3.2476 \(\int \frac{(3+5 x)^{3/2}}{\sqrt{1-2 x} (2+3 x)^5} \, dx\)

Optimal. Leaf size=151 \[ \frac{57595 \sqrt{1-2 x} \sqrt{5 x+3}}{197568 (3 x+2)}+\frac{85 \sqrt{1-2 x} \sqrt{5 x+3}}{14112 (3 x+2)^2}-\frac{43 \sqrt{1-2 x} \sqrt{5 x+3}}{504 (3 x+2)^3}+\frac{\sqrt{1-2 x} \sqrt{5 x+3}}{84 (3 x+2)^4}-\frac{78045 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{21952 \sqrt{7}} \]

[Out]

(Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(84*(2 + 3*x)^4) - (43*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(504*(2 + 3*x)^3) + (85*Sqrt
[1 - 2*x]*Sqrt[3 + 5*x])/(14112*(2 + 3*x)^2) + (57595*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(197568*(2 + 3*x)) - (78045
*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(21952*Sqrt[7])

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Rubi [A]  time = 0.0490276, antiderivative size = 151, normalized size of antiderivative = 1., 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} \[ \frac{57595 \sqrt{1-2 x} \sqrt{5 x+3}}{197568 (3 x+2)}+\frac{85 \sqrt{1-2 x} \sqrt{5 x+3}}{14112 (3 x+2)^2}-\frac{43 \sqrt{1-2 x} \sqrt{5 x+3}}{504 (3 x+2)^3}+\frac{\sqrt{1-2 x} \sqrt{5 x+3}}{84 (3 x+2)^4}-\frac{78045 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{21952 \sqrt{7}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(84*(2 + 3*x)^4) - (43*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(504*(2 + 3*x)^3) + (85*Sqrt
[1 - 2*x]*Sqrt[3 + 5*x])/(14112*(2 + 3*x)^2) + (57595*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(197568*(2 + 3*x)) - (78045
*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(21952*Sqrt[7])

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 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 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}}{\sqrt{1-2 x} (2+3 x)^5} \, dx &=\frac{\sqrt{1-2 x} \sqrt{3+5 x}}{84 (2+3 x)^4}-\frac{1}{84} \int \frac{-\frac{793}{2}-670 x}{\sqrt{1-2 x} (2+3 x)^4 \sqrt{3+5 x}} \, dx\\ &=\frac{\sqrt{1-2 x} \sqrt{3+5 x}}{84 (2+3 x)^4}-\frac{43 \sqrt{1-2 x} \sqrt{3+5 x}}{504 (2+3 x)^3}-\frac{\int \frac{-\frac{8225}{4}-3010 x}{\sqrt{1-2 x} (2+3 x)^3 \sqrt{3+5 x}} \, dx}{1764}\\ &=\frac{\sqrt{1-2 x} \sqrt{3+5 x}}{84 (2+3 x)^4}-\frac{43 \sqrt{1-2 x} \sqrt{3+5 x}}{504 (2+3 x)^3}+\frac{85 \sqrt{1-2 x} \sqrt{3+5 x}}{14112 (2+3 x)^2}-\frac{\int \frac{-\frac{126455}{8}+\frac{2975 x}{2}}{\sqrt{1-2 x} (2+3 x)^2 \sqrt{3+5 x}} \, dx}{24696}\\ &=\frac{\sqrt{1-2 x} \sqrt{3+5 x}}{84 (2+3 x)^4}-\frac{43 \sqrt{1-2 x} \sqrt{3+5 x}}{504 (2+3 x)^3}+\frac{85 \sqrt{1-2 x} \sqrt{3+5 x}}{14112 (2+3 x)^2}+\frac{57595 \sqrt{1-2 x} \sqrt{3+5 x}}{197568 (2+3 x)}-\frac{\int -\frac{4916835}{16 \sqrt{1-2 x} (2+3 x) \sqrt{3+5 x}} \, dx}{172872}\\ &=\frac{\sqrt{1-2 x} \sqrt{3+5 x}}{84 (2+3 x)^4}-\frac{43 \sqrt{1-2 x} \sqrt{3+5 x}}{504 (2+3 x)^3}+\frac{85 \sqrt{1-2 x} \sqrt{3+5 x}}{14112 (2+3 x)^2}+\frac{57595 \sqrt{1-2 x} \sqrt{3+5 x}}{197568 (2+3 x)}+\frac{78045 \int \frac{1}{\sqrt{1-2 x} (2+3 x) \sqrt{3+5 x}} \, dx}{43904}\\ &=\frac{\sqrt{1-2 x} \sqrt{3+5 x}}{84 (2+3 x)^4}-\frac{43 \sqrt{1-2 x} \sqrt{3+5 x}}{504 (2+3 x)^3}+\frac{85 \sqrt{1-2 x} \sqrt{3+5 x}}{14112 (2+3 x)^2}+\frac{57595 \sqrt{1-2 x} \sqrt{3+5 x}}{197568 (2+3 x)}+\frac{78045 \operatorname{Subst}\left (\int \frac{1}{-7-x^2} \, dx,x,\frac{\sqrt{1-2 x}}{\sqrt{3+5 x}}\right )}{21952}\\ &=\frac{\sqrt{1-2 x} \sqrt{3+5 x}}{84 (2+3 x)^4}-\frac{43 \sqrt{1-2 x} \sqrt{3+5 x}}{504 (2+3 x)^3}+\frac{85 \sqrt{1-2 x} \sqrt{3+5 x}}{14112 (2+3 x)^2}+\frac{57595 \sqrt{1-2 x} \sqrt{3+5 x}}{197568 (2+3 x)}-\frac{78045 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{3+5 x}}\right )}{21952 \sqrt{7}}\\ \end{align*}

