3.26.63 \(\int \frac {(2+3 x)^3}{(1-2 x)^{3/2} (3+5 x)^{3/2}} \, dx\) [2563]

Optimal. Leaf size=84 \[ \frac {7 (2+3 x)^2}{11 \sqrt {1-2 x} \sqrt {3+5 x}}+\frac {\sqrt {1-2 x} (30443+50985 x)}{12100 \sqrt {3+5 x}}-\frac {999 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{100 \sqrt {10}} \]

[Out]

-999/1000*arcsin(1/11*22^(1/2)*(3+5*x)^(1/2))*10^(1/2)+7/11*(2+3*x)^2/(1-2*x)^(1/2)/(3+5*x)^(1/2)+1/12100*(304
43+50985*x)*(1-2*x)^(1/2)/(3+5*x)^(1/2)

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Rubi [A]
time = 0.01, 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 = {100, 148, 56, 222} \begin {gather*} -\frac {999 \text {ArcSin}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{100 \sqrt {10}}+\frac {7 (3 x+2)^2}{11 \sqrt {1-2 x} \sqrt {5 x+3}}+\frac {\sqrt {1-2 x} (50985 x+30443)}{12100 \sqrt {5 x+3}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(7*(2 + 3*x)^2)/(11*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]) + (Sqrt[1 - 2*x]*(30443 + 50985*x))/(12100*Sqrt[3 + 5*x]) - (
999*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(100*Sqrt[10])

Rule 56

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 100

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 148

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[m, 1] ||  !SumSimplerQ[n, 1])

Rule 222

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}{(1-2 x)^{3/2} (3+5 x)^{3/2}} \, dx &=\frac {7 (2+3 x)^2}{11 \sqrt {1-2 x} \sqrt {3+5 x}}-\frac {1}{11} \int \frac {(2+3 x) \left (89+\frac {309 x}{2}\right )}{\sqrt {1-2 x} (3+5 x)^{3/2}} \, dx\\ &=\frac {7 (2+3 x)^2}{11 \sqrt {1-2 x} \sqrt {3+5 x}}+\frac {\sqrt {1-2 x} (30443+50985 x)}{12100 \sqrt {3+5 x}}-\frac {999}{200} \int \frac {1}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx\\ &=\frac {7 (2+3 x)^2}{11 \sqrt {1-2 x} \sqrt {3+5 x}}+\frac {\sqrt {1-2 x} (30443+50985 x)}{12100 \sqrt {3+5 x}}-\frac {999 \text {Subst}\left (\int \frac {1}{\sqrt {11-2 x^2}} \, dx,x,\sqrt {3+5 x}\right )}{100 \sqrt {5}}\\ &=\frac {7 (2+3 x)^2}{11 \sqrt {1-2 x} \sqrt {3+5 x}}+\frac {\sqrt {1-2 x} (30443+50985 x)}{12100 \sqrt {3+5 x}}-\frac {999 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{100 \sqrt {10}}\\ \end {align*}

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Mathematica [A]
time = 0.13, size = 74, normalized size = 0.88 \begin {gather*} \frac {612430+824990 x-326700 x^2+120879 \sqrt {10-20 x} \sqrt {3+5 x} \tan ^{-1}\left (\frac {\sqrt {\frac {5}{2}-5 x}}{\sqrt {3+5 x}}\right )}{121000 \sqrt {1-2 x} \sqrt {3+5 x}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(612430 + 824990*x - 326700*x^2 + 120879*Sqrt[10 - 20*x]*Sqrt[3 + 5*x]*ArcTan[Sqrt[5/2 - 5*x]/Sqrt[3 + 5*x]])/
(121000*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])

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Maple [A]
time = 0.09, size = 120, normalized size = 1.43

method	result	size
default	\(-\frac {\sqrt {1-2 x}\, \left (1208790 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right ) x^{2}+120879 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right ) x -653400 x^{2} \sqrt {-10 x^{2}-x +3}-362637 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )+1649980 x \sqrt {-10 x^{2}-x +3}+1224860 \sqrt {-10 x^{2}-x +3}\right )}{242000 \left (-1+2 x \right ) \sqrt {-10 x^{2}-x +3}\, \sqrt {3+5 x}}\)	\(120\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/242000*(1-2*x)^(1/2)*(1208790*10^(1/2)*arcsin(20/11*x+1/11)*x^2+120879*10^(1/2)*arcsin(20/11*x+1/11)*x-6534
00*x^2*(-10*x^2-x+3)^(1/2)-362637*10^(1/2)*arcsin(20/11*x+1/11)+1649980*x*(-10*x^2-x+3)^(1/2)+1224860*(-10*x^2
-x+3)^(1/2))/(-1+2*x)/(-10*x^2-x+3)^(1/2)/(3+5*x)^(1/2)

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Maxima [A]
time = 0.60, size = 58, normalized size = 0.69 \begin {gather*} -\frac {27 \, x^{2}}{10 \, \sqrt {-10 \, x^{2} - x + 3}} + \frac {999}{2000} \, \sqrt {10} \arcsin \left (-\frac {20}{11} \, x - \frac {1}{11}\right ) + \frac {82499 \, x}{12100 \, \sqrt {-10 \, x^{2} - x + 3}} + \frac {61243}{12100 \, \sqrt {-10 \, x^{2} - x + 3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-27/10*x^2/sqrt(-10*x^2 - x + 3) + 999/2000*sqrt(10)*arcsin(-20/11*x - 1/11) + 82499/12100*x/sqrt(-10*x^2 - x
+ 3) + 61243/12100/sqrt(-10*x^2 - x + 3)

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Fricas [A]
time = 0.54, size = 87, normalized size = 1.04 \begin {gather*} \frac {120879 \, \sqrt {10} {\left (10 \, x^{2} + x - 3\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 (32670 \, x^{2} - 82499 \, x - 61243\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{242000 \, {\left (10 \, x^{2} + x - 3\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/242000*(120879*sqrt(10)*(10*x^2 + x - 3)*arctan(1/20*sqrt(10)*(20*x + 1)*sqrt(5*x + 3)*sqrt(-2*x + 1)/(10*x^
2 + x - 3)) + 20*(32670*x^2 - 82499*x - 61243)*sqrt(5*x + 3)*sqrt(-2*x + 1))/(10*x^2 + x - 3)

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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}}{\left (1 - 2 x\right )^{\frac {3}{2}} \left (5 x + 3\right )^{\frac {3}{2}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]
time = 0.95, size = 118, normalized size = 1.40 \begin {gather*} -\frac {999}{1000} \, \sqrt {10} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right ) + \frac {{\left (6534 \, \sqrt {5} {\left (5 \, x + 3\right )} - 121687 \, \sqrt {5}\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5}}{302500 \, {\left (2 \, x - 1\right )}} - \frac {\sqrt {10} {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}{30250 \, \sqrt {5 \, x + 3}} + \frac {2 \, \sqrt {10} \sqrt {5 \, x + 3}}{15125 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-999/1000*sqrt(10)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3)) + 1/302500*(6534*sqrt(5)*(5*x + 3) - 121687*sqrt(5))*sq
rt(5*x + 3)*sqrt(-10*x + 5)/(2*x - 1) - 1/30250*sqrt(10)*(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) +
2/15125*sqrt(10)*sqrt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))

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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}{{\left (1-2\,x\right )}^{3/2}\,{\left (5\,x+3\right )}^{3/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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