3.26.7 \(\int \frac {(2+3 x)^5}{\sqrt {1-2 x} (3+5 x)^{5/2}} \, dx\) [2507]

Optimal. Leaf size=142 \[ -\frac {2 \sqrt {1-2 x} (2+3 x)^4}{165 (3+5 x)^{3/2}}-\frac {734 \sqrt {1-2 x} (2+3 x)^3}{9075 \sqrt {3+5 x}}+\frac {511 \sqrt {1-2 x} (2+3 x)^2 \sqrt {3+5 x}}{30250}-\frac {7 \sqrt {1-2 x} \sqrt {3+5 x} (938509+366420 x)}{4840000}+\frac {462357 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{40000 \sqrt {10}} \]

[Out]

462357/400000*arcsin(1/11*22^(1/2)*(3+5*x)^(1/2))*10^(1/2)-2/165*(2+3*x)^4*(1-2*x)^(1/2)/(3+5*x)^(3/2)-734/907
5*(2+3*x)^3*(1-2*x)^(1/2)/(3+5*x)^(1/2)+511/30250*(2+3*x)^2*(1-2*x)^(1/2)*(3+5*x)^(1/2)-7/4840000*(938509+3664
20*x)*(1-2*x)^(1/2)*(3+5*x)^(1/2)

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Rubi [A]
time = 0.03, antiderivative size = 142, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {100, 155, 158, 152, 56, 222} \begin {gather*} \frac {462357 \text {ArcSin}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{40000 \sqrt {10}}-\frac {2 \sqrt {1-2 x} (3 x+2)^4}{165 (5 x+3)^{3/2}}-\frac {734 \sqrt {1-2 x} (3 x+2)^3}{9075 \sqrt {5 x+3}}+\frac {511 \sqrt {1-2 x} \sqrt {5 x+3} (3 x+2)^2}{30250}-\frac {7 \sqrt {1-2 x} \sqrt {5 x+3} (366420 x+938509)}{4840000} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*Sqrt[1 - 2*x]*(2 + 3*x)^4)/(165*(3 + 5*x)^(3/2)) - (734*Sqrt[1 - 2*x]*(2 + 3*x)^3)/(9075*Sqrt[3 + 5*x]) +
(511*Sqrt[1 - 2*x]*(2 + 3*x)^2*Sqrt[3 + 5*x])/30250 - (7*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(938509 + 366420*x))/4840
000 + (462357*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(40000*Sqrt[10])

Rule 56

Int[1/(Sqrt[(a_.) + (b_.)*(x_)]*Sqrt[(c_.) + (d_.)*(x_)]), x_Symbol] :> Dist[2/Sqrt[b], Subst[Int[1/Sqrt[b*c -
a*d + d*x^2], x], x, Sqrt[a + b*x]], x] /; FreeQ[{a, b, c, d}, x] && GtQ[b*c - a*d, 0] && GtQ[b, 0]

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 152

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_) + (f_.)*(x_))*((g_.) + (h_.)*(x_)), x_Symbol]
:> Simp[(-(a*d*f*h*(n + 2) + b*c*f*h*(m + 2) - b*d*(f*g + e*h)*(m + n + 3) - b*d*f*h*(m + n + 2)*x))*(a + b*x)
^(m + 1)*((c + d*x)^(n + 1)/(b^2*d^2*(m + n + 2)*(m + n + 3))), x] + Dist[(a^2*d^2*f*h*(n + 1)*(n + 2) + a*b*d
*(n + 1)*(2*c*f*h*(m + 1) - d*(f*g + e*h)*(m + n + 3)) + b^2*(c^2*f*h*(m + 1)*(m + 2) - c*d*(f*g + e*h)*(m + 1
)*(m + n + 3) + d^2*e*g*(m + n + 2)*(m + n + 3)))/(b^2*d^2*(m + n + 2)*(m + n + 3)), Int[(a + b*x)^m*(c + d*x)
^n, x], x] /; FreeQ[{a, b, c, d, e, f, g, h, m, n}, x] && NeQ[m + n + 2, 0] && NeQ[m + n + 3, 0]

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 158

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] && IntegerQ[m]

Rule 222

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSin[Rt[-b, 2]*(x/Sqrt[a])]/Rt[-b, 2], x] /; FreeQ[{a, b}
, x] && GtQ[a, 0] && NegQ[b]

