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

Optimal. Leaf size=122 \[ \frac {4 (5 x+3)^{5/2}}{231 (1-2 x)^{3/2} (3 x+2)}+\frac {190 (5 x+3)^{3/2}}{1617 \sqrt {1-2 x} (3 x+2)}+\frac {95 \sqrt {1-2 x} \sqrt {5 x+3}}{3773 (3 x+2)}+\frac {95 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{343 \sqrt {7}} \]

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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}}{231 (1-2 x)^{3/2} (3 x+2)}+\frac {190 (5 x+3)^{3/2}}{1617 \sqrt {1-2 x} (3 x+2)}+\frac {95 \sqrt {1-2 x} \sqrt {5 x+3}}{3773 (3 x+2)}+\frac {95 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{343 \sqrt {7}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(95*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(3773*(2 + 3*x)) + (190*(3 + 5*x)^(3/2))/(1617*Sqrt[1 - 2*x]*(2 + 3*x)) + (4*
(3 + 5*x)^(5/2))/(231*(1 - 2*x)^(3/2)*(2 + 3*x)) + (95*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(343*Sqr
t[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)^{5/2} (2+3 x)^2} \, dx &=\frac {4 (3+5 x)^{5/2}}{231 (1-2 x)^{3/2} (2+3 x)}+\frac {95}{231} \int \frac {(3+5 x)^{3/2}}{(1-2 x)^{3/2} (2+3 x)^2} \, dx\\ &=\frac {190 (3+5 x)^{3/2}}{1617 \sqrt {1-2 x} (2+3 x)}+\frac {4 (3+5 x)^{5/2}}{231 (1-2 x)^{3/2} (2+3 x)}-\frac {95}{539} \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} (2+3 x)^2} \, dx\\ &=\frac {95 \sqrt {1-2 x} \sqrt {3+5 x}}{3773 (2+3 x)}+\frac {190 (3+5 x)^{3/2}}{1617 \sqrt {1-2 x} (2+3 x)}+\frac {4 (3+5 x)^{5/2}}{231 (1-2 x)^{3/2} (2+3 x)}-\frac {95}{686} \int \frac {1}{\sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}} \, dx\\ &=\frac {95 \sqrt {1-2 x} \sqrt {3+5 x}}{3773 (2+3 x)}+\frac {190 (3+5 x)^{3/2}}{1617 \sqrt {1-2 x} (2+3 x)}+\frac {4 (3+5 x)^{5/2}}{231 (1-2 x)^{3/2} (2+3 x)}-\frac {95}{343} \operatorname {Subst}\left (\int \frac {1}{-7-x^2} \, dx,x,\frac {\sqrt {1-2 x}}{\sqrt {3+5 x}}\right )\\ &=\frac {95 \sqrt {1-2 x} \sqrt {3+5 x}}{3773 (2+3 x)}+\frac {190 (3+5 x)^{3/2}}{1617 \sqrt {1-2 x} (2+3 x)}+\frac {4 (3+5 x)^{5/2}}{231 (1-2 x)^{3/2} (2+3 x)}+\frac {95 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{343 \sqrt {7}}\\ \end {align*}

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Mathematica [A]  time = 0.05, size = 86, normalized size = 0.70 \begin {gather*} -\frac {7 \sqrt {5 x+3} \left (660 x^2-310 x-549\right )+285 \sqrt {7-14 x} \left (6 x^2+x-2\right ) \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{7203 (1-2 x)^{3/2} (3 x+2)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/7203*(7*Sqrt[3 + 5*x]*(-549 - 310*x + 660*x^2) + 285*Sqrt[7 - 14*x]*(-2 + x + 6*x^2)*ArcTan[Sqrt[1 - 2*x]/(
Sqrt[7]*Sqrt[3 + 5*x])])/((1 - 2*x)^(3/2)*(2 + 3*x))

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IntegrateAlgebraic [A]  time = 0.15, size = 106, normalized size = 0.87 \begin {gather*} \frac {\left (\frac {285 (1-2 x)^2}{(5 x+3)^2}+\frac {1330 (1-2 x)}{5 x+3}+196\right ) (5 x+3)^{3/2}}{1029 (1-2 x)^{3/2} \left (\frac {1-2 x}{5 x+3}+7\right )}+\frac {95 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{343 \sqrt {7}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((3 + 5*x)^(3/2)*(196 + (285*(1 - 2*x)^2)/(3 + 5*x)^2 + (1330*(1 - 2*x))/(3 + 5*x)))/(1029*(1 - 2*x)^(3/2)*(7
+ (1 - 2*x)/(3 + 5*x))) + (95*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(343*Sqrt[7])

