3.2605 \(\int \frac{(3+5 x)^{5/2}}{(1-2 x)^{5/2} (2+3 x)^3} \, dx\)

Optimal. Leaf size=151 \[ \frac{4 (5 x+3)^{7/2}}{231 (1-2 x)^{3/2} (3 x+2)^2}+\frac{26 (5 x+3)^{5/2}}{231 \sqrt{1-2 x} (3 x+2)^2}+\frac{65 \sqrt{1-2 x} (5 x+3)^{3/2}}{3234 (3 x+2)^2}+\frac{65 \sqrt{1-2 x} \sqrt{5 x+3}}{1372 (3 x+2)}+\frac{715 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{1372 \sqrt{7}} \]

[Out]

(65*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(1372*(2 + 3*x)) + (65*Sqrt[1 - 2*x]*(3 + 5*x)^(3/2))/(3234*(2 + 3*x)^2) + (2
6*(3 + 5*x)^(5/2))/(231*Sqrt[1 - 2*x]*(2 + 3*x)^2) + (4*(3 + 5*x)^(7/2))/(231*(1 - 2*x)^(3/2)*(2 + 3*x)^2) + (
715*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(1372*Sqrt[7])

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Rubi [A]  time = 0.0385597, 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} \[ \frac{4 (5 x+3)^{7/2}}{231 (1-2 x)^{3/2} (3 x+2)^2}+\frac{26 (5 x+3)^{5/2}}{231 \sqrt{1-2 x} (3 x+2)^2}+\frac{65 \sqrt{1-2 x} (5 x+3)^{3/2}}{3234 (3 x+2)^2}+\frac{65 \sqrt{1-2 x} \sqrt{5 x+3}}{1372 (3 x+2)}+\frac{715 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{1372 \sqrt{7}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(65*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(1372*(2 + 3*x)) + (65*Sqrt[1 - 2*x]*(3 + 5*x)^(3/2))/(3234*(2 + 3*x)^2) + (2
6*(3 + 5*x)^(5/2))/(231*Sqrt[1 - 2*x]*(2 + 3*x)^2) + (4*(3 + 5*x)^(7/2))/(231*(1 - 2*x)^(3/2)*(2 + 3*x)^2) + (
715*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(1372*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)^{5/2} (2+3 x)^3} \, dx &=\frac{4 (3+5 x)^{7/2}}{231 (1-2 x)^{3/2} (2+3 x)^2}+\frac{13}{33} \int \frac{(3+5 x)^{5/2}}{(1-2 x)^{3/2} (2+3 x)^3} \, dx\\ &=\frac{26 (3+5 x)^{5/2}}{231 \sqrt{1-2 x} (2+3 x)^2}+\frac{4 (3+5 x)^{7/2}}{231 (1-2 x)^{3/2} (2+3 x)^2}-\frac{65}{231} \int \frac{(3+5 x)^{3/2}}{\sqrt{1-2 x} (2+3 x)^3} \, dx\\ &=\frac{65 \sqrt{1-2 x} (3+5 x)^{3/2}}{3234 (2+3 x)^2}+\frac{26 (3+5 x)^{5/2}}{231 \sqrt{1-2 x} (2+3 x)^2}+\frac{4 (3+5 x)^{7/2}}{231 (1-2 x)^{3/2} (2+3 x)^2}-\frac{65}{196} \int \frac{\sqrt{3+5 x}}{\sqrt{1-2 x} (2+3 x)^2} \, dx\\ &=\frac{65 \sqrt{1-2 x} \sqrt{3+5 x}}{1372 (2+3 x)}+\frac{65 \sqrt{1-2 x} (3+5 x)^{3/2}}{3234 (2+3 x)^2}+\frac{26 (3+5 x)^{5/2}}{231 \sqrt{1-2 x} (2+3 x)^2}+\frac{4 (3+5 x)^{7/2}}{231 (1-2 x)^{3/2} (2+3 x)^2}-\frac{715 \int \frac{1}{\sqrt{1-2 x} (2+3 x) \sqrt{3+5 x}} \, dx}{2744}\\ &=\frac{65 \sqrt{1-2 x} \sqrt{3+5 x}}{1372 (2+3 x)}+\frac{65 \sqrt{1-2 x} (3+5 x)^{3/2}}{3234 (2+3 x)^2}+\frac{26 (3+5 x)^{5/2}}{231 \sqrt{1-2 x} (2+3 x)^2}+\frac{4 (3+5 x)^{7/2}}{231 (1-2 x)^{3/2} (2+3 x)^2}-\frac{715 \operatorname{Subst}\left (\int \frac{1}{-7-x^2} \, dx,x,\frac{\sqrt{1-2 x}}{\sqrt{3+5 x}}\right )}{1372}\\ &=\frac{65 \sqrt{1-2 x} \sqrt{3+5 x}}{1372 (2+3 x)}+\frac{65 \sqrt{1-2 x} (3+5 x)^{3/2}}{3234 (2+3 x)^2}+\frac{26 (3+5 x)^{5/2}}{231 \sqrt{1-2 x} (2+3 x)^2}+\frac{4 (3+5 x)^{7/2}}{231 (1-2 x)^{3/2} (2+3 x)^2}+\frac{715 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{3+5 x}}\right )}{1372 \sqrt{7}}\\ \end{align*}

