3.23.87 \(\int \sqrt {1-2 x} (2+3 x)^3 (3+5 x)^{5/2} \, dx\) [2287]

Optimal. Leaf size=172 \[ \frac {394818523 \sqrt {1-2 x} \sqrt {3+5 x}}{8192000}-\frac {35892593 (1-2 x)^{3/2} \sqrt {3+5 x}}{819200}-\frac {3262963 (1-2 x)^{3/2} (3+5 x)^{3/2}}{307200}-\frac {296633 (1-2 x)^{3/2} (3+5 x)^{5/2}}{128000}-\frac {3}{70} (1-2 x)^{3/2} (2+3 x)^2 (3+5 x)^{7/2}-\frac {3 (1-2 x)^{3/2} (3+5 x)^{7/2} (1963+1140 x)}{8000}+\frac {4343003753 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{8192000 \sqrt {10}} \]

[Out]

-3262963/307200*(1-2*x)^(3/2)*(3+5*x)^(3/2)-296633/128000*(1-2*x)^(3/2)*(3+5*x)^(5/2)-3/70*(1-2*x)^(3/2)*(2+3*
x)^2*(3+5*x)^(7/2)-3/8000*(1-2*x)^(3/2)*(3+5*x)^(7/2)*(1963+1140*x)[phone]/81920000*arcsin(1/11*22^(1/2)*(
3+5*x)^(1/2))*10^(1/2)-35892593/819200*(1-2*x)^(3/2)*(3+5*x)^(1/2)[phone]/8192000*(1-2*x)^(1/2)*(3+5*x)^(1/
2)

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Rubi [A]
time = 0.04, antiderivative size = 172, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 5, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.192, Rules used = {102, 152, 52, 56, 222} \begin {gather*} \frac {4343003753 \text {ArcSin}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{8192000 \sqrt {10}}-\frac {3}{70} (1-2 x)^{3/2} (3 x+2)^2 (5 x+3)^{7/2}-\frac {3 (1-2 x)^{3/2} (1140 x+1963) (5 x+3)^{7/2}}{8000}-\frac {296633 (1-2 x)^{3/2} (5 x+3)^{5/2}}{128000}-\frac {3262963 (1-2 x)^{3/2} (5 x+3)^{3/2}}{307200}-\frac {35892593 (1-2 x)^{3/2} \sqrt {5 x+3}}{819200}+\frac {394818523 \sqrt {1-2 x} \sqrt {5 x+3}}{8192000} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(394818523*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/8192000 - (35892593*(1 - 2*x)^(3/2)*Sqrt[3 + 5*x])/819200 - (3262963*(
1 - 2*x)^(3/2)*(3 + 5*x)^(3/2))/307200 - (296633*(1 - 2*x)^(3/2)*(3 + 5*x)^(5/2))/128000 - (3*(1 - 2*x)^(3/2)*
(2 + 3*x)^2*(3 + 5*x)^(7/2))/70 - (3*(1 - 2*x)^(3/2)*(3 + 5*x)^(7/2)*(1963 + 1140*x))/8000 + (4343003753*ArcSi
n[Sqrt[2/11]*Sqrt[3 + 5*x]])/(8192000*Sqrt[10])

Rule 52

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(a + b*x)^(m + 1)*((c + d*x)^n/(b*(
m + n + 1))), x] + Dist[n*((b*c - a*d)/(b*(m + n + 1))), Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a
, b, c, d}, x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !IntegerQ[n] || (G
tQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 0] && IntLinearQ[a, b, c, d, m, n, x]

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 102

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)/(d*f*(m + n + p + 1))), x] + Dist[1/(d*f*(m + n + p + 1)), I
nt[(a + b*x)^(m - 2)*(c + d*x)^n*(e + f*x)^p*Simp[a^2*d*f*(m + n + p + 1) - b*(b*c*e*(m - 1) + a*(d*e*(n + 1)
+ c*f*(p + 1))) + b*(a*d*f*(2*m + n + p) - b*(d*e*(m + n) + c*f*(m + p)))*x, x], x], x] /; FreeQ[{a, b, c, d,
e, f, n, p}, x] && GtQ[m, 1] && NeQ[m + n + p + 1, 0] && IntegerQ[m]

