sqrt {3+5 x}}+\frac {2719 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{34375}+\frac {2818 \sqrt {1-2 x} (2+3 x)^{3/2} \sqrt {3+5 x}}{6875}+\frac {5753 \int \frac {1}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{31250}+\frac {47342 \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx}{171875}\\ &=-\frac {2 \sqrt {1-2 x} (2+3 x)^{7/2}}{15 (3+5 x)^{3/2}}-\frac {458 \sqrt {1-2 x} (2+3 x)^{5/2}}{825 \sqrt {3+5 x}}+\frac {2719 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{34375}+\frac {2818 \sqrt {1-2 x} (2+3 x)^{3/2} \sqrt {3+5 x}}{6875}-\frac {47342 E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{15625 \sqrt {33}}-\frac {523 \sqrt {\frac {11}{3}} F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{15625}\\ \end {align*}

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Mathematica [A]
time = 4.59, size = 107, normalized size = 0.57 \begin {gather*} \frac {\frac {10 \sqrt {1-2 x} \sqrt {2+3 x} \left (37273+221200 x+398475 x^2+222750 x^3\right )}{(3+5 x)^{3/2}}+94684 \sqrt {2} E\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )|-\frac {33}{2}\right )+95165 \sqrt {2} F\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )|-\frac {33}{2}\right )}{1031250} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((10*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*(37273 + 221200*x + 398475*x^2 + 222750*x^3))/(3 + 5*x)^(3/2) + 94684*Sqrt[2]
*EllipticE[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2] + 95165*Sqrt[2]*EllipticF[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]]
, -33/2])/1031250

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Maple [A]
time = 0.10, size = 225, normalized size = 1.19

method	result	size
default	\(-\frac {\left (949245 \sqrt {2}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}-473420 \sqrt {2}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}+569547 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )-284052 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )-13365000 x^{5}-26136000 x^{4}-12801750 x^{3}+3521120 x^{2}+4051270 x +745460\right ) \sqrt {1-2 x}\, \sqrt {2+3 x}}{1031250 \left (6 x^{2}+x -2\right ) \left (3+5 x \right )^{\frac {3}{2}}}\)	\(225\)
elliptic	\(\frac {\sqrt {-\left (3+5 x \right ) \left (-1+2 x \right ) \left (2+3 x \right )}\, \left (\frac {54 x \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{625}+\frac {159 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{3125}+\frac {69467 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{1443750 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}+\frac {47342 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \left (-\frac {\EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{15}-\frac {3 \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{5}\right )}{721875 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}-\frac {2 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{46875 \left (x +\frac {3}{5}\right )^{2}}-\frac {656 \left (-30 x^{2}-5 x +10\right )}{103125 \sqrt {\left (x +\frac {3}{5}\right ) \left (-30 x^{2}-5 x +10\right )}}\right )}{\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}\)	\(264\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/1031250*(949245*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)-473420*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)+
569547*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))-284052*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))-13365000*x^5-261
36000*x^4-12801750*x^3+3521120*x^2+4051270*x+745460)*(1-2*x)^(1/2)*(2+3*x)^(1/2)/(6*x^2+x-2)/(3+5*x)^(3/2)

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

[Out]

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

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Fricas [A]
time = 0.11, size = 50, normalized size = 0.26 \begin {gather*} \frac {{\left (222750 \, x^{3} + 398475 \, x^{2} + 221200 \, x + 37273\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}}{103125 \, {\left (25 \, x^{2} + 30 \, x + 9\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/103125*(222750*x^3 + 398475*x^2 + 221200*x + 37273)*sqrt(5*x + 3)*sqrt(3*x + 2)*sqrt(-2*x + 1)/(25*x^2 + 30*
x + 9)

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

[Out]

Exception raised: SystemError >> excessive stack use: stack is 3877 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)^(1/2)/(3+5*x)^(5/2),x, algorithm="giac")

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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