3.24.79 \(\int \frac {(2+3 x)^5 \sqrt {3+5 x}}{(1-2 x)^{3/2}} \, dx\)

Optimal. Leaf size=168 \[ \frac {\sqrt {5 x+3} (3 x+2)^5}{\sqrt {1-2 x}}+\frac {33}{20} \sqrt {1-2 x} \sqrt {5 x+3} (3 x+2)^4+\frac {10389 \sqrt {1-2 x} \sqrt {5 x+3} (3 x+2)^3}{1600}+\frac {847637 \sqrt {1-2 x} \sqrt {5 x+3} (3 x+2)^2}{32000}+\frac {49 \sqrt {1-2 x} \sqrt {5 x+3} (36265980 x+87394471)}{5120000}-\frac {35439958001 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{5120000 \sqrt {10}} \]

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Rubi [A]  time = 0.05, antiderivative size = 168, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.192, Rules used = {97, 153, 147, 54, 216} \begin {gather*} \frac {\sqrt {5 x+3} (3 x+2)^5}{\sqrt {1-2 x}}+\frac {33}{20} \sqrt {1-2 x} \sqrt {5 x+3} (3 x+2)^4+\frac {10389 \sqrt {1-2 x} \sqrt {5 x+3} (3 x+2)^3}{1600}+\frac {847637 \sqrt {1-2 x} \sqrt {5 x+3} (3 x+2)^2}{32000}+\frac {49 \sqrt {1-2 x} \sqrt {5 x+3} (36265980 x+87394471)}{5120000}-\frac {35439958001 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{5120000 \sqrt {10}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(847637*Sqrt[1 - 2*x]*(2 + 3*x)^2*Sqrt[3 + 5*x])/32000 + (10389*Sqrt[1 - 2*x]*(2 + 3*x)^3*Sqrt[3 + 5*x])/1600
+ (33*Sqrt[1 - 2*x]*(2 + 3*x)^4*Sqrt[3 + 5*x])/20 + ((2 + 3*x)^5*Sqrt[3 + 5*x])/Sqrt[1 - 2*x] + (49*Sqrt[1 - 2
*x]*Sqrt[3 + 5*x]*(87394471 + 36265980*x))/5120000 - (35439958001*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(5120000*S
qrt[10])

Rule 54

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 97

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 147

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 153

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 216

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 {3+5 x}}{(1-2 x)^{3/2}} \, dx &=\frac {(2+3 x)^5 \sqrt {3+5 x}}{\sqrt {1-2 x}}-\int \frac {(2+3 x)^4 \left (50+\frac {165 x}{2}\right )}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx\\ &=\frac {33}{20} \sqrt {1-2 x} (2+3 x)^4 \sqrt {3+5 x}+\frac {(2+3 x)^5 \sqrt {3+5 x}}{\sqrt {1-2 x}}+\frac {1}{50} \int \frac {\left (-\frac {15775}{2}-\frac {51945 x}{4}\right ) (2+3 x)^3}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx\\ &=\frac {10389 \sqrt {1-2 x} (2+3 x)^3 \sqrt {3+5 x}}{1600}+\frac {33}{20} \sqrt {1-2 x} (2+3 x)^4 \sqrt {3+5 x}+\frac {(2+3 x)^5 \sqrt {3+5 x}}{\sqrt {1-2 x}}-\frac {\int \frac {(2+3 x)^2 \left (\frac {1937285}{2}+\frac {12714555 x}{8}\right )}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx}{2000}\\ &=\frac {847637 \sqrt {1-2 x} (2+3 x)^2 \sqrt {3+5 x}}{32000}+\frac {10389 \sqrt {1-2 x} (2+3 x)^3 \sqrt {3+5 x}}{1600}+\frac {33}{20} \sqrt {1-2 x} (2+3 x)^4 \sqrt {3+5 x}+\frac {(2+3 x)^5 \sqrt {3+5 x}}{\sqrt {1-2 x}}+\frac {\int \frac {\left (-\frac {681095835}{8}-\frac {2221291275 x}{16}\right ) (2+3 x)}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx}{60000}\\ &=\frac {847637 \sqrt {1-2 x} (2+3 x)^2 \sqrt {3+5 x}}{32000}+\frac {10389 \sqrt {1-2 x} (2+3 x)^3 \sqrt {3+5 x}}{1600}+\frac {33}{20} \sqrt {1-2 x} (2+3 x)^4 \sqrt {3+5 x}+\frac {(2+3 x)^5 \sqrt {3+5 x}}{\sqrt {1-2 x}}+\frac {49 \sqrt {1-2 x} \sqrt {3+5 x} (87394471+36265980 x)}{5120000}-\frac {35439958001 \int \frac {1}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx}{10240000}\\ &=\frac {847637 \sqrt {1-2 x} (2+3 x)^2 \sqrt {3+5 x}}{32000}+\frac {10389 \sqrt {1-2 x} (2+3 x)^3 \sqrt {3+5 x}}{1600}+\frac {33}{20} \sqrt {1-2 x} (2+3 x)^4 \sqrt {3+5 x}+\frac {(2+3 x)^5 \sqrt {3+5 x}}{\sqrt {1-2 x}}+\frac {49 \sqrt {1-2 x} \sqrt {3+5 x} (87394471+36265980 x)}{5120000}-\frac {35439958001 \operatorname {Subst}\left (\int \frac {1}{\sqrt {11-2 x^2}} \, dx,x,\sqrt {3+5 x}\right )}{5120000 \sqrt {5}}\\ &=\frac {847637 \sqrt {1-2 x} (2+3 x)^2 \sqrt {3+5 x}}{32000}+\frac {10389 \sqrt {1-2 x} (2+3 x)^3 \sqrt {3+5 x}}{1600}+\frac {33}{20} \sqrt {1-2 x} (2+3 x)^4 \sqrt {3+5 x}+\frac {(2+3 x)^5 \sqrt {3+5 x}}{\sqrt {1-2 x}}+\frac {49 \sqrt {1-2 x} \sqrt {3+5 x} (87394471+36265980 x)}{5120000}-\frac {35439958001 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{5120000 \sqrt {10}}\\ \end {align*}

