3.2497 \(\int \frac{1}{\sqrt{1-2 x} (2+3 x)^4 \sqrt{3+5 x}} \, dx\)

Optimal. Leaf size=122 \[ \frac{19415 \sqrt{1-2 x} \sqrt{5 x+3}}{2744 (3 x+2)}+\frac{185 \sqrt{1-2 x} \sqrt{5 x+3}}{196 (3 x+2)^2}+\frac{\sqrt{1-2 x} \sqrt{5 x+3}}{7 (3 x+2)^3}-\frac{222185 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{2744 \sqrt{7}} \]

[Out]

(Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(7*(2 + 3*x)^3) + (185*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(196*(2 + 3*x)^2) + (19415*S
qrt[1 - 2*x]*Sqrt[3 + 5*x])/(2744*(2 + 3*x)) - (222185*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(2744*Sq
rt[7])

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Rubi [A]  time = 0.037073, antiderivative size = 122, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.192, Rules used = {103, 151, 12, 93, 204} \[ \frac{19415 \sqrt{1-2 x} \sqrt{5 x+3}}{2744 (3 x+2)}+\frac{185 \sqrt{1-2 x} \sqrt{5 x+3}}{196 (3 x+2)^2}+\frac{\sqrt{1-2 x} \sqrt{5 x+3}}{7 (3 x+2)^3}-\frac{222185 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{2744 \sqrt{7}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(7*(2 + 3*x)^3) + (185*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(196*(2 + 3*x)^2) + (19415*S
qrt[1 - 2*x]*Sqrt[3 + 5*x])/(2744*(2 + 3*x)) - (222185*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(2744*Sq
rt[7])

Rule 103

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[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*(m + 1) - b*(d*e*(m + n + 2) +
c*f*(m + p + 2)) - b*d*f*(m + n + p + 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && LtQ[m, -1] &&
IntegerQ[m] && (IntegerQ[n] || IntegersQ[2*n, 2*p])

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{1}{\sqrt{1-2 x} (2+3 x)^4 \sqrt{3+5 x}} \, dx &=\frac{\sqrt{1-2 x} \sqrt{3+5 x}}{7 (2+3 x)^3}+\frac{1}{21} \int \frac{\frac{105}{2}-60 x}{\sqrt{1-2 x} (2+3 x)^3 \sqrt{3+5 x}} \, dx\\ &=\frac{\sqrt{1-2 x} \sqrt{3+5 x}}{7 (2+3 x)^3}+\frac{185 \sqrt{1-2 x} \sqrt{3+5 x}}{196 (2+3 x)^2}+\frac{1}{294} \int \frac{\frac{12015}{4}-2775 x}{\sqrt{1-2 x} (2+3 x)^2 \sqrt{3+5 x}} \, dx\\ &=\frac{\sqrt{1-2 x} \sqrt{3+5 x}}{7 (2+3 x)^3}+\frac{185 \sqrt{1-2 x} \sqrt{3+5 x}}{196 (2+3 x)^2}+\frac{19415 \sqrt{1-2 x} \sqrt{3+5 x}}{2744 (2+3 x)}+\frac{\int \frac{666555}{8 \sqrt{1-2 x} (2+3 x) \sqrt{3+5 x}} \, dx}{2058}\\ &=\frac{\sqrt{1-2 x} \sqrt{3+5 x}}{7 (2+3 x)^3}+\frac{185 \sqrt{1-2 x} \sqrt{3+5 x}}{196 (2+3 x)^2}+\frac{19415 \sqrt{1-2 x} \sqrt{3+5 x}}{2744 (2+3 x)}+\frac{222185 \int \frac{1}{\sqrt{1-2 x} (2+3 x) \sqrt{3+5 x}} \, dx}{5488}\\ &=\frac{\sqrt{1-2 x} \sqrt{3+5 x}}{7 (2+3 x)^3}+\frac{185 \sqrt{1-2 x} \sqrt{3+5 x}}{196 (2+3 x)^2}+\frac{19415 \sqrt{1-2 x} \sqrt{3+5 x}}{2744 (2+3 x)}+\frac{222185 \operatorname{Subst}\left (\int \frac{1}{-7-x^2} \, dx,x,\frac{\sqrt{1-2 x}}{\sqrt{3+5 x}}\right )}{2744}\\ &=\frac{\sqrt{1-2 x} \sqrt{3+5 x}}{7 (2+3 x)^3}+\frac{185 \sqrt{1-2 x} \sqrt{3+5 x}}{196 (2+3 x)^2}+\frac{19415 \sqrt{1-2 x} \sqrt{3+5 x}}{2744 (2+3 x)}-\frac{222185 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{3+5 x}}\right )}{2744 \sqrt{7}}\\ \end{align*}