Rubi steps

\begin {align*} \int \frac {(2+3 x)^5}{\sqrt {1-2 x} (3+5 x)^{5/2}} \, dx &=-\frac {2 \sqrt {1-2 x} (2+3 x)^4}{165 (3+5 x)^{3/2}}-\frac {2}{165} \int \frac {\left (-115-\frac {261 x}{2}\right ) (2+3 x)^3}{\sqrt {1-2 x} (3+5 x)^{3/2}} \, dx\\ &=-\frac {2 \sqrt {1-2 x} (2+3 x)^4}{165 (3+5 x)^{3/2}}-\frac {734 \sqrt {1-2 x} (2+3 x)^3}{9075 \sqrt {3+5 x}}-\frac {4 \int \frac {(2+3 x)^2 \left (-3087+\frac {4599 x}{4}\right )}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx}{9075}\\ &=-\frac {2 \sqrt {1-2 x} (2+3 x)^4}{165 (3+5 x)^{3/2}}-\frac {734 \sqrt {1-2 x} (2+3 x)^3}{9075 \sqrt {3+5 x}}+\frac {511 \sqrt {1-2 x} (2+3 x)^2 \sqrt {3+5 x}}{30250}+\frac {2 \int \frac {(2+3 x) \left (\frac {662697}{4}+\frac {1923705 x}{8}\right )}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx}{136125}\\ &=-\frac {2 \sqrt {1-2 x} (2+3 x)^4}{165 (3+5 x)^{3/2}}-\frac {734 \sqrt {1-2 x} (2+3 x)^3}{9075 \sqrt {3+5 x}}+\frac {511 \sqrt {1-2 x} (2+3 x)^2 \sqrt {3+5 x}}{30250}-\frac {7 \sqrt {1-2 x} \sqrt {3+5 x} (938509+366420 x)}{4840000}+\frac {462357 \int \frac {1}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx}{80000}\\ &=-\frac {2 \sqrt {1-2 x} (2+3 x)^4}{165 (3+5 x)^{3/2}}-\frac {734 \sqrt {1-2 x} (2+3 x)^3}{9075 \sqrt {3+5 x}}+\frac {511 \sqrt {1-2 x} (2+3 x)^2 \sqrt {3+5 x}}{30250}-\frac {7 \sqrt {1-2 x} \sqrt {3+5 x} (938509+366420 x)}{4840000}+\frac {462357 \text {Subst}\left (\int \frac {1}{\sqrt {11-2 x^2}} \, dx,x,\sqrt {3+5 x}\right )}{40000 \sqrt {5}}\\ &=-\frac {2 \sqrt {1-2 x} (2+3 x)^4}{165 (3+5 x)^{3/2}}-\frac {734 \sqrt {1-2 x} (2+3 x)^3}{9075 \sqrt {3+5 x}}+\frac {511 \sqrt {1-2 x} (2+3 x)^2 \sqrt {3+5 x}}{30250}-\frac {7 \sqrt {1-2 x} \sqrt {3+5 x} (938509+366420 x)}{4840000}+\frac {462357 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{40000 \sqrt {10}}\\ \end {align*}

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Mathematica [A]
time = 0.20, size = 74, normalized size = 0.52 \begin {gather*} -\frac {\sqrt {1-2 x} \left (199549721+795297410 x+1030526145 x^2+502791300 x^3+117612000 x^4\right )}{14520000 (3+5 x)^{3/2}}-\frac {462357 \tan ^{-1}\left (\frac {\sqrt {\frac {5}{2}-5 x}}{\sqrt {3+5 x}}\right )}{40000 \sqrt {10}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/14520000*(Sqrt[1 - 2*x]*(199549721 + 795297410*x + 1030526145*x^2 + 502791300*x^3 + 117612000*x^4))/(3 + 5*
x)^(3/2) - (462357*ArcTan[Sqrt[5/2 - 5*x]/Sqrt[3 + 5*x]])/(40000*Sqrt[10])

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Maple [A]
time = 0.08, size = 147, normalized size = 1.04

method	result	size
default	\(\frac {\left (-2352240000 x^{4} \sqrt {-10 x^{2}-x +3}[phone] \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right ) x^{2}-10055826000 x^{3} \sqrt {-10 x^{2}-x +3}[phone] \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right ) x -20610522900 x^{2} \sqrt {-10 x^{2}-x +3}[phone] \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )-15905948200 x \sqrt {-10 x^{2}-x +3}-3990994420 \sqrt {-10 x^{2}-x +3}\right ) \sqrt {1-2 x}}{290400000 \sqrt {-10 x^{2}-x +3}\, \left (3+5 x \right )^{\frac {3}{2}}}\)	\(147\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/290400000*(-2352240000*x^4*(-10*x^2-x+3)^(1/2)[phone]*10^(1/2)*arcsin(20/11*x+1/11)*x^2-10055826000*x^3*
(-10*x^2-x+3)^(1/2)[phone]*10^(1/2)*arcsin(20/11*x+1/11)*x-20610522900*x^2*(-10*x^2-x+3)^(1/2)[phone]*
10^(1/2)*arcsin(20/11*x+1/11)-15905948200*x*(-10*x^2-x+3)^(1/2)-3990994420*(-10*x^2-x+3)^(1/2))*(1-2*x)^(1/2)/
(-10*x^2-x+3)^(1/2)/(3+5*x)^(3/2)

