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

Optimal. Leaf size=84 \[ -\frac {2 \sqrt {1-2 x} (3 x+2)^2}{165 (5 x+3)^{3/2}}-\frac {\sqrt {1-2 x} (9405 x+5831)}{18150 \sqrt {5 x+3}}+\frac {81 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{50 \sqrt {10}} \]

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Rubi [A]  time = 0.02, antiderivative size = 84, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {98, 143, 54, 216} \begin {gather*} -\frac {2 \sqrt {1-2 x} (3 x+2)^2}{165 (5 x+3)^{3/2}}-\frac {\sqrt {1-2 x} (9405 x+5831)}{18150 \sqrt {5 x+3}}+\frac {81 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{50 \sqrt {10}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*Sqrt[1 - 2*x]*(2 + 3*x)^2)/(165*(3 + 5*x)^(3/2)) - (Sqrt[1 - 2*x]*(5831 + 9405*x))/(18150*Sqrt[3 + 5*x]) +
(81*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(50*Sqrt[10])

Rule 54

Int[1/(Sqrt[(a_.) + (b_.)*(x_)]*Sqrt[(c_.) + (d_.)*(x_)]), x_Symbol] :> Dist[2/Sqrt[b], Subst[Int[1/Sqrt[b*c -
a*d + d*x^2], x], x, Sqrt[a + b*x]], x] /; FreeQ[{a, b, c, d}, x] && GtQ[b*c - a*d, 0] && GtQ[b, 0]

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 143

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_) + (f_.)*(x_))*((g_.) + (h_.)*(x_)), x_Symbol] :
> Simp[((b^2*d*e*g - a^2*d*f*h*m - a*b*(d*(f*g + e*h) - c*f*h*(m + 1)) + b*f*h*(b*c - a*d)*(m + 1)*x)*(a + b*x
)^(m + 1)*(c + d*x)^(n + 1))/(b^2*d*(b*c - a*d)*(m + 1)), x] + Dist[(a*d*f*h*m + b*(d*(f*g + e*h) - c*f*h*(m +
2)))/(b^2*d), Int[(a + b*x)^(m + 1)*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, e, f, g, h, m, n}, x] && EqQ[m
+ n + 2, 0] && NeQ[m, -1] &&  !(SumSimplerQ[n, 1] &&  !SumSimplerQ[m, 1])

Rule 216

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSin[(Rt[-b, 2]*x)/Sqrt[a]]/Rt[-b, 2], x] /; FreeQ[{a, b}
, x] && GtQ[a, 0] && NegQ[b]

Rubi steps

\begin {align*} \int \frac {(2+3 x)^3}{\sqrt {1-2 x} (3+5 x)^{5/2}} \, dx &=-\frac {2 \sqrt {1-2 x} (2+3 x)^2}{165 (3+5 x)^{3/2}}-\frac {2}{165} \int \frac {\left (-109-\frac {285 x}{2}\right ) (2+3 x)}{\sqrt {1-2 x} (3+5 x)^{3/2}} \, dx\\ &=-\frac {2 \sqrt {1-2 x} (2+3 x)^2}{165 (3+5 x)^{3/2}}-\frac {\sqrt {1-2 x} (5831+9405 x)}{18150 \sqrt {3+5 x}}+\frac {81}{100} \int \frac {1}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx\\ &=-\frac {2 \sqrt {1-2 x} (2+3 x)^2}{165 (3+5 x)^{3/2}}-\frac {\sqrt {1-2 x} (5831+9405 x)}{18150 \sqrt {3+5 x}}+\frac {81 \operatorname {Subst}\left (\int \frac {1}{\sqrt {11-2 x^2}} \, dx,x,\sqrt {3+5 x}\right )}{50 \sqrt {5}}\\ &=-\frac {2 \sqrt {1-2 x} (2+3 x)^2}{165 (3+5 x)^{3/2}}-\frac {\sqrt {1-2 x} (5831+9405 x)}{18150 \sqrt {3+5 x}}+\frac {81 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{50 \sqrt {10}}\\ \end {align*}

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Mathematica [A]  time = 0.07, size = 69, normalized size = 0.82 \begin {gather*} \frac {\sqrt {1-2 x} \left (-\frac {10 \left (49005 x^2+60010 x+18373\right )}{(5 x+3)^{3/2}}-\frac {29403 \sqrt {10} \sinh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {2 x-1}\right )}{\sqrt {2 x-1}}\right )}{181500} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[1 - 2*x]*((-10*(18373 + 60010*x + 49005*x^2))/(3 + 5*x)^(3/2) - (29403*Sqrt[10]*ArcSinh[Sqrt[5/11]*Sqrt[
-1 + 2*x]])/Sqrt[-1 + 2*x]))/181500

