3.30.26 \(\int \frac {(2+3 x)^{9/2}}{(1-2 x)^{3/2} (3+5 x)^{3/2}} \, dx\) [2926]

Optimal. Leaf size=191 \[ -\frac {37 \sqrt {1-2 x} (2+3 x)^{5/2}}{605 \sqrt {3+5 x}}+\frac {7 (2+3 x)^{7/2}}{11 \sqrt {1-2 x} \sqrt {3+5 x}}+\frac {502941 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{151250}+\frac {10851 \sqrt {1-2 x} (2+3 x)^{3/2} \sqrt {3+5 x}}{15125}+\frac {2911577 \sqrt {\frac {3}{11}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{34375}+\frac {175111 \sqrt {\frac {3}{11}} F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{68750} \]

[Out]

2911577/378125*EllipticE(1/7*21^(1/2)*(1-2*x)^(1/2),1/33*1155^(1/2))*33^(1/2)+175111/756250*EllipticF(1/7*21^(
1/2)*(1-2*x)^(1/2),1/33*1155^(1/2))*33^(1/2)+7/11*(2+3*x)^(7/2)/(1-2*x)^(1/2)/(3+5*x)^(1/2)-37/605*(2+3*x)^(5/
2)*(1-2*x)^(1/2)/(3+5*x)^(1/2)+10851/15125*(2+3*x)^(3/2)*(1-2*x)^(1/2)*(3+5*x)^(1/2)+502941/151250*(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 = 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 {175111 \sqrt {\frac {3}{11}} F\left (\text {ArcSin}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{68750}+\frac {2911577 \sqrt {\frac {3}{11}} E\left (\text {ArcSin}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{34375}+\frac {7 (3 x+2)^{7/2}}{11 \sqrt {1-2 x} \sqrt {5 x+3}}-\frac {37 \sqrt {1-2 x} (3 x+2)^{5/2}}{605 \sqrt {5 x+3}}+\frac {10851 \sqrt {1-2 x} \sqrt {5 x+3} (3 x+2)^{3/2}}{15125}+\frac {502941 \sqrt {1-2 x} \sqrt {5 x+3} \sqrt {3 x+2}}{151250} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(2 + 3*x)^(9/2)/((1 - 2*x)^(3/2)*(3 + 5*x)^(3/2)),x]

[Out]

(-37*Sqrt[1 - 2*x]*(2 + 3*x)^(5/2))/(605*Sqrt[3 + 5*x]) + (7*(2 + 3*x)^(7/2))/(11*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])
+ (502941*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x])/151250 + (10851*Sqrt[1 - 2*x]*(2 + 3*x)^(3/2)*Sqrt[3 + 5
*x])/15125 + (2911577*Sqrt[3/11]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/34375 + (175111*Sqrt[3/11]
*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/68750

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)^{9/2}}{(1-2 x)^{3/2} (3+5 x)^{3/2}} \, dx &=\frac {7 (2+3 x)^{7/2}}{11 \sqrt {1-2 x} \sqrt {3+5 x}}-\frac {1}{11} \int \frac {(2+3 x)^{5/2} \left (\frac {367}{2}+312 x\right )}{\sqrt {1-2 x} (3+5 x)^{3/2}} \, dx\\ &=-\frac {37 \sqrt {1-2 x} (2+3 x)^{5/2}}{605 \sqrt {3+5 x}}+\frac {7 (2+3 x)^{7/2}}{11 \sqrt {1-2 x} \sqrt {3+5 x}}-\frac {2}{605} \int \frac {(2+3 x)^{3/2} \left (\frac {13173}{4}+\frac {10851 x}{2}\right )}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx\\ &=-\frac {37 \sqrt {1-2 x} (2+3 x)^{5/2}}{605 \sqrt {3+5 x}}+\frac {7 (2+3 x)^{7/2}}{11 \sqrt {1-2 x} \sqrt {3+5 x}}+\frac {10851 \sqrt {1-2 x} (2+3 x)^{3/2} \sqrt {3+5 x}}{15125}+\frac {2 \int \frac {\left (-\frac {929925}{4}-\frac {1508823 x}{4}\right ) \sqrt {2+3 x}}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx}{15125}\\ &=-\frac {37 \sqrt {1-2 x} (2+3 x)^{5/2}}{605 \sqrt {3+5 x}}+\frac {7 (2+3 x)^{7/2}}{11 \sqrt {1-2 x} \sqrt {3+5 x}}+\frac {502941 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{151250}+\frac {10851 \sqrt {1-2 x} (2+3 x)^{3/2} \sqrt {3+5 x}}{15125}-\frac {2 \int \frac {\frac {66357261}{8}+\frac {26204193 x}{2}}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{226875}\\ &=-\frac {37 \sqrt {1-2 x} (2+3 x)^{5/2}}{605 \sqrt {3+5 x}}+\frac {7 (2+3 x)^{7/2}}{11 \sqrt {1-2 x} \sqrt {3+5 x}}+\frac {502941 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{151250}+\frac {10851 \sqrt {1-2 x} (2+3 x)^{3/2} \sqrt {3+5 x}}{15125}-\frac {525333 \int \frac {1}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{137500}-\frac {8734731 \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx}{378125}\\ &=-\frac {37 \sqrt {1-2 x} (2+3 x)^{5/2}}{605 \sqrt {3+5 x}}+\frac {7 (2+3 x)^{7/2}}{11 \sqrt {1-2 x} \sqrt {3+5 x}}+\frac {502941 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{151250}+\frac {10851 \sqrt {1-2 x} (2+3 x)^{3/2} \sqrt {3+5 x}}{15125}+\frac {2911577 \sqrt {\frac {3}{11}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{34375}+\frac {175111 \sqrt {\frac {3}{11}} F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{68750}\\ \end {align*}

