3.24.93 \(\int \frac {(1-2 x)^{5/2} \sqrt {3+5 x}}{(2+3 x)^2} \, dx\) [2393]

Optimal. Leaf size=135 \[ -\frac {43}{30} \sqrt {1-2 x} \sqrt {3+5 x}-\frac {1}{3} (1-2 x)^{3/2} \sqrt {3+5 x}-\frac {(1-2 x)^{5/2} \sqrt {3+5 x}}{3 (2+3 x)}-\frac {2119 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{90 \sqrt {10}}-\frac {35}{9} \sqrt {7} \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right ) \]

[Out]

-35/9*arctan(1/7*(1-2*x)^(1/2)*7^(1/2)/(3+5*x)^(1/2))*7^(1/2)-2119/900*arcsin(1/11*22^(1/2)*(3+5*x)^(1/2))*10^
(1/2)-1/3*(1-2*x)^(3/2)*(3+5*x)^(1/2)-1/3*(1-2*x)^(5/2)*(3+5*x)^(1/2)/(2+3*x)-43/30*(1-2*x)^(1/2)*(3+5*x)^(1/2
)

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Rubi [A]
time = 0.04, antiderivative size = 135, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.269, Rules used = {99, 159, 163, 56, 222, 95, 210} \begin {gather*} -\frac {2119 \text {ArcSin}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{90 \sqrt {10}}-\frac {35}{9} \sqrt {7} \text {ArcTan}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )-\frac {\sqrt {5 x+3} (1-2 x)^{5/2}}{3 (3 x+2)}-\frac {1}{3} \sqrt {5 x+3} (1-2 x)^{3/2}-\frac {43}{30} \sqrt {5 x+3} \sqrt {1-2 x} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-43*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/30 - ((1 - 2*x)^(3/2)*Sqrt[3 + 5*x])/3 - ((1 - 2*x)^(5/2)*Sqrt[3 + 5*x])/(3*
(2 + 3*x)) - (2119*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(90*Sqrt[10]) - (35*Sqrt[7]*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]
*Sqrt[3 + 5*x])])/9

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 95

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 99

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/(b*(m + 1))), x] - Dist[1/(b*(m + 1)), Int[(a + b*x)^(m + 1)*(c + d*x)^(n
- 1)*(e + f*x)^(p - 1)*Simp[d*e*n + c*f*p + d*f*(n + p)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && LtQ[m
, -1] && GtQ[n, 0] && GtQ[p, 0] && (IntegersQ[2*m, 2*n, 2*p] || IntegersQ[m, n + p] || IntegersQ[p, m + n])

Rule 159

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

Rule 163

Int[(((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)))/((a_.) + (b_.)*(x_)), x_Symbol]
:> Dist[h/b, Int[(c + d*x)^n*(e + f*x)^p, x], x] + Dist[(b*g - a*h)/b, Int[(c + d*x)^n*((e + f*x)^p/(a + b*x)
), x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x]

Rule 210

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^(-1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])
], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

