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

Optimal. Leaf size=122 \[ \frac {4 (5 x+3)^{5/2}}{77 \sqrt {1-2 x} (3 x+2)^2}-\frac {25 \sqrt {1-2 x} (5 x+3)^{3/2}}{1078 (3 x+2)^2}-\frac {75 \sqrt {1-2 x} \sqrt {5 x+3}}{1372 (3 x+2)}-\frac {825 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{1372 \sqrt {7}} \]

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Rubi [A]  time = 0.03, antiderivative size = 122, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {96, 94, 93, 204} \begin {gather*} \frac {4 (5 x+3)^{5/2}}{77 \sqrt {1-2 x} (3 x+2)^2}-\frac {25 \sqrt {1-2 x} (5 x+3)^{3/2}}{1078 (3 x+2)^2}-\frac {75 \sqrt {1-2 x} \sqrt {5 x+3}}{1372 (3 x+2)}-\frac {825 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{1372 \sqrt {7}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-75*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(1372*(2 + 3*x)) - (25*Sqrt[1 - 2*x]*(3 + 5*x)^(3/2))/(1078*(2 + 3*x)^2) + (
4*(3 + 5*x)^(5/2))/(77*Sqrt[1 - 2*x]*(2 + 3*x)^2) - (825*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(1372*
Sqrt[7])

Rule 93

Int[(((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_))/((e_.) + (f_.)*(x_)), x_Symbol] :> With[{q = Denomin
ator[m]}, Dist[q, Subst[Int[x^(q*(m + 1) - 1)/(b*e - a*f - (d*e - c*f)*x^q), x], x, (a + b*x)^(1/q)/(c + d*x)^
(1/q)], x]] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[m + n + 1, 0] && RationalQ[n] && LtQ[-1, m, 0] && SimplerQ[
a + b*x, c + d*x]

Rule 94

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 + 1))/((m + 1)*(b*e - a*f)), x] - Dist[(n*(d*e - c*f))/((m + 1)*(b*e - a*
f)), Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, m, p}, x] && EqQ[
m + n + p + 2, 0] && GtQ[n, 0] &&  !(SumSimplerQ[p, 1] &&  !SumSimplerQ[m, 1])

Rule 96

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(b*(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[(a*d*f*(m + 1)
+ b*c*f*(n + 1) + b*d*e*(p + 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, x], x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && EqQ[Simplify[m + n + p + 3], 0] && (LtQ[m, -1] || Sum
SimplerQ[m, 1])

Rule 204

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTan[(Rt[-b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[-b, 2]), x] /
; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rubi steps

\begin {align*} \int \frac {(3+5 x)^{3/2}}{(1-2 x)^{3/2} (2+3 x)^3} \, dx &=\frac {4 (3+5 x)^{5/2}}{77 \sqrt {1-2 x} (2+3 x)^2}+\frac {25}{77} \int \frac {(3+5 x)^{3/2}}{\sqrt {1-2 x} (2+3 x)^3} \, dx\\ &=-\frac {25 \sqrt {1-2 x} (3+5 x)^{3/2}}{1078 (2+3 x)^2}+\frac {4 (3+5 x)^{5/2}}{77 \sqrt {1-2 x} (2+3 x)^2}+\frac {75}{196} \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} (2+3 x)^2} \, dx\\ &=-\frac {75 \sqrt {1-2 x} \sqrt {3+5 x}}{1372 (2+3 x)}-\frac {25 \sqrt {1-2 x} (3+5 x)^{3/2}}{1078 (2+3 x)^2}+\frac {4 (3+5 x)^{5/2}}{77 \sqrt {1-2 x} (2+3 x)^2}+\frac {825 \int \frac {1}{\sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}} \, dx}{2744}\\ &=-\frac {75 \sqrt {1-2 x} \sqrt {3+5 x}}{1372 (2+3 x)}-\frac {25 \sqrt {1-2 x} (3+5 x)^{3/2}}{1078 (2+3 x)^2}+\frac {4 (3+5 x)^{5/2}}{77 \sqrt {1-2 x} (2+3 x)^2}+\frac {825 \operatorname {Subst}\left (\int \frac {1}{-7-x^2} \, dx,x,\frac {\sqrt {1-2 x}}{\sqrt {3+5 x}}\right )}{1372}\\ &=-\frac {75 \sqrt {1-2 x} \sqrt {3+5 x}}{1372 (2+3 x)}-\frac {25 \sqrt {1-2 x} (3+5 x)^{3/2}}{1078 (2+3 x)^2}+\frac {4 (3+5 x)^{5/2}}{77 \sqrt {1-2 x} (2+3 x)^2}-\frac {825 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{1372 \sqrt {7}}\\ \end {align*}