Rule 152

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_) + (f_.)*(x_))*((g_.) + (h_.)*(x_)), x_Symbol]
:> Simp[(-(a*d*f*h*(n + 2) + b*c*f*h*(m + 2) - b*d*(f*g + e*h)*(m + n + 3) - b*d*f*h*(m + n + 2)*x))*(a + b*x)
^(m + 1)*((c + d*x)^(n + 1)/(b^2*d^2*(m + n + 2)*(m + n + 3))), x] + Dist[(a^2*d^2*f*h*(n + 1)*(n + 2) + a*b*d
*(n + 1)*(2*c*f*h*(m + 1) - d*(f*g + e*h)*(m + n + 3)) + b^2*(c^2*f*h*(m + 1)*(m + 2) - c*d*(f*g + e*h)*(m + 1
)*(m + n + 3) + d^2*e*g*(m + n + 2)*(m + n + 3)))/(b^2*d^2*(m + n + 2)*(m + n + 3)), Int[(a + b*x)^m*(c + d*x)
^n, x], x] /; FreeQ[{a, b, c, d, e, f, g, h, m, n}, x] && NeQ[m + n + 2, 0] && NeQ[m + n + 3, 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 \sqrt {1-2 x} (2+3 x)^3 (3+5 x)^{5/2} \, dx &=-\frac {3}{70} (1-2 x)^{3/2} (2+3 x)^2 (3+5 x)^{7/2}-\frac {1}{70} \int \left (-385-\frac {1197 x}{2}\right ) \sqrt {1-2 x} (2+3 x) (3+5 x)^{5/2} \, dx\\ &=-\frac {3}{70} (1-2 x)^{3/2} (2+3 x)^2 (3+5 x)^{7/2}-\frac {3 (1-2 x)^{3/2} (3+5 x)^{7/2} (1963+1140 x)}{8000}+\frac {296633 \int \sqrt {1-2 x} (3+5 x)^{5/2} \, dx}{16000}\\ &=-\frac {296633 (1-2 x)^{3/2} (3+5 x)^{5/2}}{128000}-\frac {3}{70} (1-2 x)^{3/2} (2+3 x)^2 (3+5 x)^{7/2}-\frac {3 (1-2 x)^{3/2} (3+5 x)^{7/2} (1963+1140 x)}{8000}+\frac {3262963 \int \sqrt {1-2 x} (3+5 x)^{3/2} \, dx}{51200}\\ &=-\frac {3262963 (1-2 x)^{3/2} (3+5 x)^{3/2}}{307200}-\frac {296633 (1-2 x)^{3/2} (3+5 x)^{5/2}}{128000}-\frac {3}{70} (1-2 x)^{3/2} (2+3 x)^2 (3+5 x)^{7/2}-\frac {3 (1-2 x)^{3/2} (3+5 x)^{7/2} (1963+1140 x)}{8000}+\frac {35892593 \int \sqrt {1-2 x} \sqrt {3+5 x} \, dx}{204800}\\ &=-\frac {35892593 (1-2 x)^{3/2} \sqrt {3+5 x}}{819200}-\frac {3262963 (1-2 x)^{3/2} (3+5 x)^{3/2}}{307200}-\frac {296633 (1-2 x)^{3/2} (3+5 x)^{5/2}}{128000}-\frac {3}{70} (1-2 x)^{3/2} (2+3 x)^2 (3+5 x)^{7/2}-\frac {3 (1-2 x)^{3/2} (3+5 x)^{7/2} (1963+1140 x)}{8000}+\frac {394818523 \int \frac {\sqrt {1-2 x}}{\sqrt {3+5 x}} \, dx}{1638400}\\ &=\frac {394818523 \sqrt {1-2 x} \sqrt {3+5 x}}{8192000}-\frac {35892593 (1-2 x)^{3/2} \sqrt {3+5 x}}{819200}-\frac {3262963 (1-2 x)^{3/2} (3+5 x)^{3/2}}{307200}-\frac {296633 (1-2 x)^{3/2} (3+5 x)^{5/2}}{128000}-\frac {3}{70} (1-2 x)^{3/2} (2+3 x)^2 (3+5 x)^{7/2}-\frac {3 (1-2 x)^{3/2} (3+5 x)^{7/2} (1963+1140 x)}{8000}+\frac {4343003753 \int \frac {1}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx}{16384000}\\ &=\frac {394818523 \sqrt {1-2 x} \sqrt {3+5 x}}{8192000}-\frac {35892593 (1-2 x)^{3/2} \sqrt {3+5 x}}{819200}-\frac {3262963 (1-2 x)^{3/2} (3+5 x)^{3/2}}{307200}-\frac {296633 (1-2 x)^{3/2} (3+5 x)^{5/2}}{128000}-\frac {3}{70} (1-2 x)^{3/2} (2+3 x)^2 (3+5 x)^{7/2}-\frac {3 (1-2 x)^{3/2} (3+5 x)^{7/2} (1963+1140 x)}{8000}+\frac {4343003753 \text {Subst}\left (\int \frac {1}{\sqrt {11-2 x^2}} \, dx,x,\sqrt {3+5 x}\right )}{8192000 \sqrt {5}}\\ &=\frac {394818523 \sqrt {1-2 x} \sqrt {3+5 x}}{8192000}-\frac {35892593 (1-2 x)^{3/2} \sqrt {3+5 x}}{819200}-\frac {3262963 (1-2 x)^{3/2} (3+5 x)^{3/2}}{307200}-\frac {296633 (1-2 x)^{3/2} (3+5 x)^{5/2}}{128000}-\frac {3}{70} (1-2 x)^{3/2} (2+3 x)^2 (3+5 x)^{7/2}-\frac {3 (1-2 x)^{3/2} (3+5 x)^{7/2} (1963+1140 x)}{8000}+\frac {4343003753 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{8192000 \sqrt {10}}\\ \end {align*}

