3.29.35 \(\int \frac {(3+5 x)^{5/2}}{\sqrt {1-2 x} (2+3 x)^{3/2}} \, dx\) [2835]

Optimal. Leaf size=129 \[ -\frac {205}{189} \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}+\frac {2 \sqrt {1-2 x} (3+5 x)^{3/2}}{21 \sqrt {2+3 x}}-\frac {974}{189} \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )-\frac {41}{189} \sqrt {\frac {11}{3}} F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right ) \]

[Out]

-974/567*EllipticE(1/7*21^(1/2)*(1-2*x)^(1/2),1/33*1155^(1/2))*33^(1/2)-41/567*EllipticF(1/7*21^(1/2)*(1-2*x)^
(1/2),1/33*1155^(1/2))*33^(1/2)+2/21*(3+5*x)^(3/2)*(1-2*x)^(1/2)/(2+3*x)^(1/2)-205/189*(1-2*x)^(1/2)*(2+3*x)^(
1/2)*(3+5*x)^(1/2)

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Rubi [A]
time = 0.03, antiderivative size = 129, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.179, Rules used = {100, 159, 164, 114, 120} \begin {gather*} -\frac {41}{189} \sqrt {\frac {11}{3}} F\left (\text {ArcSin}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )-\frac {974}{189} \sqrt {\frac {11}{3}} E\left (\text {ArcSin}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )+\frac {2 \sqrt {1-2 x} (5 x+3)^{3/2}}{21 \sqrt {3 x+2}}-\frac {205}{189} \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-205*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x])/189 + (2*Sqrt[1 - 2*x]*(3 + 5*x)^(3/2))/(21*Sqrt[2 + 3*x]) -
(974*Sqrt[11/3]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/189 - (41*Sqrt[11/3]*EllipticF[ArcSin[Sqrt[
3/7]*Sqrt[1 - 2*x]], 35/33])/189

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 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 {(3+5 x)^{5/2}}{\sqrt {1-2 x} (2+3 x)^{3/2}} \, dx &=\frac {2 \sqrt {1-2 x} (3+5 x)^{3/2}}{21 \sqrt {2+3 x}}-\frac {2}{21} \int \frac {\left (-45-\frac {205 x}{2}\right ) \sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx\\ &=-\frac {205}{189} \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}+\frac {2 \sqrt {1-2 x} (3+5 x)^{3/2}}{21 \sqrt {2+3 x}}+\frac {2}{189} \int \frac {\frac {6295}{4}+2435 x}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx\\ &=-\frac {205}{189} \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}+\frac {2 \sqrt {1-2 x} (3+5 x)^{3/2}}{21 \sqrt {2+3 x}}+\frac {451}{378} \int \frac {1}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx+\frac {974}{189} \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx\\ &=-\frac {205}{189} \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}+\frac {2 \sqrt {1-2 x} (3+5 x)^{3/2}}{21 \sqrt {2+3 x}}-\frac {974}{189} \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )-\frac {41}{189} \sqrt {\frac {11}{3}} F\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 = 3.66, size = 97, normalized size = 0.75 \begin {gather*} \frac {-\frac {6 \sqrt {1-2 x} \sqrt {3+5 x} (356+525 x)}{\sqrt {2+3 x}}+1948 \sqrt {2} E\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )|-\frac {33}{2}\right )-595 \sqrt {2} F\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )|-\frac {33}{2}\right )}{1134} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((-6*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(356 + 525*x))/Sqrt[2 + 3*x] + 1948*Sqrt[2]*EllipticE[ArcSin[Sqrt[2/11]*Sqrt[
3 + 5*x]], -33/2] - 595*Sqrt[2]*EllipticF[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2])/1134

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Maple [A]
time = 0.10, size = 138, normalized size = 1.07

method	result	size
default	\(-\frac {\sqrt {3+5 x}\, \sqrt {2+3 x}\, \sqrt {1-2 x}\, \left (1353 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )-1948 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )+31500 x^{3}+24510 x^{2}-7314 x -6408\right )}{1134 \left (30 x^{3}+23 x^{2}-7 x -6\right )}\)	\(138\)
elliptic	\(\frac {\sqrt {-\left (3+5 x \right ) \left (-1+2 x \right ) \left (2+3 x \right )}\, \left (-\frac {2 \left (-30 x^{2}-3 x +9\right )}{189 \sqrt {\left (\frac {2}{3}+x \right ) \left (-30 x^{2}-3 x +9\right )}}+\frac {6295 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{7938 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}+\frac {4870 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \left (-\frac {\EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{15}-\frac {3 \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{5}\right )}{3969 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}-\frac {25 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{27}\right )}{\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}\)	\(220\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/1134*(3+5*x)^(1/2)*(2+3*x)^(1/2)*(1-2*x)^(1/2)*(1353*2^(1/2)*(2+3*x)^(1/2)*(-3-5*x)^(1/2)*(1-2*x)^(1/2)*Ell
ipticF(1/7*(28+42*x)^(1/2),1/2*70^(1/2))-1948*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))+31500*x^3+24510*x^2-7314*x-6408)/(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((3+5*x)^(5/2)/(2+3*x)^(3/2)/(1-2*x)^(1/2),x, algorithm="maxima")

[Out]

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

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Fricas [A]
time = 0.29, size = 28, normalized size = 0.22 \begin {gather*} -\frac {{\left (525 \, x + 356\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{189 \, \sqrt {3 \, x + 2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/189*(525*x + 356)*sqrt(5*x + 3)*sqrt(-2*x + 1)/sqrt(3*x + 2)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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