Mathematica [A]  time = 0.0555103, size = 95, normalized size = 0.63 \[ -\frac{7 \sqrt{5 x+3} \left (10260 x^3+1620 x^2-13627 x-6732\right )+2145 \sqrt{7-14 x} (2 x-1) (3 x+2)^2 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{28812 (1-2 x)^{3/2} (3 x+2)^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(7*Sqrt[3 + 5*x]*(-6732 - 13627*x + 1620*x^2 + 10260*x^3) + 2145*Sqrt[7 - 14*x]*(-1 + 2*x)*(2 + 3*x)^2*ArcTan
[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(28812*(1 - 2*x)^(3/2)*(2 + 3*x)^2)

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Maple [B]  time = 0.013, size = 257, normalized size = 1.7 \begin{align*} -{\frac{1}{57624\, \left ( 2+3\,x \right ) ^{2} \left ( 2\,x-1 \right ) ^{2}} \left ( 77220\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{4}+25740\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{3}-49335\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{2}+143640\,{x}^{3}\sqrt{-10\,{x}^{2}-x+3}-8580\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) x+22680\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}+8580\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) -190778\,x\sqrt{-10\,{x}^{2}-x+3}-94248\,\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)^(5/2)/(2+3*x)^3,x)

[Out]

-1/57624*(77220*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^4+25740*7^(1/2)*arctan(1/14*(37*x
+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^3-49335*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^2+143
640*x^3*(-10*x^2-x+3)^(1/2)-8580*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x+22680*x^2*(-10*x
^2-x+3)^(1/2)+8580*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))-190778*x*(-10*x^2-x+3)^(1/2)-942
48*(-10*x^2-x+3)^(1/2))*(1-2*x)^(1/2)*(3+5*x)^(1/2)/(2+3*x)^2/(2*x-1)^2/(-10*x^2-x+3)^(1/2)

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Maxima [A]  time = 3.49885, size = 232, normalized size = 1.54 \begin{align*} -\frac{715}{19208} \, \sqrt{7} \arcsin \left (\frac{37 \, x}{11 \,{\left | 3 \, x + 2 \right |}} + \frac{20}{11 \,{\left | 3 \, x + 2 \right |}}\right ) + \frac{475 \, x}{686 \, \sqrt{-10 \, x^{2} - x + 3}} - \frac{215}{4116 \, \sqrt{-10 \, x^{2} - x + 3}} + \frac{17375 \, x}{2646 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}} - \frac{1}{1134 \,{\left (9 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} x^{2} + 12 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} x + 4 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}\right )}} + \frac{1}{36 \,{\left (3 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} x + 2 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}\right )}} + \frac{60695}{15876 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-715/19208*sqrt(7)*arcsin(37/11*x/abs(3*x + 2) + 20/11/abs(3*x + 2)) + 475/686*x/sqrt(-10*x^2 - x + 3) - 215/4
116/sqrt(-10*x^2 - x + 3) + 17375/2646*x/(-10*x^2 - x + 3)^(3/2) - 1/1134/(9*(-10*x^2 - x + 3)^(3/2)*x^2 + 12*
(-10*x^2 - x + 3)^(3/2)*x + 4*(-10*x^2 - x + 3)^(3/2)) + 1/36/(3*(-10*x^2 - x + 3)^(3/2)*x + 2*(-10*x^2 - x +
3)^(3/2)) + 60695/15876/(-10*x^2 - x + 3)^(3/2)

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Fricas [A]  time = 1.51072, size = 339, normalized size = 2.25 \begin{align*} \frac{2145 \, \sqrt{7}{\left (36 \, x^{4} + 12 \, x^{3} - 23 \, 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 (10260 \, x^{3} + 1620 \, x^{2} - 13627 \, x - 6732\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}{57624 \,{\left (36 \, x^{4} + 12 \, x^{3} - 23 \, x^{2} - 4 \, x + 4\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/57624*(2145*sqrt(7)*(36*x^4 + 12*x^3 - 23*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*(10260*x^3 + 1620*x^2 - 13627*x - 6732)*sqrt(5*x + 3)*sqrt(-2*x + 1))/(36*x^4
+ 12*x^3 - 23*x^2 - 4*x + 4)

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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)**(5/2)/(2+3*x)**3,x)

[Out]

Exception raised: ValueError

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Giac [B]  time = 4.04074, size = 400, normalized size = 2.65 \begin{align*} -\frac{143}{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{22 \,{\left (104 \, \sqrt{5}{\left (5 \, x + 3\right )} - 957 \, \sqrt{5}\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5}}{180075 \,{\left (2 \, x - 1\right )}^{2}} + \frac{11 \,{\left (223 \, \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} + 80920 \, \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 )}}{4802 \,{\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{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-143/38416*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)))) - 22/180075*(104*sqrt(5)*(5*x + 3) - 957*sqrt(5))*sq
rt(5*x + 3)*sqrt(-10*x + 5)/(2*x - 1)^2 + 11/4802*(223*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 + 80920*sqrt(10)*((sqrt(2)*sqrt(-10*x + 5) - s
qrt(22))/sqrt(5*x + 3) - 4*sqrt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))))/(((sqrt(2)*sqrt(-10*x + 5) - s
qrt(22))/sqrt(5*x + 3) - 4*sqrt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22)))^2 + 280)^2