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Mathematica [A]
time = 0.25, size = 93, normalized size = 0.54 \begin {gather*} \frac {10 \sqrt {1-2 x} \left (-37594707201-77366257275 x+48496951780 x^2+339459234400 x^3+646978128000 x^4+659577600000 x^5+360115200000 x^6+82944000000 x^7\right )-91203078813 \sqrt {30+50 x} \tan ^{-1}\left (\frac {\sqrt {\frac {5}{2}-5 x}}{\sqrt {3+5 x}}\right )}{1720320000 \sqrt {3+5 x}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(10*Sqrt[1 - 2*x]*(-37594707201 - 77366257275*x + 48496951780*x^2 + 339459234400*x^3 + 646978128000*x^4 + 6595
77600000*x^5 + 360115200000*x^6 + 82944000000*x^7) - 91203078813*Sqrt[30 + 50*x]*ArcTan[Sqrt[5/2 - 5*x]/Sqrt[3
+ 5*x]])/(1720320000*Sqrt[3 + 5*x])

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Maple [A]
time = 0.12, size = 155, normalized size = 0.90

method	result	size
risch	\(-\frac {\left (16588800000 x^{6}[phone] x^{5}[phone] x^{4}[phone] x^{3}[phone] x^{2}-4902803980 x -12531569067\right ) \sqrt {3+5 x}\, \left (-1+2 x \right ) \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{172032000 \sqrt {-\left (3+5 x \right ) \left (-1+2 x \right )}\, \sqrt {1-2 x}}+\frac {4343003753 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right ) \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{163840000 \sqrt {1-2 x}\, \sqrt {3+5 x}}\)	\(118\)
default	\(\frac {\sqrt {3+5 x}\, \sqrt {1-2 x}\, \left (331776000000 \sqrt {-10 x^{2}-x +3}\, x^{6}[phone] x^{5} \sqrt {-10 x^{2}-x +3}[phone] x^{4} \sqrt {-10 x^{2}-x +3}[phone] x^{3} \sqrt {-10 x^{2}-x +3}[phone] x^{2} \sqrt {-10 x^{2}-x +3}[phone] \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )-98056079600 x \sqrt {-10 x^{2}-x +3}-250631381340 \sqrt {-10 x^{2}-x +3}\right )}{3440640000 \sqrt {-10 x^{2}-x +3}}\)	\(155\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3440640000*(3+5*x)^(1/2)*(1-2*x)^(1/2)*(331776000000*(-10*x^2-x+3)^(1/2)*x^6+1241395200000*x^5*(-10*x^2-x+3)
^(1/2)[phone]*x^4*(-10*x^2-x+3)^(1/2)[phone]*x^3*(-10*x^2-x+3)^(1/2)[phone]*x^2*(-10*x^2-x
+3)^(1/2)[phone]*10^(1/2)*arcsin(20/11*x+1/11)-98056079600*x*(-10*x^2-x+3)^(1/2)-250631381340*(-10*x^2-x+
3)^(1/2))/(-10*x^2-x+3)^(1/2)