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Mathematica [A]  time = 0.05, size = 85, normalized size = 0.70 \begin {gather*} \frac {7 \sqrt {5 x+3} \left (2550 x^2+2245 x+396\right )-825 \sqrt {7-14 x} (3 x+2)^2 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{9604 \sqrt {1-2 x} (3 x+2)^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(7*Sqrt[3 + 5*x]*(396 + 2245*x + 2550*x^2) - 825*Sqrt[7 - 14*x]*(2 + 3*x)^2*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt
[3 + 5*x])])/(9604*Sqrt[1 - 2*x]*(2 + 3*x)^2)

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IntegrateAlgebraic [A]  time = 3.07, size = 203, normalized size = 1.66 \begin {gather*} -\frac {25 \sqrt {11-2 (5 x+3)} \left (102 \sqrt {5} (5 x+3)^{5/2}-163 \sqrt {5} (5 x+3)^{3/2}-33 \sqrt {5} \sqrt {5 x+3}\right )}{1372 (2 (5 x+3)-11) (3 (5 x+3)+1)^2}-\frac {825 \tan ^{-1}\left (\frac {\sqrt {\frac {2}{34+\sqrt {1155}}} \sqrt {5 x+3}}{\sqrt {11}-\sqrt {11-2 (5 x+3)}}\right )}{1372 \sqrt {7}}-\frac {825 \tan ^{-1}\left (\frac {\sqrt {68+2 \sqrt {1155}} \sqrt {5 x+3}}{\sqrt {11}-\sqrt {11-2 (5 x+3)}}\right )}{1372 \sqrt {7}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-25*Sqrt[11 - 2*(3 + 5*x)]*(-33*Sqrt[5]*Sqrt[3 + 5*x] - 163*Sqrt[5]*(3 + 5*x)^(3/2) + 102*Sqrt[5]*(3 + 5*x)^(
5/2)))/(1372*(-11 + 2*(3 + 5*x))*(1 + 3*(3 + 5*x))^2) - (825*ArcTan[(Sqrt[2/(34 + Sqrt[1155])]*Sqrt[3 + 5*x])/
(Sqrt[11] - Sqrt[11 - 2*(3 + 5*x)])])/(1372*Sqrt[7]) - (825*ArcTan[(Sqrt[68 + 2*Sqrt[1155]]*Sqrt[3 + 5*x])/(Sq
rt[11] - Sqrt[11 - 2*(3 + 5*x)])])/(1372*Sqrt[7])

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fricas [A]  time = 1.27, size = 101, normalized size = 0.83 \begin {gather*} -\frac {825 \, \sqrt {7} {\left (18 \, x^{3} + 15 \, x^{2} - 4 \, x - 4\right )} \arctan \left (\frac {\sqrt {7} {\left (37 \, x + 20\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{14 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) + 14 \, {\left (2550 \, x^{2} + 2245 \, x + 396\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{19208 \, {\left (18 \, x^{3} + 15 \, x^{2} - 4 \, x - 4\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)^3,x, algorithm="fricas")

[Out]

-1/19208*(825*sqrt(7)*(18*x^3 + 15*x^2 - 4*x - 4)*arctan(1/14*sqrt(7)*(37*x + 20)*sqrt(5*x + 3)*sqrt(-2*x + 1)
/(10*x^2 + x - 3)) + 14*(2550*x^2 + 2245*x + 396)*sqrt(5*x + 3)*sqrt(-2*x + 1))/(18*x^3 + 15*x^2 - 4*x - 4)