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IntegrateAlgebraic [A]  time = 0.13, size = 109, normalized size = 1.30 \begin {gather*} -\frac {\sqrt {1-2 x} \left (\frac {20 (1-2 x)^2}{(5 x+3)^2}+\frac {1220 (1-2 x)}{5 x+3}+22047\right )}{18150 \sqrt {5 x+3} \left (\frac {5 (1-2 x)}{5 x+3}+2\right )}-\frac {81 \tan ^{-1}\left (\frac {\sqrt {\frac {5}{2}} \sqrt {1-2 x}}{\sqrt {5 x+3}}\right )}{50 \sqrt {10}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/18150*(Sqrt[1 - 2*x]*(22047 + (20*(1 - 2*x)^2)/(3 + 5*x)^2 + (1220*(1 - 2*x))/(3 + 5*x)))/(Sqrt[3 + 5*x]*(2
+ (5*(1 - 2*x))/(3 + 5*x))) - (81*ArcTan[(Sqrt[5/2]*Sqrt[1 - 2*x])/Sqrt[3 + 5*x]])/(50*Sqrt[10])

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fricas [A]  time = 1.16, size = 91, normalized size = 1.08 \begin {gather*} -\frac {29403 \, \sqrt {10} {\left (25 \, x^{2} + 30 \, x + 9\right )} \arctan \left (\frac {\sqrt {10} {\left (20 \, x + 1\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{20 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) + 20 \, {\left (49005 \, x^{2} + 60010 \, x + 18373\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{363000 \, {\left (25 \, x^{2} + 30 \, x + 9\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/363000*(29403*sqrt(10)*(25*x^2 + 30*x + 9)*arctan(1/20*sqrt(10)*(20*x + 1)*sqrt(5*x + 3)*sqrt(-2*x + 1)/(10
*x^2 + x - 3)) + 20*(49005*x^2 + 60010*x + 18373)*sqrt(5*x + 3)*sqrt(-2*x + 1))/(25*x^2 + 30*x + 9)

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giac [B]  time = 1.14, size = 158, normalized size = 1.88 \begin {gather*} -\frac {1}{3630000} \, \sqrt {10} {\left (\frac {{\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{3}}{{\left (5 \, x + 3\right )}^{\frac {3}{2}}} + \frac {2412 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}{\sqrt {5 \, x + 3}}\right )} - \frac {27}{1250} \, \sqrt {5} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5} + \frac {81}{500} \, \sqrt {10} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right ) + \frac {\sqrt {10} {\left (5 \, x + 3\right )}^{\frac {3}{2}} {\left (\frac {603 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{2}}{5 \, x + 3} + 4\right )}}{226875 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3630000*sqrt(10)*((sqrt(2)*sqrt(-10*x + 5) - sqrt(22))^3/(5*x + 3)^(3/2) + 2412*(sqrt(2)*sqrt(-10*x + 5) -
sqrt(22))/sqrt(5*x + 3)) - 27/1250*sqrt(5)*sqrt(5*x + 3)*sqrt(-10*x + 5) + 81/500*sqrt(10)*arcsin(1/11*sqrt(22
)*sqrt(5*x + 3)) + 1/226875*sqrt(10)*(5*x + 3)^(3/2)*(603*(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))^2/(5*x + 3) + 4
)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))^3

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maple [A]  time = 0.01, size = 113, normalized size = 1.35 \begin {gather*} \frac {\left (735075 \sqrt {10}\, x^{2} \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )-980100 \sqrt {-10 x^{2}-x +3}\, x^{2}+882090 \sqrt {10}\, x \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )-1200200 \sqrt {-10 x^{2}-x +3}\, x +264627 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )-367460 \sqrt {-10 x^{2}-x +3}\right ) \sqrt {-2 x +1}}{363000 \sqrt {-10 x^{2}-x +3}\, \left (5 x +3\right )^{\frac {3}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/363000*(735075*10^(1/2)*x^2*arcsin(20/11*x+1/11)+882090*10^(1/2)*x*arcsin(20/11*x+1/11)-980100*(-10*x^2-x+3)
^(1/2)*x^2+264627*10^(1/2)*arcsin(20/11*x+1/11)-1200200*(-10*x^2-x+3)^(1/2)*x-367460*(-10*x^2-x+3)^(1/2))*(-2*
x+1)^(1/2)/(-10*x^2-x+3)^(1/2)/(5*x+3)^(3/2)

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maxima [A]  time = 1.26, size = 76, normalized size = 0.90 \begin {gather*} \frac {81}{1000} \, \sqrt {5} \sqrt {2} \arcsin \left (\frac {20}{11} \, x + \frac {1}{11}\right ) - \frac {27}{250} \, \sqrt {-10 \, x^{2} - x + 3} - \frac {2 \, \sqrt {-10 \, x^{2} - x + 3}}{4125 \, {\left (25 \, x^{2} + 30 \, x + 9\right )}} - \frac {602 \, \sqrt {-10 \, x^{2} - x + 3}}{45375 \, {\left (5 \, x + 3\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

81/1000*sqrt(5)*sqrt(2)*arcsin(20/11*x + 1/11) - 27/250*sqrt(-10*x^2 - x + 3) - 2/4125*sqrt(-10*x^2 - x + 3)/(
25*x^2 + 30*x + 9) - 602/45375*sqrt(-10*x^2 - x + 3)/(5*x + 3)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\left (3\,x+2\right )}^3}{\sqrt {1-2\,x}\,{\left (5\,x+3\right )}^{5/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (3 x + 2\right )^{3}}{\sqrt {1 - 2 x} \left (5 x + 3\right )^{\frac {5}{2}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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