Mathematica [A]  time = 0.0483581, size = 74, normalized size = 0.61 \[ \frac{\frac{63 \sqrt{1-2 x} \sqrt{5 x+3} \left (19415 x^2+26750 x+9248\right )}{(3 x+2)^3}-222185 \sqrt{7} \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{19208} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((63*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(9248 + 26750*x + 19415*x^2))/(2 + 3*x)^3 - 222185*Sqrt[7]*ArcTan[Sqrt[1 - 2*
x]/(Sqrt[7]*Sqrt[3 + 5*x])])/19208

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Maple [B]  time = 0.016, size = 202, normalized size = 1.7 \begin{align*}{\frac{1}{38416\, \left ( 2+3\,x \right ) ^{3}}\sqrt{1-2\,x}\sqrt{3+5\,x} \left ( 5998995\,\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}+7998660\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) x+2446290\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}+1777480\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) +3370500\,x\sqrt{-10\,{x}^{2}-x+3}+1165248\,\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(1/(2+3*x)^4/(1-2*x)^(1/2)/(3+5*x)^(1/2),x)

[Out]

1/38416*(1-2*x)^(1/2)*(3+5*x)^(1/2)*(5998995*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^3+11
997990*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^2+7998660*7^(1/2)*arctan(1/14*(37*x+20)*7^
(1/2)/(-10*x^2-x+3)^(1/2))*x+2446290*x^2*(-10*x^2-x+3)^(1/2)+1777480*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-1
0*x^2-x+3)^(1/2))+3370500*x*(-10*x^2-x+3)^(1/2)+1165248*(-10*x^2-x+3)^(1/2))/(-10*x^2-x+3)^(1/2)/(2+3*x)^3

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Maxima [A]  time = 3.51482, size = 144, normalized size = 1.18 \begin{align*} \frac{222185}{38416} \, \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}}{7 \,{\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}} + \frac{185 \, \sqrt{-10 \, x^{2} - x + 3}}{196 \,{\left (9 \, x^{2} + 12 \, x + 4\right )}} + \frac{19415 \, \sqrt{-10 \, x^{2} - x + 3}}{2744 \,{\left (3 \, x + 2\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(2+3*x)^4/(1-2*x)^(1/2)/(3+5*x)^(1/2),x, algorithm="maxima")

[Out]

222185/38416*sqrt(7)*arcsin(37/11*x/abs(3*x + 2) + 20/11/abs(3*x + 2)) + 1/7*sqrt(-10*x^2 - x + 3)/(27*x^3 + 5
4*x^2 + 36*x + 8) + 185/196*sqrt(-10*x^2 - x + 3)/(9*x^2 + 12*x + 4) + 19415/2744*sqrt(-10*x^2 - x + 3)/(3*x +
2)

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Fricas [A]  time = 1.99714, size = 308, normalized size = 2.52 \begin{align*} -\frac{222185 \, \sqrt{7}{\left (27 \, x^{3} + 54 \, x^{2} + 36 \, 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 ) - 126 \,{\left (19415 \, x^{2} + 26750 \, x + 9248\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}{38416 \,{\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(2+3*x)^4/(1-2*x)^(1/2)/(3+5*x)^(1/2),x, algorithm="fricas")

[Out]

-1/38416*(222185*sqrt(7)*(27*x^3 + 54*x^2 + 36*x + 8)*arctan(1/14*sqrt(7)*(37*x + 20)*sqrt(5*x + 3)*sqrt(-2*x
+ 1)/(10*x^2 + x - 3)) - 126*(19415*x^2 + 26750*x + 9248)*sqrt(5*x + 3)*sqrt(-2*x + 1))/(27*x^3 + 54*x^2 + 36*
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(1/(2+3*x)**4/(1-2*x)**(1/2)/(3+5*x)**(1/2),x)

[Out]

Exception raised: ValueError

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Giac [B]  time = 2.88152, size = 429, normalized size = 3.52 \begin{align*} \frac{44437}{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{495 \,{\left (937 \, \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} + 333760 \, \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} + 35170240 \, \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 )}}{1372 \,{\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(1/(2+3*x)^4/(1-2*x)^(1/2)/(3+5*x)^(1/2),x, algorithm="giac")

[Out]

44437/76832*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)))) + 495/1372*(937*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 + 333760*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 + 351702
40*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) - s
qrt(22))))/(((sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) - 4*sqrt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - s
qrt(22)))^2 + 280)^3