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giac [B]  time = 2.25, size = 278, normalized size = 2.28 \begin {gather*} \frac {165}{38416} \, \sqrt {70} \sqrt {10} {\left (\pi + 2 \, \arctan \left (-\frac {\sqrt {70} \sqrt {5 \, x + 3} {\left (\frac {{\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}\right )\right )} - \frac {44 \, \sqrt {5} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5}}{1715 \, {\left (2 \, x - 1\right )}} - \frac {11 \, \sqrt {10} {\left (13 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{3} + \frac {6280 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}{\sqrt {5 \, x + 3}} - \frac {25120 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}}{98 \, {\left ({\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{2} + 280\right )}^{2}} \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)^3,x, algorithm="giac")

[Out]

165/38416*sqrt(70)*sqrt(10)*(pi + 2*arctan(-1/140*sqrt(70)*sqrt(5*x + 3)*((sqrt(2)*sqrt(-10*x + 5) - sqrt(22))
^2/(5*x + 3) - 4)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22)))) - 44/1715*sqrt(5)*sqrt(5*x + 3)*sqrt(-10*x + 5)/(2*x
- 1) - 11/98*sqrt(10)*(13*((sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) - 4*sqrt(5*x + 3)/(sqrt(2)*sqrt(
-10*x + 5) - sqrt(22)))^3 + 6280*(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) - 25120*sqrt(5*x + 3)/(sqr
t(2)*sqrt(-10*x + 5) - sqrt(22)))/(((sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) - 4*sqrt(5*x + 3)/(sqrt
(2)*sqrt(-10*x + 5) - sqrt(22)))^2 + 280)^2

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maple [B]  time = 0.02, size = 209, normalized size = 1.71 \begin {gather*} \frac {\left (14850 \sqrt {7}\, x^{3} \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+12375 \sqrt {7}\, x^{2} \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )-35700 \sqrt {-10 x^{2}-x +3}\, x^{2}-3300 \sqrt {7}\, x \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )-31430 \sqrt {-10 x^{2}-x +3}\, x -3300 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )-5544 \sqrt {-10 x^{2}-x +3}\right ) \sqrt {-2 x +1}\, \sqrt {5 x +3}}{19208 \left (3 x +2\right )^{2} \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((5*x+3)^(3/2)/(-2*x+1)^(3/2)/(3*x+2)^3,x)

[Out]

1/19208*(14850*7^(1/2)*x^3*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))+12375*7^(1/2)*x^2*arctan(1/14*(3
7*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))-3300*7^(1/2)*x*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))-35700*(
-10*x^2-x+3)^(1/2)*x^2-3300*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))-31430*(-10*x^2-x+3)^(1/
2)*x-5544*(-10*x^2-x+3)^(1/2))*(-2*x+1)^(1/2)*(5*x+3)^(1/2)/(3*x+2)^2/(2*x-1)/(-10*x^2-x+3)^(1/2)

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maxima [A]  time = 1.29, size = 143, normalized size = 1.17 \begin {gather*} \frac {825}{19208} \, \sqrt {7} \arcsin \left (\frac {37 \, x}{11 \, {\left | 3 \, x + 2 \right |}} + \frac {20}{11 \, {\left | 3 \, x + 2 \right |}}\right ) + \frac {2125 \, x}{2058 \, \sqrt {-10 \, x^{2} - x + 3}} + \frac {625}{4116 \, \sqrt {-10 \, x^{2} - x + 3}} - \frac {1}{126 \, {\left (9 \, \sqrt {-10 \, x^{2} - x + 3} x^{2} + 12 \, \sqrt {-10 \, x^{2} - x + 3} x + 4 \, \sqrt {-10 \, x^{2} - x + 3}\right )}} + \frac {235}{1764 \, {\left (3 \, \sqrt {-10 \, x^{2} - x + 3} x + 2 \, \sqrt {-10 \, x^{2} - x + 3}\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)^3,x, algorithm="maxima")

[Out]

825/19208*sqrt(7)*arcsin(37/11*x/abs(3*x + 2) + 20/11/abs(3*x + 2)) + 2125/2058*x/sqrt(-10*x^2 - x + 3) + 625/
4116/sqrt(-10*x^2 - x + 3) - 1/126/(9*sqrt(-10*x^2 - x + 3)*x^2 + 12*sqrt(-10*x^2 - x + 3)*x + 4*sqrt(-10*x^2
- x + 3)) + 235/1764/(3*sqrt(-10*x^2 - x + 3)*x + 2*sqrt(-10*x^2 - x + 3))

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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 )}^3} \,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)^3),x)

[Out]

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

[Out]

Timed out

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