3.30.6 \(\int \frac {(3+5 x)^{3/2}}{(1-2 x)^{3/2} (2+3 x)^{7/2}} \, dx\) [2906]

Optimal. Leaf size=191 \[ \frac {11 \sqrt {3+5 x}}{7 \sqrt {1-2 x} (2+3 x)^{5/2}}-\frac {163 \sqrt {1-2 x} \sqrt {3+5 x}}{245 (2+3 x)^{5/2}}-\frac {458 \sqrt {1-2 x} \sqrt {3+5 x}}{1715 (2+3 x)^{3/2}}+\frac {338 \sqrt {1-2 x} \sqrt {3+5 x}}{12005 \sqrt {2+3 x}}-\frac {338 \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{12005}-\frac {992 \sqrt {\frac {11}{3}} F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{12005} \]

[Out]

-338/36015*EllipticE(1/7*21^(1/2)*(1-2*x)^(1/2),1/33*1155^(1/2))*33^(1/2)-992/36015*EllipticF(1/7*21^(1/2)*(1-
2*x)^(1/2),1/33*1155^(1/2))*33^(1/2)+11/7*(3+5*x)^(1/2)/(2+3*x)^(5/2)/(1-2*x)^(1/2)-163/245*(1-2*x)^(1/2)*(3+5
*x)^(1/2)/(2+3*x)^(5/2)-458/1715*(1-2*x)^(1/2)*(3+5*x)^(1/2)/(2+3*x)^(3/2)+338/12005*(1-2*x)^(1/2)*(3+5*x)^(1/
2)/(2+3*x)^(1/2)

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Rubi [A]
time = 0.04, 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 = {100, 157, 164, 114, 120} \begin {gather*} -\frac {992 \sqrt {\frac {11}{3}} F\left (\text {ArcSin}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{12005}-\frac {338 \sqrt {\frac {11}{3}} E\left (\text {ArcSin}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{12005}+\frac {338 \sqrt {1-2 x} \sqrt {5 x+3}}{12005 \sqrt {3 x+2}}-\frac {458 \sqrt {1-2 x} \sqrt {5 x+3}}{1715 (3 x+2)^{3/2}}-\frac {163 \sqrt {1-2 x} \sqrt {5 x+3}}{245 (3 x+2)^{5/2}}+\frac {11 \sqrt {5 x+3}}{7 \sqrt {1-2 x} (3 x+2)^{5/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(11*Sqrt[3 + 5*x])/(7*Sqrt[1 - 2*x]*(2 + 3*x)^(5/2)) - (163*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(245*(2 + 3*x)^(5/2))
- (458*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(1715*(2 + 3*x)^(3/2)) + (338*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(12005*Sqrt[2
+ 3*x]) - (338*Sqrt[11/3]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/12005 - (992*Sqrt[11/3]*EllipticF
[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/12005

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 157

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 + 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*
g - b*(d*e + c*f)*g + b*c*e*h)*(m + 1) - (b*g - a*h)*(d*e*(n + 1) + c*f*(p + 1)) - d*f*(b*g - a*h)*(m + n + p
+ 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x] && LtQ[m, -1] && 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)^{3/2}}{(1-2 x)^{3/2} (2+3 x)^{7/2}} \, dx &=\frac {11 \sqrt {3+5 x}}{7 \sqrt {1-2 x} (2+3 x)^{5/2}}-\frac {1}{7} \int \frac {-\frac {379}{2}-325 x}{\sqrt {1-2 x} (2+3 x)^{7/2} \sqrt {3+5 x}} \, dx\\ &=\frac {11 \sqrt {3+5 x}}{7 \sqrt {1-2 x} (2+3 x)^{5/2}}-\frac {163 \sqrt {1-2 x} \sqrt {3+5 x}}{245 (2+3 x)^{5/2}}-\frac {2}{245} \int \frac {-\frac {1401}{2}-\frac {2445 x}{2}}{\sqrt {1-2 x} (2+3 x)^{5/2} \sqrt {3+5 x}} \, dx\\ &=\frac {11 \sqrt {3+5 x}}{7 \sqrt {1-2 x} (2+3 x)^{5/2}}-\frac {163 \sqrt {1-2 x} \sqrt {3+5 x}}{245 (2+3 x)^{5/2}}-\frac {458 \sqrt {1-2 x} \sqrt {3+5 x}}{1715 (2+3 x)^{3/2}}-\frac {4 \int \frac {-\frac {4749}{4}-\frac {3435 x}{2}}{\sqrt {1-2 x} (2+3 x)^{3/2} \sqrt {3+5 x}} \, dx}{5145}\\ &=\frac {11 \sqrt {3+5 x}}{7 \sqrt {1-2 x} (2+3 x)^{5/2}}-\frac {163 \sqrt {1-2 x} \sqrt {3+5 x}}{245 (2+3 x)^{5/2}}-\frac {458 \sqrt {1-2 x} \sqrt {3+5 x}}{1715 (2+3 x)^{3/2}}+\frac {338 \sqrt {1-2 x} \sqrt {3+5 x}}{12005 \sqrt {2+3 x}}-\frac {8 \int \frac {-\frac {9705}{4}-\frac {2535 x}{4}}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{36015}\\ &=\frac {11 \sqrt {3+5 x}}{7 \sqrt {1-2 x} (2+3 x)^{5/2}}-\frac {163 \sqrt {1-2 x} \sqrt {3+5 x}}{245 (2+3 x)^{5/2}}-\frac {458 \sqrt {1-2 x} \sqrt {3+5 x}}{1715 (2+3 x)^{3/2}}+\frac {338 \sqrt {1-2 x} \sqrt {3+5 x}}{12005 \sqrt {2+3 x}}+\frac {338 \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx}{12005}+\frac {5456 \int \frac {1}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{12005}\\ &=\frac {11 \sqrt {3+5 x}}{7 \sqrt {1-2 x} (2+3 x)^{5/2}}-\frac {163 \sqrt {1-2 x} \sqrt {3+5 x}}{245 (2+3 x)^{5/2}}-\frac {458 \sqrt {1-2 x} \sqrt {3+5 x}}{1715 (2+3 x)^{3/2}}+\frac {338 \sqrt {1-2 x} \sqrt {3+5 x}}{12005 \sqrt {2+3 x}}-\frac {338 \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{12005}-\frac {992 \sqrt {\frac {11}{3}} F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{12005}\\ \end {align*}