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Mathematica [A]  time = 0.15, size = 93, normalized size = 0.55 \begin {gather*} \frac {-10 \sqrt {2 x-1} \sqrt {5 x+3} \left (124416000 x^5+613267200 x^4+1429191360 x^3+2297649240 x^2+3810769458 x-5389783159\right )-35439958001 \sqrt {10} (2 x-1) \sinh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {2 x-1}\right )}{51200000 \sqrt {-(1-2 x)^2}} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(-10*Sqrt[-1 + 2*x]*Sqrt[3 + 5*x]*(-5389783159 + 3810769458*x + 2297649240*x^2 + 1429191360*x^3 + 613267200*x^
4 + 124416000*x^5) - 35439958001*Sqrt[10]*(-1 + 2*x)*ArcSinh[Sqrt[5/11]*Sqrt[-1 + 2*x]])/(51200000*Sqrt[-(1 -
2*x)^2])

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IntegrateAlgebraic [A]  time = 0.25, size = 157, normalized size = 0.93 \begin {gather*} \frac {\sqrt {5 x+3} \left (\frac {22149973750625 (1-2 x)^5}{(5 x+3)^5}+\frac {41346618350500 (1-2 x)^4}{(5 x+3)^4}+\frac {30242112212480 (1-2 x)^3}{(5 x+3)^3}+\frac {10665887306480 (1-2 x)^2}{(5 x+3)^2}+\frac {1737920671984 (1-2 x)}{5 x+3}[phone]\right )}{5120000 \sqrt {1-2 x} \left (\frac {5 (1-2 x)}{5 x+3}+2\right )^5}+\frac {35439958001 \tan ^{-1}\left (\frac {\sqrt {\frac {5}{2}} \sqrt {1-2 x}}{\sqrt {5 x+3}}\right )}{5120000 \sqrt {10}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[3 + 5*x]*(86051840000 + (22149973750625*(1 - 2*x)^5)/(3 + 5*x)^5 + (41346618350500*(1 - 2*x)^4)/(3 + 5*x
)^4 + (30242112212480*(1 - 2*x)^3)/(3 + 5*x)^3 + (10665887306480*(1 - 2*x)^2)/(3 + 5*x)^2 + (1737920671984*(1
- 2*x))/(3 + 5*x)))/(5120000*Sqrt[1 - 2*x]*(2 + (5*(1 - 2*x))/(3 + 5*x))^5) + (35439958001*ArcTan[(Sqrt[5/2]*S
qrt[1 - 2*x])/Sqrt[3 + 5*x]])/(5120000*Sqrt[10])

