3.2547 \(\int \frac{(3+5 x)^{5/2}}{(1-2 x)^{3/2} (2+3 x)^4} \, dx\)

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

[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])

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Rubi [A]  time = 0.0405661, antiderivative size = 151, normalized size of antiderivative = 1., 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} \[ \frac{4 (5 x+3)^{7/2}}{77 \sqrt{1-2 x} (3 x+2)^3}-\frac{\sqrt{1-2 x} (5 x+3)^{5/2}}{77 (3 x+2)^3}-\frac{5 \sqrt{1-2 x} (5 x+3)^{3/2}}{196 (3 x+2)^2}-\frac{165 \sqrt{1-2 x} \sqrt{5 x+3}}{2744 (3 x+2)}-\frac{1815 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{2744 \sqrt{7}} \]

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 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 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 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 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*}

Mathematica [A]  time = 0.0560872, size = 90, normalized size = 0.6 \[ \frac{7 \sqrt{5 x+3} \left (24670 x^3+37405 x^2+17666 x+2448\right )-1815 \sqrt{7-14 x} (3 x+2)^3 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{19208 \sqrt{1-2 x} (3 x+2)^3} \]

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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Maple [B]  time = 0.013, size = 257, normalized size = 1.7 \begin{align*}{\frac{1}{38416\, \left ( 2+3\,x \right ) ^{3} \left ( 2\,x-1 \right ) } \left ( 98010\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{4}+147015\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{3}+32670\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{2}-345380\,{x}^{3}\sqrt{-10\,{x}^{2}-x+3}-36300\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) x-523670\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}-14520\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) -247324\,x\sqrt{-10\,{x}^{2}-x+3}-34272\,\sqrt{-10\,{x}^{2}-x+3} \right ) \sqrt{1-2\,x}\sqrt{3+5\,x}{\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/38416*(98010*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^4+147015*7^(1/2)*arctan(1/14*(37*x
+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^3+32670*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^2-345
380*x^3*(-10*x^2-x+3)^(1/2)-36300*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x-523670*x^2*(-10
*x^2-x+3)^(1/2)-14520*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))-247324*x*(-10*x^2-x+3)^(1/2)-
34272*(-10*x^2-x+3)^(1/2))*(1-2*x)^(1/2)*(3+5*x)^(1/2)/(2+3*x)^3/(2*x-1)/(-10*x^2-x+3)^(1/2)

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Maxima [A]  time = 2.95583, size = 285, normalized size = 1.89 \begin{align*} \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{align*}

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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Fricas [A]  time = 1.88679, size = 344, normalized size = 2.28 \begin{align*} -\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{align*}

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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Sympy [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

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]

Exception raised: ValueError

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Giac [B]  time = 3.85944, size = 464, normalized size = 3.07 \begin{align*} \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 \,{\left (137 \, \sqrt{10}{\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 \, \sqrt{10}{\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} + 25636800 \, \sqrt{10}{\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 )}\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{align*}

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*(137*sqrt(10)*((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(10)*((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(10)*((sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sq
rt(5*x + 3) - 4*sqrt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))))/(((sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sq
rt(5*x + 3) - 4*sqrt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22)))^2 + 280)^3