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 {(1-2 x)^{5/2} \sqrt {3+5 x}}{(2+3 x)^2} \, dx &=-\frac {(1-2 x)^{5/2} \sqrt {3+5 x}}{3 (2+3 x)}+\frac {1}{3} \int \frac {\left (-\frac {25}{2}-30 x\right ) (1-2 x)^{3/2}}{(2+3 x) \sqrt {3+5 x}} \, dx\\ &=-\frac {1}{3} (1-2 x)^{3/2} \sqrt {3+5 x}-\frac {(1-2 x)^{5/2} \sqrt {3+5 x}}{3 (2+3 x)}+\frac {1}{90} \int \frac {(-765-1935 x) \sqrt {1-2 x}}{(2+3 x) \sqrt {3+5 x}} \, dx\\ &=-\frac {43}{30} \sqrt {1-2 x} \sqrt {3+5 x}-\frac {1}{3} (1-2 x)^{3/2} \sqrt {3+5 x}-\frac {(1-2 x)^{5/2} \sqrt {3+5 x}}{3 (2+3 x)}+\frac {\int \frac {-13410-\frac {95355 x}{2}}{\sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}} \, dx}{1350}\\ &=-\frac {43}{30} \sqrt {1-2 x} \sqrt {3+5 x}-\frac {1}{3} (1-2 x)^{3/2} \sqrt {3+5 x}-\frac {(1-2 x)^{5/2} \sqrt {3+5 x}}{3 (2+3 x)}-\frac {2119}{180} \int \frac {1}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx+\frac {245}{18} \int \frac {1}{\sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}} \, dx\\ &=-\frac {43}{30} \sqrt {1-2 x} \sqrt {3+5 x}-\frac {1}{3} (1-2 x)^{3/2} \sqrt {3+5 x}-\frac {(1-2 x)^{5/2} \sqrt {3+5 x}}{3 (2+3 x)}+\frac {245}{9} \text {Subst}\left (\int \frac {1}{-7-x^2} \, dx,x,\frac {\sqrt {1-2 x}}{\sqrt {3+5 x}}\right )-\frac {2119 \text {Subst}\left (\int \frac {1}{\sqrt {11-2 x^2}} \, dx,x,\sqrt {3+5 x}\right )}{90 \sqrt {5}}\\ &=-\frac {43}{30} \sqrt {1-2 x} \sqrt {3+5 x}-\frac {1}{3} (1-2 x)^{3/2} \sqrt {3+5 x}-\frac {(1-2 x)^{5/2} \sqrt {3+5 x}}{3 (2+3 x)}-\frac {2119 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{90 \sqrt {10}}-\frac {35}{9} \sqrt {7} \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )\\ \end {align*}

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Mathematica [A]
time = 0.28, size = 108, normalized size = 0.80 \begin {gather*} \frac {1}{900} \left (-\frac {30 \sqrt {1-2 x} \left (348+817 x+335 x^2-100 x^3\right )}{(2+3 x) \sqrt {3+5 x}}+2119 \sqrt {10} \tan ^{-1}\left (\frac {\sqrt {\frac {5}{2}-5 x}}{\sqrt {3+5 x}}\right )-3500 \sqrt {7} \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((-30*Sqrt[1 - 2*x]*(348 + 817*x + 335*x^2 - 100*x^3))/((2 + 3*x)*Sqrt[3 + 5*x]) + 2119*Sqrt[10]*ArcTan[Sqrt[5
/2 - 5*x]/Sqrt[3 + 5*x]] - 3500*Sqrt[7]*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/900

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Maple [A]
time = 0.11, size = 163, normalized size = 1.21

method	result	size
risch	\(-\frac {\sqrt {3+5 x}\, \left (-1+2 x \right ) \left (20 x^{2}-79 x -116\right ) \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{30 \left (2+3 x \right ) \sqrt {-\left (3+5 x \right ) \left (-1+2 x \right )}\, \sqrt {1-2 x}}-\frac {\left (\frac {2119 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )}{1800}-\frac {35 \sqrt {7}\, \arctan \left (\frac {9 \left (\frac {20}{3}+\frac {37 x}{3}\right ) \sqrt {7}}{14 \sqrt {-90 \left (\frac {2}{3}+x \right )^{2}+67+111 x}}\right )}{18}\right ) \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{\sqrt {1-2 x}\, \sqrt {3+5 x}}\)	\(138\)
default	\(-\frac {\sqrt {1-2 x}\, \sqrt {3+5 x}\, \left (6357 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right ) x -10500 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x -1200 x^{2} \sqrt {-10 x^{2}-x +3}+4238 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )-7000 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+4740 x \sqrt {-10 x^{2}-x +3}+6960 \sqrt {-10 x^{2}-x +3}\right )}{1800 \sqrt {-10 x^{2}-x +3}\, \left (2+3 x \right )}\)	\(163\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/1800*(1-2*x)^(1/2)*(3+5*x)^(1/2)*(6357*10^(1/2)*arcsin(20/11*x+1/11)*x-10500*7^(1/2)*arctan(1/14*(37*x+20)*
7^(1/2)/(-10*x^2-x+3)^(1/2))*x-1200*x^2*(-10*x^2-x+3)^(1/2)+4238*10^(1/2)*arcsin(20/11*x+1/11)-7000*7^(1/2)*ar
ctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))+4740*x*(-10*x^2-x+3)^(1/2)+6960*(-10*x^2-x+3)^(1/2))/(-10*x^2
-x+3)^(1/2)/(2+3*x)