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Mathematica [A]
time = 7.66, size = 132, normalized size = 0.69 \begin {gather*} \frac {10 \sqrt {2+3 x} \sqrt {3+5 x} \left (2892883+3684629 x-2188890 x^2-490050 x^3\right )-11646308 \sqrt {2-4 x} (3+5 x) E\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )|-\frac {33}{2}\right )+5867645 \sqrt {2-4 x} (3+5 x) F\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )|-\frac {33}{2}\right )}{1512500 \sqrt {1-2 x} (3+5 x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(2 + 3*x)^(9/2)/((1 - 2*x)^(3/2)*(3 + 5*x)^(3/2)),x]

[Out]

(10*Sqrt[2 + 3*x]*Sqrt[3 + 5*x]*(2892883 + 3684629*x - 2188890*x^2 - 490050*x^3) - 11646308*Sqrt[2 - 4*x]*(3 +
5*x)*EllipticE[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2] + 5867645*Sqrt[2 - 4*x]*(3 + 5*x)*EllipticF[ArcSin[Sq
rt[2/11]*Sqrt[3 + 5*x]], -33/2])/(1512500*Sqrt[1 - 2*x]*(3 + 5*x))

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Maple [A]
time = 0.10, size = 143, normalized size = 0.75

method	result	size
default	\(\frac {\sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {3+5 x}\, \left (5778663 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )-11646308 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )[phone] x^{4}[phone] x^{3}-66761070 x^{2}-160479070 x -57857660\right )}{45375000 x^{3}[phone] x^{2}-10587500 x -9075000}\)	\(143\)
elliptic	\(\frac {\sqrt {-\left (3+5 x \right ) \left (-1+2 x \right ) \left (2+3 x \right )}\, \left (-\frac {2 \left (-20-30 x \right ) \left (\frac {900367}{1210000}+\frac {1500641 x}{1210000}\right )}{\sqrt {\left (x^{2}+\frac {1}{10} x -\frac {3}{10}\right ) \left (-20-30 x \right )}}+\frac {81 x \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{250}+\frac {3537 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{2500}-\frac {7373029 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{2117500 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}-\frac {2911577 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \left (-\frac {\EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{15}-\frac {3 \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{5}\right )}{529375 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}\right )}{\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}\)	\(240\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2+3*x)^(9/2)/(1-2*x)^(3/2)/(3+5*x)^(3/2),x,method=_RETURNVERBOSE)

[Out]

1/1512500*(2+3*x)^(1/2)*(1-2*x)^(1/2)*(3+5*x)^(1/2)*(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)*Elli
pticE(1/7*(28+42*x)^(1/2),1/2*70^(1/2))[phone]*x^4+75467700*x^3-66761070*x^2-160479070*x-57857660)/(30*x^3+2
3*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)^(9/2)/(1-2*x)^(3/2)/(3+5*x)^(3/2),x, algorithm="maxima")

[Out]

integrate((3*x + 2)^(9/2)/((5*x + 3)^(3/2)*(-2*x + 1)^(3/2)), x)

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Fricas [A]
time = 0.30, size = 48, normalized size = 0.25 \begin {gather*} \frac {{\left (490050 \, x^{3} + 2188890 \, x^{2} - 3684629 \, x - 2892883\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}}{151250 \, {\left (10 \, x^{2} + x - 3\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2+3*x)^(9/2)/(1-2*x)^(3/2)/(3+5*x)^(3/2),x, algorithm="fricas")

[Out]

1/151250*(490050*x^3 + 2188890*x^2 - 3684629*x - 2892883)*sqrt(5*x + 3)*sqrt(3*x + 2)*sqrt(-2*x + 1)/(10*x^2 +
x - 3)

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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)**(9/2)/(1-2*x)**(3/2)/(3+5*x)**(3/2),x)

[Out]

Exception raised: SystemError >> excessive stack use: stack is 7317 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)^(9/2)/(1-2*x)^(3/2)/(3+5*x)^(3/2),x, algorithm="giac")

[Out]

integrate((3*x + 2)^(9/2)/((5*x + 3)^(3/2)*(-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 (3\,x+2\right )}^{9/2}}{{\left (1-2\,x\right )}^{3/2}\,{\left (5\,x+3\right )}^{3/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((3*x + 2)^(9/2)/((1 - 2*x)^(3/2)*(5*x + 3)^(3/2)),x)

[Out]

int((3*x + 2)^(9/2)/((1 - 2*x)^(3/2)*(5*x + 3)^(3/2)), x)

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