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Mathematica [A]
time = 7.78, size = 104, normalized size = 0.54 \begin {gather*} \frac {2 \left (-\frac {3 \sqrt {3+5 x} \left (-2909-10266 x-7083 x^2+3042 x^3\right )}{\sqrt {1-2 x} (2+3 x)^{5/2}}+\sqrt {2} \left (169 E\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )|-\frac {33}{2}\right )+8015 F\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )|-\frac {33}{2}\right )\right )\right )}{36015} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*((-3*Sqrt[3 + 5*x]*(-2909 - 10266*x - 7083*x^2 + 3042*x^3))/(Sqrt[1 - 2*x]*(2 + 3*x)^(5/2)) + Sqrt[2]*(169*
EllipticE[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2] + 8015*EllipticF[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2]))
)/36015

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(307\) vs. \(2(139)=278\).
time = 0.10, size = 308, normalized size = 1.61

method	result	size
elliptic	\(\frac {\sqrt {-\left (3+5 x \right ) \left (-1+2 x \right ) \left (2+3 x \right )}\, \left (-\frac {44 \left (-30 x^{2}-38 x -12\right )}{2401 \sqrt {\left (-\frac {1}{2}+x \right ) \left (-30 x^{2}-38 x -12\right )}}+\frac {1294 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{50421 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}+\frac {338 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \left (-\frac {\EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{15}-\frac {3 \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{5}\right )}{50421 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}+\frac {2 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{6615 \left (\frac {2}{3}+x \right )^{3}}-\frac {128 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{15435 \left (\frac {2}{3}+x \right )^{2}}+\frac {-\frac {1996}{2401} x^{2}-\frac {998}{12005} x +\frac {2994}{12005}}{\sqrt {\left (\frac {2}{3}+x \right ) \left (-30 x^{2}-3 x +9\right )}}\right )}{\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}\)	\(277\)
default	\(-\frac {2 \sqrt {3+5 x}\, \sqrt {1-2 x}\, \left (73656 \sqrt {2}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x^{2} \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}-1521 \sqrt {2}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x^{2} \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}+98208 \sqrt {2}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}-2028 \sqrt {2}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}+32736 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )-676 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )-45630 x^{4}+78867 x^{3}+217737 x^{2}+136029 x +26181\right )}{36015 \left (2+3 x \right )^{\frac {5}{2}} \left (10 x^{2}+x -3\right )}\)	\(308\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/36015*(3+5*x)^(1/2)*(1-2*x)^(1/2)*(73656*2^(1/2)*EllipticF(1/7*(28+42*x)^(1/2),1/2*70^(1/2))*x^2*(2+3*x)^(1
/2)*(-3-5*x)^(1/2)*(1-2*x)^(1/2)-1521*2^(1/2)*EllipticE(1/7*(28+42*x)^(1/2),1/2*70^(1/2))*x^2*(2+3*x)^(1/2)*(-
3-5*x)^(1/2)*(1-2*x)^(1/2)+98208*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)-2028*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)+32736*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
))-676*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))-45630*x^
4+78867*x^3+217737*x^2+136029*x+26181)/(2+3*x)^(5/2)/(10*x^2+x-3)

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

[Out]

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

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Fricas [A]
time = 0.29, size = 60, normalized size = 0.31 \begin {gather*} \frac {2 \, {\left (3042 \, x^{3} - 7083 \, x^{2} - 10266 \, x - 2909\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}}{12005 \, {\left (54 \, x^{4} + 81 \, x^{3} + 18 \, x^{2} - 20 \, x - 8\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/12005*(3042*x^3 - 7083*x^2 - 10266*x - 2909)*sqrt(5*x + 3)*sqrt(3*x + 2)*sqrt(-2*x + 1)/(54*x^4 + 81*x^3 + 1
8*x^2 - 20*x - 8)

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

[Out]

integrate((5*x + 3)^(3/2)/((3*x + 2)^(7/2)*(-2*x + 1)^(3/2)), 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 )}^{3/2}}{{\left (1-2\,x\right )}^{3/2}\,{\left (3\,x+2\right )}^{7/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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