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Maxima [A]
time = 0.66, size = 108, normalized size = 0.76 \begin {gather*} -\frac {81}{250} \, \sqrt {-10 \, x^{2} - x + 3} x^{2} + \frac {462357}{800000} \, \sqrt {5} \sqrt {2} \arcsin \left (\frac {20}{11} \, x + \frac {1}{11}\right ) - \frac {9963}{10000} \, \sqrt {-10 \, x^{2} - x + 3} x - \frac {305343}{200000} \, \sqrt {-10 \, x^{2} - x + 3} - \frac {2 \, \sqrt {-10 \, x^{2} - x + 3}}{103125 \, {\left (25 \, x^{2} + 30 \, x + 9\right )}} - \frac {998 \, \sqrt {-10 \, x^{2} - x + 3}}{1134375 \, {\left (5 \, x + 3\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-81/250*sqrt(-10*x^2 - x + 3)*x^2 + 462357/800000*sqrt(5)*sqrt(2)*arcsin(20/11*x + 1/11) - 9963/10000*sqrt(-10
*x^2 - x + 3)*x - 305343/200000*sqrt(-10*x^2 - x + 3) - 2/103125*sqrt(-10*x^2 - x + 3)/(25*x^2 + 30*x + 9) - 9
98/1134375*sqrt(-10*x^2 - x + 3)/(5*x + 3)

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Fricas [A]
time = 0.59, size = 101, normalized size = 0.71 \begin {gather*} -\frac {167835591 \, \sqrt {10} {\left (25 \, x^{2} + 30 \, x + 9\right )} \arctan \left (\frac {\sqrt {10} {\left (20 \, x + 1\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{20 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) + 20 \, {\left (117612000 \, x^{4} + 502791300 \, x^{3} + 1030526145 \, x^{2} + 795297410 \, x + 199549721\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{290400000 \, {\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)^5/(3+5*x)^(5/2)/(1-2*x)^(1/2),x, algorithm="fricas")

[Out]

-1/290400000*(167835591*sqrt(10)*(25*x^2 + 30*x + 9)*arctan(1/20*sqrt(10)*(20*x + 1)*sqrt(5*x + 3)*sqrt(-2*x +
1)/(10*x^2 + x - 3)) + 20*(117612000*x^4 + 502791300*x^3 + 1030526145*x^2 + 795297410*x + 199549721)*sqrt(5*x
+ 3)*sqrt(-2*x + 1))/(25*x^2 + 30*x + 9)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]
time = 1.08, size = 184, normalized size = 1.30 \begin {gather*} -\frac {27}{1000000} \, {\left (12 \, {\left (8 \, \sqrt {5} {\left (5 \, x + 3\right )} + 75 \, \sqrt {5}\right )} {\left (5 \, x + 3\right )} + 7745 \, \sqrt {5}\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5} - \frac {1}{90750000} \, \sqrt {10} {\left (\frac {{\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{3}}{{\left (5 \, x + 3\right )}^{\frac {3}{2}}} + \frac {3996 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}{\sqrt {5 \, x + 3}}\right )} + \frac {462357}{400000} \, \sqrt {10} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right ) + \frac {\sqrt {10} {\left (5 \, x + 3\right )}^{\frac {3}{2}} {\left (\frac {999 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{2}}{5 \, x + 3} + 4\right )}}{5671875 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-27/1000000*(12*(8*sqrt(5)*(5*x + 3) + 75*sqrt(5))*(5*x + 3) + 7745*sqrt(5))*sqrt(5*x + 3)*sqrt(-10*x + 5) - 1
/90750000*sqrt(10)*((sqrt(2)*sqrt(-10*x + 5) - sqrt(22))^3/(5*x + 3)^(3/2) + 3996*(sqrt(2)*sqrt(-10*x + 5) - s
qrt(22))/sqrt(5*x + 3)) + 462357/400000*sqrt(10)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3)) + 1/5671875*sqrt(10)*(5*x
+ 3)^(3/2)*(999*(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))^2/(5*x + 3) + 4)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))^3

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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