3.24.98 \(\int \frac {(3+5 x)^{3/2}}{(1-2 x)^{3/2} (2+3 x)^4} \, dx\)

Optimal. Leaf size=151 \[ -\frac {415 \sqrt {1-2 x} \sqrt {5 x+3}}{8232 (3 x+2)}-\frac {145 \sqrt {1-2 x} \sqrt {5 x+3}}{588 (3 x+2)^2}-\frac {2 \sqrt {1-2 x} \sqrt {5 x+3}}{3 (3 x+2)^3}+\frac {11 \sqrt {5 x+3}}{7 \sqrt {1-2 x} (3 x+2)^3}-\frac {2805 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{2744 \sqrt {7}} \]

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Rubi [A]  time = 0.05, antiderivative size = 151, 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 = {98, 151, 12, 93, 204} \begin {gather*} -\frac {415 \sqrt {1-2 x} \sqrt {5 x+3}}{8232 (3 x+2)}-\frac {145 \sqrt {1-2 x} \sqrt {5 x+3}}{588 (3 x+2)^2}-\frac {2 \sqrt {1-2 x} \sqrt {5 x+3}}{3 (3 x+2)^3}+\frac {11 \sqrt {5 x+3}}{7 \sqrt {1-2 x} (3 x+2)^3}-\frac {2805 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{2744 \sqrt {7}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(3 + 5*x)^(3/2)/((1 - 2*x)^(3/2)*(2 + 3*x)^4),x]

[Out]

(11*Sqrt[3 + 5*x])/(7*Sqrt[1 - 2*x]*(2 + 3*x)^3) - (2*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(3*(2 + 3*x)^3) - (145*Sqrt
[1 - 2*x]*Sqrt[3 + 5*x])/(588*(2 + 3*x)^2) - (415*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(8232*(2 + 3*x)) - (2805*ArcTan
[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(2744*Sqrt[7])

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

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 98

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 151

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] && IntegerQ[m]

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)^4} \, dx &=\frac {11 \sqrt {3+5 x}}{7 \sqrt {1-2 x} (2+3 x)^3}-\frac {1}{7} \int \frac {-239-\frac {815 x}{2}}{\sqrt {1-2 x} (2+3 x)^4 \sqrt {3+5 x}} \, dx\\ &=\frac {11 \sqrt {3+5 x}}{7 \sqrt {1-2 x} (2+3 x)^3}-\frac {2 \sqrt {1-2 x} \sqrt {3+5 x}}{3 (2+3 x)^3}-\frac {1}{147} \int \frac {-\frac {2275}{2}-1960 x}{\sqrt {1-2 x} (2+3 x)^3 \sqrt {3+5 x}} \, dx\\ &=\frac {11 \sqrt {3+5 x}}{7 \sqrt {1-2 x} (2+3 x)^3}-\frac {2 \sqrt {1-2 x} \sqrt {3+5 x}}{3 (2+3 x)^3}-\frac {145 \sqrt {1-2 x} \sqrt {3+5 x}}{588 (2+3 x)^2}-\frac {\int \frac {-\frac {12565}{4}-5075 x}{\sqrt {1-2 x} (2+3 x)^2 \sqrt {3+5 x}} \, dx}{2058}\\ &=\frac {11 \sqrt {3+5 x}}{7 \sqrt {1-2 x} (2+3 x)^3}-\frac {2 \sqrt {1-2 x} \sqrt {3+5 x}}{3 (2+3 x)^3}-\frac {145 \sqrt {1-2 x} \sqrt {3+5 x}}{588 (2+3 x)^2}-\frac {415 \sqrt {1-2 x} \sqrt {3+5 x}}{8232 (2+3 x)}-\frac {\int -\frac {58905}{8 \sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}} \, dx}{14406}\\ &=\frac {11 \sqrt {3+5 x}}{7 \sqrt {1-2 x} (2+3 x)^3}-\frac {2 \sqrt {1-2 x} \sqrt {3+5 x}}{3 (2+3 x)^3}-\frac {145 \sqrt {1-2 x} \sqrt {3+5 x}}{588 (2+3 x)^2}-\frac {415 \sqrt {1-2 x} \sqrt {3+5 x}}{8232 (2+3 x)}+\frac {2805 \int \frac {1}{\sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}} \, dx}{5488}\\ &=\frac {11 \sqrt {3+5 x}}{7 \sqrt {1-2 x} (2+3 x)^3}-\frac {2 \sqrt {1-2 x} \sqrt {3+5 x}}{3 (2+3 x)^3}-\frac {145 \sqrt {1-2 x} \sqrt {3+5 x}}{588 (2+3 x)^2}-\frac {415 \sqrt {1-2 x} \sqrt {3+5 x}}{8232 (2+3 x)}+\frac {2805 \operatorname {Subst}\left (\int \frac {1}{-7-x^2} \, dx,x,\frac {\sqrt {1-2 x}}{\sqrt {3+5 x}}\right )}{2744}\\ &=\frac {11 \sqrt {3+5 x}}{7 \sqrt {1-2 x} (2+3 x)^3}-\frac {2 \sqrt {1-2 x} \sqrt {3+5 x}}{3 (2+3 x)^3}-\frac {145 \sqrt {1-2 x} \sqrt {3+5 x}}{588 (2+3 x)^2}-\frac {415 \sqrt {1-2 x} \sqrt {3+5 x}}{8232 (2+3 x)}-\frac {2805 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{2744 \sqrt {7}}\\ \end {align*}