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fricas [A]  time = 0.85, size = 101, normalized size = 0.83 \begin {gather*} \frac {285 \, \sqrt {7} {\left (12 \, x^{3} - 4 \, x^{2} - 5 \, x + 2\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 (660 \, x^{2} - 310 \, x - 549\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{14406 \, {\left (12 \, x^{3} - 4 \, x^{2} - 5 \, x + 2\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/14406*(285*sqrt(7)*(12*x^3 - 4*x^2 - 5*x + 2)*arctan(1/14*sqrt(7)*(37*x + 20)*sqrt(5*x + 3)*sqrt(-2*x + 1)/(
10*x^2 + x - 3)) - 14*(660*x^2 - 310*x - 549)*sqrt(5*x + 3)*sqrt(-2*x + 1))/(12*x^3 - 4*x^2 - 5*x + 2)

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giac [B]  time = 2.04, size = 232, normalized size = 1.90 \begin {gather*} -\frac {19}{9604} \, \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 {66 \, \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 )}}{343 \, {\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 )}} - \frac {2 \, {\left (116 \, \sqrt {5} {\left (5 \, x + 3\right )} - 1023 \, \sqrt {5}\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5}}{25725 \, {\left (2 \, x - 1\right )}^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-19/9604*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)))) + 66/343*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)))/(((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) - 2/25725*(116*sqrt(5)*(5*x +
3) - 1023*sqrt(5))*sqrt(5*x + 3)*sqrt(-10*x + 5)/(2*x - 1)^2

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maple [B]  time = 0.01, size = 209, normalized size = 1.71 \begin {gather*} -\frac {\left (3420 \sqrt {7}\, x^{3} \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )-1140 \sqrt {7}\, x^{2} \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+9240 \sqrt {-10 x^{2}-x +3}\, x^{2}-1425 \sqrt {7}\, x \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )-4340 \sqrt {-10 x^{2}-x +3}\, x +570 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )-7686 \sqrt {-10 x^{2}-x +3}\right ) \sqrt {-2 x +1}\, \sqrt {5 x +3}}{14406 \left (3 x +2\right ) \left (2 x -1\right )^{2} \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)^(5/2)/(3*x+2)^2,x)

[Out]

-1/14406*(3420*7^(1/2)*x^3*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))-1140*7^(1/2)*x^2*arctan(1/14*(37
*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))-1425*7^(1/2)*x*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))+9240*(-1
0*x^2-x+3)^(1/2)*x^2+570*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))-4340*(-10*x^2-x+3)^(1/2)*x
-7686*(-10*x^2-x+3)^(1/2))*(-2*x+1)^(1/2)*(5*x+3)^(1/2)/(3*x+2)/(2*x-1)^2/(-10*x^2-x+3)^(1/2)

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maxima [A]  time = 1.13, size = 121, normalized size = 0.99 \begin {gather*} -\frac {95}{4802} \, \sqrt {7} \arcsin \left (\frac {37 \, x}{11 \, {\left | 3 \, x + 2 \right |}} + \frac {20}{11 \, {\left | 3 \, x + 2 \right |}}\right ) + \frac {550 \, x}{1029 \, \sqrt {-10 \, x^{2} - x + 3}} - \frac {20}{1029 \, \sqrt {-10 \, x^{2} - x + 3}} + \frac {1825 \, x}{441 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}} + \frac {1}{189 \, {\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 {3250}{1323 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-95/4802*sqrt(7)*arcsin(37/11*x/abs(3*x + 2) + 20/11/abs(3*x + 2)) + 550/1029*x/sqrt(-10*x^2 - x + 3) - 20/102
9/sqrt(-10*x^2 - x + 3) + 1825/441*x/(-10*x^2 - x + 3)^(3/2) + 1/189/(3*(-10*x^2 - x + 3)^(3/2)*x + 2*(-10*x^2
- x + 3)^(3/2)) + 3250/1323/(-10*x^2 - x + 3)^(3/2)

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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 )}^{5/2}\,{\left (3\,x+2\right )}^2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Timed out

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