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fricas [A]  time = 0.72, size = 96, normalized size = 0.57 \begin {gather*} \frac {35439958001 \, \sqrt {10} {\left (2 \, x - 1\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 (124416000 \, x^{5} + 613267200 \, x^{4} + 1429191360 \, x^{3} + 2297649240 \, x^{2} + 3810769458 \, x - 5389783159\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{102400000 \, {\left (2 \, x - 1\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/102400000*(35439958001*sqrt(10)*(2*x - 1)*arctan(1/20*sqrt(10)*(20*x + 1)*sqrt(5*x + 3)*sqrt(-2*x + 1)/(10*x
^2 + x - 3)) + 20*(124416000*x^5 + 613267200*x^4 + 1429191360*x^3 + 2297649240*x^2 + 3810769458*x - 5389783159
)*sqrt(5*x + 3)*sqrt(-2*x + 1))/(2*x - 1)

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giac [A]  time = 1.00, size = 110, normalized size = 0.65 \begin {gather*} -\frac {35439958001}{51200000} \, \sqrt {10} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right ) + \frac {{\left (6 \, {\left (12 \, {\left (8 \, {\left (36 \, {\left (48 \, \sqrt {5} {\left (5 \, x + 3\right )} + 463 \, \sqrt {5}\right )} {\left (5 \, x + 3\right )} + 140711 \, \sqrt {5}\right )} {\left (5 \, x + 3\right )} + 10847547 \, \sqrt {5}\right )} {\left (5 \, x + 3\right )} + 1789896455 \, \sqrt {5}\right )} {\left (5 \, x + 3\right )} - 177199790005 \, \sqrt {5}\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5}}{640000000 \, {\left (2 \, x - 1\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-35439958001/51200000*sqrt(10)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3)) + 1/640000000*(6*(12*(8*(36*(48*sqrt(5)*(5*
x + 3) + 463*sqrt(5))*(5*x + 3) + 140711*sqrt(5))*(5*x + 3) + 10847547*sqrt(5))*(5*x + 3) + 1789896455*sqrt(5)
)*(5*x + 3) - 177199790005*sqrt(5))*sqrt(5*x + 3)*sqrt(-10*x + 5)/(2*x - 1)

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maple [A]  time = 0.02, size = 157, normalized size = 0.93 \begin {gather*} -\frac {\left (-2488320000 \sqrt {-10 x^{2}-x +3}\, x^{5}-12265344000 \sqrt {-10 x^{2}-x +3}\, x^{4}-28583827200 \sqrt {-10 x^{2}-x +3}\, x^{3}-45952984800 \sqrt {-10 x^{2}-x +3}\, x^{2}[phone] \sqrt {10}\, x \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )-76215389160 \sqrt {-10 x^{2}-x +3}\, x -35439958001 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )[phone] \sqrt {-10 x^{2}-x +3}\right ) \sqrt {-2 x +1}\, \sqrt {5 x +3}}{102400000 \left (2 x -1\right ) \sqrt {-10 x^{2}-x +3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/102400000*(-2488320000*(-10*x^2-x+3)^(1/2)*x^5-12265344000*(-10*x^2-x+3)^(1/2)*x^4-28583827200*(-10*x^2-x+3
)^(1/2)*x^3+70879916002*10^(1/2)*x*arcsin(20/11*x+1/11)-45952984800*(-10*x^2-x+3)^(1/2)*x^2-35439958001*10^(1/
2)*arcsin(20/11*x+1/11)-76215389160*(-10*x^2-x+3)^(1/2)*x+107795663180*(-10*x^2-x+3)^(1/2))*(-2*x+1)^(1/2)*(5*
x+3)^(1/2)/(2*x-1)/(-10*x^2-x+3)^(1/2)

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maxima [A]  time = 1.27, size = 111, normalized size = 0.66 \begin {gather*} -\frac {243}{200} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x^{2} - \frac {103599}{16000} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x - \frac {35439958001}{102400000} \, \sqrt {5} \sqrt {2} \arcsin \left (\frac {20}{11} \, x + \frac {1}{11}\right ) - \frac {1086219}{64000} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} + \frac {80155719}{256000} \, \sqrt {-10 \, x^{2} - x + 3} x + \frac {2961355719}{5120000} \, \sqrt {-10 \, x^{2} - x + 3} - \frac {16807 \, \sqrt {-10 \, x^{2} - x + 3}}{32 \, {\left (2 \, x - 1\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-243/200*(-10*x^2 - x + 3)^(3/2)*x^2 - 103599/16000*(-10*x^2 - x + 3)^(3/2)*x - 35439958001/102400000*sqrt(5)*
sqrt(2)*arcsin(20/11*x + 1/11) - 1086219/64000*(-10*x^2 - x + 3)^(3/2) + 80155719/256000*sqrt(-10*x^2 - x + 3)
*x + 2961355719/5120000*sqrt(-10*x^2 - x + 3) - 16807/32*sqrt(-10*x^2 - x + 3)/(2*x - 1)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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