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Mathematica [A]  time = 0.06, size = 90, normalized size = 0.60 \begin {gather*} \frac {7 \sqrt {5 x+3} \left (2490 x^3+6135 x^2+3782 x+576\right )-2805 \sqrt {7-14 x} (3 x+2)^3 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{19208 \sqrt {1-2 x} (3 x+2)^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(3 + 5*x)^(3/2)/((1 - 2*x)^(3/2)*(2 + 3*x)^4),x]

[Out]

(7*Sqrt[3 + 5*x]*(576 + 3782*x + 6135*x^2 + 2490*x^3) - 2805*Sqrt[7 - 14*x]*(2 + 3*x)^3*ArcTan[Sqrt[1 - 2*x]/(
Sqrt[7]*Sqrt[3 + 5*x])])/(19208*Sqrt[1 - 2*x]*(2 + 3*x)^3)

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IntegrateAlgebraic [A]  time = 3.32, size = 219, normalized size = 1.45 \begin {gather*} -\frac {5 \sqrt {11-2 (5 x+3)} \left (498 \sqrt {5} (5 x+3)^{7/2}+1653 \sqrt {5} (5 x+3)^{5/2}-4454 \sqrt {5} (5 x+3)^{3/2}-561 \sqrt {5} \sqrt {5 x+3}\right )}{2744 (2 (5 x+3)-11) (3 (5 x+3)+1)^3}-\frac {2805 \tan ^{-1}\left (\frac {\sqrt {\frac {2}{34+\sqrt {1155}}} \sqrt {5 x+3}}{\sqrt {11}-\sqrt {11-2 (5 x+3)}}\right )}{2744 \sqrt {7}}-\frac {2805 \tan ^{-1}\left (\frac {\sqrt {68+2 \sqrt {1155}} \sqrt {5 x+3}}{\sqrt {11}-\sqrt {11-2 (5 x+3)}}\right )}{2744 \sqrt {7}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(3 + 5*x)^(3/2)/((1 - 2*x)^(3/2)*(2 + 3*x)^4),x]

[Out]

(-5*Sqrt[11 - 2*(3 + 5*x)]*(-561*Sqrt[5]*Sqrt[3 + 5*x] - 4454*Sqrt[5]*(3 + 5*x)^(3/2) + 1653*Sqrt[5]*(3 + 5*x)
^(5/2) + 498*Sqrt[5]*(3 + 5*x)^(7/2)))/(2744*(-11 + 2*(3 + 5*x))*(1 + 3*(3 + 5*x))^3) - (2805*ArcTan[(Sqrt[2/(
34 + Sqrt[1155])]*Sqrt[3 + 5*x])/(Sqrt[11] - Sqrt[11 - 2*(3 + 5*x)])])/(2744*Sqrt[7]) - (2805*ArcTan[(Sqrt[68
+ 2*Sqrt[1155]]*Sqrt[3 + 5*x])/(Sqrt[11] - Sqrt[11 - 2*(3 + 5*x)])])/(2744*Sqrt[7])

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fricas [A]  time = 1.21, size = 116, normalized size = 0.77 \begin {gather*} -\frac {2805 \, \sqrt {7} {\left (54 \, x^{4} + 81 \, x^{3} + 18 \, x^{2} - 20 \, x - 8\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 (2490 \, x^{3} + 6135 \, x^{2} + 3782 \, x + 576\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{38416 \, {\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)^4,x, algorithm="fricas")

[Out]

