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

Optimal. Leaf size=151 \[ \frac {4 (5 x+3)^{7/2}}{77 \sqrt {1-2 x} (3 x+2)^3}-\frac {\sqrt {1-2 x} (5 x+3)^{5/2}}{77 (3 x+2)^3}-\frac {5 \sqrt {1-2 x} (5 x+3)^{3/2}}{196 (3 x+2)^2}-\frac {165 \sqrt {1-2 x} \sqrt {5 x+3}}{2744 (3 x+2)}-\frac {1815 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{2744 \sqrt {7}} \]

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Rubi [A]  time = 0.04, antiderivative size = 151, normalized size of antiderivative = 1.00, number of steps used = 6, 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)^{7/2}}{77 \sqrt {1-2 x} (3 x+2)^3}-\frac {\sqrt {1-2 x} (5 x+3)^{5/2}}{77 (3 x+2)^3}-\frac {5 \sqrt {1-2 x} (5 x+3)^{3/2}}{196 (3 x+2)^2}-\frac {165 \sqrt {1-2 x} \sqrt {5 x+3}}{2744 (3 x+2)}-\frac {1815 \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)^(5/2)/((1 - 2*x)^(3/2)*(2 + 3*x)^4),x]

[Out]

(-165*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(2744*(2 + 3*x)) - (5*Sqrt[1 - 2*x]*(3 + 5*x)^(3/2))/(196*(2 + 3*x)^2) - (S
qrt[1 - 2*x]*(3 + 5*x)^(5/2))/(77*(2 + 3*x)^3) + (4*(3 + 5*x)^(7/2))/(77*Sqrt[1 - 2*x]*(2 + 3*x)^3) - (1815*Ar
cTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(2744*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)^{5/2}}{(1-2 x)^{3/2} (2+3 x)^4} \, dx &=\frac {4 (3+5 x)^{7/2}}{77 \sqrt {1-2 x} (2+3 x)^3}+\frac {3}{11} \int \frac {(3+5 x)^{5/2}}{\sqrt {1-2 x} (2+3 x)^4} \, dx\\ &=-\frac {\sqrt {1-2 x} (3+5 x)^{5/2}}{77 (2+3 x)^3}+\frac {4 (3+5 x)^{7/2}}{77 \sqrt {1-2 x} (2+3 x)^3}+\frac {5}{14} \int \frac {(3+5 x)^{3/2}}{\sqrt {1-2 x} (2+3 x)^3} \, dx\\ &=-\frac {5 \sqrt {1-2 x} (3+5 x)^{3/2}}{196 (2+3 x)^2}-\frac {\sqrt {1-2 x} (3+5 x)^{5/2}}{77 (2+3 x)^3}+\frac {4 (3+5 x)^{7/2}}{77 \sqrt {1-2 x} (2+3 x)^3}+\frac {165}{392} \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} (2+3 x)^2} \, dx\\ &=-\frac {165 \sqrt {1-2 x} \sqrt {3+5 x}}{2744 (2+3 x)}-\frac {5 \sqrt {1-2 x} (3+5 x)^{3/2}}{196 (2+3 x)^2}-\frac {\sqrt {1-2 x} (3+5 x)^{5/2}}{77 (2+3 x)^3}+\frac {4 (3+5 x)^{7/2}}{77 \sqrt {1-2 x} (2+3 x)^3}+\frac {1815 \int \frac {1}{\sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}} \, dx}{5488}\\ &=-\frac {165 \sqrt {1-2 x} \sqrt {3+5 x}}{2744 (2+3 x)}-\frac {5 \sqrt {1-2 x} (3+5 x)^{3/2}}{196 (2+3 x)^2}-\frac {\sqrt {1-2 x} (3+5 x)^{5/2}}{77 (2+3 x)^3}+\frac {4 (3+5 x)^{7/2}}{77 \sqrt {1-2 x} (2+3 x)^3}+\frac {1815 \operatorname {Subst}\left (\int \frac {1}{-7-x^2} \, dx,x,\frac {\sqrt {1-2 x}}{\sqrt {3+5 x}}\right )}{2744}\\ &=-\frac {165 \sqrt {1-2 x} \sqrt {3+5 x}}{2744 (2+3 x)}-\frac {5 \sqrt {1-2 x} (3+5 x)^{3/2}}{196 (2+3 x)^2}-\frac {\sqrt {1-2 x} (3+5 x)^{5/2}}{77 (2+3 x)^3}+\frac {4 (3+5 x)^{7/2}}{77 \sqrt {1-2 x} (2+3 x)^3}-\frac {1815 \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 (24670 x^3+37405 x^2+17666 x+2448\right )-1815 \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)^(5/2)/((1 - 2*x)^(3/2)*(2 + 3*x)^4),x]

[Out]