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Maxima [A]
time = 0.52, size = 90, normalized size = 0.67 \begin {gather*} \frac {2}{9} \, \sqrt {-10 \, x^{2} - x + 3} x - \frac {2119}{1800} \, \sqrt {10} \arcsin \left (\frac {20}{11} \, x + \frac {1}{11}\right ) + \frac {35}{18} \, \sqrt {7} \arcsin \left (\frac {37 \, x}{11 \, {\left | 3 \, x + 2 \right |}} + \frac {20}{11 \, {\left | 3 \, x + 2 \right |}}\right ) - \frac {277}{270} \, \sqrt {-10 \, x^{2} - x + 3} - \frac {49 \, \sqrt {-10 \, x^{2} - x + 3}}{27 \, {\left (3 \, x + 2\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/9*sqrt(-10*x^2 - x + 3)*x - 2119/1800*sqrt(10)*arcsin(20/11*x + 1/11) + 35/18*sqrt(7)*arcsin(37/11*x/abs(3*x
+ 2) + 20/11/abs(3*x + 2)) - 277/270*sqrt(-10*x^2 - x + 3) - 49/27*sqrt(-10*x^2 - x + 3)/(3*x + 2)

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Fricas [A]
time = 0.54, size = 126, normalized size = 0.93 \begin {gather*} -\frac {3500 \, \sqrt {7} {\left (3 \, 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 ) - 2119 \, \sqrt {10} {\left (3 \, x + 2\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 ) - 60 \, {\left (20 \, x^{2} - 79 \, x - 116\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{1800 \, {\left (3 \, x + 2\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/1800*(3500*sqrt(7)*(3*x + 2)*arctan(1/14*sqrt(7)*(37*x + 20)*sqrt(5*x + 3)*sqrt(-2*x + 1)/(10*x^2 + x - 3))
- 2119*sqrt(10)*(3*x + 2)*arctan(1/20*sqrt(10)*(20*x + 1)*sqrt(5*x + 3)*sqrt(-2*x + 1)/(10*x^2 + x - 3)) - 60
*(20*x^2 - 79*x - 116)*sqrt(5*x + 3)*sqrt(-2*x + 1))/(3*x + 2)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 292 vs. \(2 (99) = 198\).
time = 0.64, size = 292, normalized size = 2.16 \begin {gather*} \frac {7}{36} \, \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 {1}{1350} \, {\left (12 \, \sqrt {5} {\left (5 \, x + 3\right )} - 313 \, \sqrt {5}\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5} - \frac {2119}{1800} \, \sqrt {10} {\left (\pi + 2 \, \arctan \left (-\frac {\sqrt {5 \, x + 3} {\left (\frac {{\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{4 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}\right )\right )} - \frac {1078 \, \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 )}}{27 \, {\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 )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

7/36*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)))) + 1/1350*(12*sqrt(5)*(5*x + 3) - 313*sqrt(5))*sqrt(5*x + 3
)*sqrt(-10*x + 5) - 2119/1800*sqrt(10)*(pi + 2*arctan(-1/4*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)))) - 1078/27*sqrt(10)*((sqrt(2)*sqrt(-10*x + 5) - sqrt(2
2))/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)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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