Mathematica [A]  time = 0.0540792, size = 79, normalized size = 0.52 \[ \frac{\frac{7 \sqrt{1-2 x} \sqrt{5 x+3} \left (172785 x^3+346760 x^2+226348 x+48240\right )}{(3 x+2)^4}-78045 \sqrt{7} \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{153664} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((7*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(48240 + 226348*x + 346760*x^2 + 172785*x^3))/(2 + 3*x)^4 - 78045*Sqrt[7]*ArcT
an[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/153664

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Maple [B]  time = 0.013, size = 250, normalized size = 1.7 \begin{align*}{\frac{1}{307328\, \left ( 2+3\,x \right ) ^{4}}\sqrt{1-2\,x}\sqrt{3+5\,x} \left ( 6321645\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{4}[phone]\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{3}[phone]\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{2}+2418990\,{x}^{3}\sqrt{-10\,{x}^{2}-x+3}+7492320\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) x+4854640\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}+1248720\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) +3168872\,x\sqrt{-10\,{x}^{2}-x+3}+675360\,\sqrt{-10\,{x}^{2}-x+3} \right ){\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)^(3/2)/(2+3*x)^5/(1-2*x)^(1/2),x)

[Out]

1/307328*(3+5*x)^(1/2)*(1-2*x)^(1/2)*(6321645*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^4+1
6857720*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^3+16857720*7^(1/2)*arctan(1/14*(37*x+20)*
7^(1/2)/(-10*x^2-x+3)^(1/2))*x^2+2418990*x^3*(-10*x^2-x+3)^(1/2)+7492320*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)
/(-10*x^2-x+3)^(1/2))*x+4854640*x^2*(-10*x^2-x+3)^(1/2)+1248720*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2
-x+3)^(1/2))+3168872*x*(-10*x^2-x+3)^(1/2)+675360*(-10*x^2-x+3)^(1/2))/(-10*x^2-x+3)^(1/2)/(2+3*x)^4

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Maxima [A]  time = 3.10527, size = 193, normalized size = 1.28 \begin{align*} \frac{78045}{307328} \, \sqrt{7} \arcsin \left (\frac{37 \, x}{11 \,{\left | 3 \, x + 2 \right |}} + \frac{20}{11 \,{\left | 3 \, x + 2 \right |}}\right ) + \frac{\sqrt{-10 \, x^{2} - x + 3}}{84 \,{\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )}} - \frac{43 \, \sqrt{-10 \, x^{2} - x + 3}}{504 \,{\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}} + \frac{85 \, \sqrt{-10 \, x^{2} - x + 3}}{14112 \,{\left (9 \, x^{2} + 12 \, x + 4\right )}} + \frac{57595 \, \sqrt{-10 \, x^{2} - x + 3}}{197568 \,{\left (3 \, x + 2\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

78045/307328*sqrt(7)*arcsin(37/11*x/abs(3*x + 2) + 20/11/abs(3*x + 2)) + 1/84*sqrt(-10*x^2 - x + 3)/(81*x^4 +
216*x^3 + 216*x^2 + 96*x + 16) - 43/504*sqrt(-10*x^2 - x + 3)/(27*x^3 + 54*x^2 + 36*x + 8) + 85/14112*sqrt(-10
*x^2 - x + 3)/(9*x^2 + 12*x + 4) + 57595/197568*sqrt(-10*x^2 - x + 3)/(3*x + 2)

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Fricas [A]  time = 1.74397, size = 360, normalized size = 2.38 \begin{align*} -\frac{78045 \, \sqrt{7}{\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\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 (172785 \, x^{3} + 346760 \, x^{2} + 226348 \, x + 48240\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}{307328 \,{\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/307328*(78045*sqrt(7)*(81*x^4 + 216*x^3 + 216*x^2 + 96*x + 16)*arctan(1/14*sqrt(7)*(37*x + 20)*sqrt(5*x + 3
)*sqrt(-2*x + 1)/(10*x^2 + x - 3)) - 14*(172785*x^3 + 346760*x^2 + 226348*x + 48240)*sqrt(5*x + 3)*sqrt(-2*x +
1))/(81*x^4 + 216*x^3 + 216*x^2 + 96*x + 16)

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

[Out]

Exception raised: ValueError

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Giac [B]  time = 3.20061, size = 512, normalized size = 3.39 \begin{align*} \frac{15609}{614656} \, \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{605 \,{\left (129 \, \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 )}^{7} + 132440 \, \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} - 21026880 \, \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} - 2510681600 \, \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 )}}{10976 \,{\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 )}^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

15609/614656*sqrt(70)*sqrt(10)*(pi + 2*arctan(-1/140*sqrt(70)*sqrt(5*x + 3)*((sqrt(2)*sqrt(-10*x + 5) - sqrt(2
2))^2/(5*x + 3) - 4)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22)))) - 605/10976*(129*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)))^7 + 132440*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 - 2102
6880*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 - 2510681600*sqrt(10)*((sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) - 4*sqrt(5*x + 3)/(sqr
t(2)*sqrt(-10*x + 5) - sqrt(22))))/(((sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) - 4*sqrt(5*x + 3)/(sqr
t(2)*sqrt(-10*x + 5) - sqrt(22)))^2 + 280)^4