(7*Sqrt[3 + 5*x]*(2448 + 17666*x + 37405*x^2 + 24670*x^3) - 1815*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 = 0.27, size = 122, normalized size = 0.81 \begin {gather*} -\frac {121 \sqrt {5 x+3} \left (\frac {15 (1-2 x)^3}{(5 x+3)^3}+\frac {280 (1-2 x)^2}{(5 x+3)^2}+\frac {1617 (1-2 x)}{5 x+3}-1568\right )}{2744 \sqrt {1-2 x} \left (\frac {1-2 x}{5 x+3}+7\right )^3}-\frac {1815 \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]

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

[Out]

(-121*Sqrt[3 + 5*x]*(-1568 + (15*(1 - 2*x)^3)/(3 + 5*x)^3 + (280*(1 - 2*x)^2)/(3 + 5*x)^2 + (1617*(1 - 2*x))/(
3 + 5*x)))/(2744*Sqrt[1 - 2*x]*(7 + (1 - 2*x)/(3 + 5*x))^3) - (1815*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x
])])/(2744*Sqrt[7])

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fricas [A]  time = 1.42, size = 116, normalized size = 0.77 \begin {gather*} -\frac {1815 \, \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 (24670 \, x^{3} + 37405 \, x^{2} + 17666 \, x + 2448\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)^(5/2)/(1-2*x)^(3/2)/(2+3*x)^4,x, algorithm="fricas")

[Out]

-1/38416*(1815*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*(24670*x^3 + 37405*x^2 + 17666*x + 2448)*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 = 3.19, size = 336, normalized size = 2.23 \begin {gather*} \frac {363}{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 {484 \, \sqrt {5} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5}}{12005 \, {\left (2 \, x - 1\right )}} - \frac {121 \, \sqrt {10} {\left (137 \, {\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} + 105280 \, {\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 {25636800 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}{\sqrt {5 \, x + 3}} - \frac {102547200 \, \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)^(5/2)/(1-2*x)^(3/2)/(2+3*x)^4,x, algorithm="giac")

[Out]

363/76832*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)))) - 484/12005*sqrt(5)*sqrt(5*x + 3)*sqrt(-10*x + 5)/(2*
x - 1) - 121/9604*sqrt(10)*(137*((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 + 105280*((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 + 25636800*(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) - 10254
7200*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 = 0.01, size = 257, normalized size = 1.70 \begin {gather*} \frac {\left (98010 \sqrt {7}\, x^{4} \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+147015 \sqrt {7}\, x^{3} \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )-345380 \sqrt {-10 x^{2}-x +3}\, x^{3}+32670 \sqrt {7}\, x^{2} \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )-523670 \sqrt {-10 x^{2}-x +3}\, x^{2}-36300 \sqrt {7}\, x \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )-247324 \sqrt {-10 x^{2}-x +3}\, x -14520 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )-34272 \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)^(5/2)/(-2*x+1)^(3/2)/(3*x+2)^4,x)

[Out]

1/38416*(98010*7^(1/2)*x^4*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))+147015*7^(1/2)*x^3*arctan(1/14*(
37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))+32670*7^(1/2)*x^2*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))-345
380*(-10*x^2-x+3)^(1/2)*x^3-36300*7^(1/2)*x*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))-523670*(-10*x^2
-x+3)^(1/2)*x^2-14520*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))-247324*(-10*x^2-x+3)^(1/2)*x-
34272*(-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 = 1.24, size = 211, normalized size = 1.40 \begin {gather*} \frac {1815}{38416} \, \sqrt {7} \arcsin \left (\frac {37 \, x}{11 \, {\left | 3 \, x + 2 \right |}} + \frac {20}{11 \, {\left | 3 \, x + 2 \right |}}\right ) + \frac {61675 \, x}{37044 \, \sqrt {-10 \, x^{2} - x + 3}} + \frac {14335}{74088 \, \sqrt {-10 \, x^{2} - x + 3}} + \frac {1}{567 \, {\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 {83}{2268 \, {\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 {3175}{10584 \, {\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)^(5/2)/(1-2*x)^(3/2)/(2+3*x)^4,x, algorithm="maxima")

[Out]

1815/38416*sqrt(7)*arcsin(37/11*x/abs(3*x + 2) + 20/11/abs(3*x + 2)) + 61675/37044*x/sqrt(-10*x^2 - x + 3) + 1
4335/74088/sqrt(-10*x^2 - x + 3) + 1/567/(27*sqrt(-10*x^2 - x + 3)*x^3 + 54*sqrt(-10*x^2 - x + 3)*x^2 + 36*sqr
t(-10*x^2 - x + 3)*x + 8*sqrt(-10*x^2 - x + 3)) - 83/2268/(9*sqrt(-10*x^2 - x + 3)*x^2 + 12*sqrt(-10*x^2 - x +
3)*x + 4*sqrt(-10*x^2 - x + 3)) + 3175/10584/(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 )}^{5/2}}{{\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)^(5/2)/((1 - 2*x)^(3/2)*(3*x + 2)^4),x)

[Out]

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

[Out]

Timed out

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