-1/38416*(2805*sqrt(7)*(54*x^4 + 81*x^3 + 18*x^2 - 20*x - 8)*arctan(1/14*sqrt(7)*(37*x + 20)*sqrt(5*x + 3)*sqr
t(-2*x + 1)/(10*x^2 + x - 3)) + 14*(2490*x^3 + 6135*x^2 + 3782*x + 576)*sqrt(5*x + 3)*sqrt(-2*x + 1))/(54*x^4
+ 81*x^3 + 18*x^2 - 20*x - 8)

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giac [B]  time = 2.83, size = 336, normalized size = 2.23 \begin {gather*} \frac {561}{76832} \, \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 {88 \, \sqrt {5} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5}}{12005 \, {\left (2 \, x - 1\right )}} - \frac {11 \, \sqrt {10} {\left (1849 \, {\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 )}^{5} + 1386560 \, {\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 {15601600 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}{\sqrt {5 \, x + 3}} - \frac {62406400 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}}{9604 \, {\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 )}^{3}} \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)^4,x, algorithm="giac")

[Out]

561/76832*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)))) - 88/12005*sqrt(5)*sqrt(5*x + 3)*sqrt(-10*x + 5)/(2*x
- 1) - 11/9604*sqrt(10)*(1849*((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)))^5 + 1386560*((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 + 15601600*(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) - 62406
400*sqrt(5*x + 3)/(sqrt(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)^3

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maple [B]  time = 0.02, size = 257, normalized size = 1.70 \begin {gather*} \frac {\left (151470 \sqrt {7}\, x^{4} \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+227205 \sqrt {7}\, x^{3} \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )-34860 \sqrt {-10 x^{2}-x +3}\, x^{3}+50490 \sqrt {7}\, x^{2} \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )-85890 \sqrt {-10 x^{2}-x +3}\, x^{2}-56100 \sqrt {7}\, x \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )-52948 \sqrt {-10 x^{2}-x +3}\, x -22440 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )-8064 \sqrt {-10 x^{2}-x +3}\right ) \sqrt {-2 x +1}\, \sqrt {5 x +3}}{38416 \left (3 x +2\right )^{3} \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)^4,x)

[Out]

1/38416*(151470*7^(1/2)*x^4*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))+227205*7^(1/2)*x^3*arctan(1/14*
(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))+50490*7^(1/2)*x^2*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))-34
860*(-10*x^2-x+3)^(1/2)*x^3-56100*7^(1/2)*x*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))-85890*(-10*x^2-
x+3)^(1/2)*x^2-22440*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))-52948*(-10*x^2-x+3)^(1/2)*x-80
64*(-10*x^2-x+3)^(1/2))*(-2*x+1)^(1/2)*(5*x+3)^(1/2)/(3*x+2)^3/(2*x-1)/(-10*x^2-x+3)^(1/2)

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maxima [A]  time = 1.28, size = 211, normalized size = 1.40 \begin {gather*} \frac {2805}{38416} \, \sqrt {7} \arcsin \left (\frac {37 \, x}{11 \, {\left | 3 \, x + 2 \right |}} + \frac {20}{11 \, {\left | 3 \, x + 2 \right |}}\right ) + \frac {2075 \, x}{12348 \, \sqrt {-10 \, x^{2} - x + 3}} + \frac {4415}{24696 \, \sqrt {-10 \, x^{2} - x + 3}} - \frac {1}{189 \, {\left (27 \, \sqrt {-10 \, x^{2} - x + 3} x^{3} + 54 \, \sqrt {-10 \, x^{2} - x + 3} x^{2} + 36 \, \sqrt {-10 \, x^{2} - x + 3} x + 8 \, \sqrt {-10 \, x^{2} - x + 3}\right )}} + \frac {53}{756 \, {\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 {275}{1176 \, {\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)^4,x, algorithm="maxima")

[Out]

2805/38416*sqrt(7)*arcsin(37/11*x/abs(3*x + 2) + 20/11/abs(3*x + 2)) + 2075/12348*x/sqrt(-10*x^2 - x + 3) + 44
15/24696/sqrt(-10*x^2 - x + 3) - 1/189/(27*sqrt(-10*x^2 - x + 3)*x^3 + 54*sqrt(-10*x^2 - x + 3)*x^2 + 36*sqrt(
-10*x^2 - x + 3)*x + 8*sqrt(-10*x^2 - x + 3)) + 53/756/(9*sqrt(-10*x^2 - x + 3)*x^2 + 12*sqrt(-10*x^2 - x + 3)
*x + 4*sqrt(-10*x^2 - x + 3)) - 275/1176/(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 )}^4} \,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)^4),x)

[Out]

int((5*x + 3)^(3/2)/((1 - 2*x)^(3/2)*(3*x + 2)^4), 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)**4,x)

[Out]

Timed out

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