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Maxima [A]
time = 0.51, size = 121, normalized size = 0.70 \begin {gather*} -\frac {135}{14} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x^{4} - \frac {3933}{112} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x^{3} - \frac {121887}{2240} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x^{2} - \frac {8474351}{179200} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x - \frac {55355473}{2150400} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} + \frac {35892593}{409600} \, \sqrt {-10 \, x^{2} - x + 3} x - \frac {4343003753}{163840000} \, \sqrt {10} \arcsin \left (-\frac {20}{11} \, x - \frac {1}{11}\right ) + \frac {35892593}{8192000} \, \sqrt {-10 \, x^{2} - x + 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="maxima")

[Out]

-135/14*(-10*x^2 - x + 3)^(3/2)*x^4 - 3933/112*(-10*x^2 - x + 3)^(3/2)*x^3 - 121887/2240*(-10*x^2 - x + 3)^(3/
2)*x^2 - 8474351/179200*(-10*x^2 - x + 3)^(3/2)*x - 55355473/2150400*(-10*x^2 - x + 3)^(3/2) + 35892593/409600
*sqrt(-10*x^2 - x + 3)*x - 4343003753/163840000*sqrt(10)*arcsin(-20/11*x - 1/11) + 35892593/8192000*sqrt(-10*x
^2 - x + 3)

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Fricas [A]
time = 0.69, size = 87, normalized size = 0.51 \begin {gather*} \frac {1}{172032000} \, {\left (16588800000 \, x^{6} + 62069760000 \, x^{5} + 94673664000 \, x^{4} + 72591427200 \, x^{3} + 24336990560 \, x^{2} - 4902803980 \, x - 12531569067\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1} - \frac {4343003753}{163840000} \, \sqrt {10} \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 ) \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]

1/172032000*(16588800000*x^6 + 62069760000*x^5 + 94673664000*x^4 + 72591427200*x^3 + 24336990560*x^2 - 4902803
980*x - 12531569067)*sqrt(5*x + 3)*sqrt(-2*x + 1) - 4343003753/163840000*sqrt(10)*arctan(1/20*sqrt(10)*(20*x +
1)*sqrt(5*x + 3)*sqrt(-2*x + 1)/(10*x^2 + x - 3))

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Sympy [A]
time = 110.16, size = 1047, normalized size = 6.09 \begin {gather*} - \frac {41503 \sqrt {2} \left (\begin {cases} \frac {121 \sqrt {5} \left (- \frac {\sqrt {5} \sqrt {1 - 2 x} \sqrt {10 x + 6} \cdot \left (20 x + 1\right )}{121} + \operatorname {asin}{\left (\frac {\sqrt {55} \sqrt {1 - 2 x}}{11} \right )}\right )}{200} & \text {for}\: \sqrt {1 - 2 x} > - \frac {\sqrt {55}}{5} \wedge \sqrt {1 - 2 x} < \frac {\sqrt {55}}{5} \end {cases}\right )}{64} + \frac {91091 \sqrt {2} \left (\begin {cases} \frac {1331 \sqrt {5} \left (- \frac {5 \sqrt {5} \left (1 - 2 x\right )^{\frac {3}{2}} \left (10 x + 6\right )^{\frac {3}{2}}}{7986} - \frac {\sqrt {5} \sqrt {1 - 2 x} \sqrt {10 x + 6} \cdot \left (20 x + 1\right )}{1936} + \frac {\operatorname {asin}{\left (\frac {\sqrt {55} \sqrt {1 - 2 x}}{11} \right )}}{16}\right )}{125} & \text {for}\: \sqrt {1 - 2 x} > - \frac {\sqrt {55}}{5} \wedge \sqrt {1 - 2 x} < \frac {\sqrt {55}}{5} \end {cases}\right )}{64} - \frac {39977 \sqrt {2} \left (\begin {cases} \frac {14641 \sqrt {5} \left (- \frac {5 \sqrt {5} \left (1 - 2 x\right )^{\frac {3}{2}} \left (10 x + 6\right )^{\frac {3}{2}}}{7986} - \frac {\sqrt {5} \sqrt {1 - 2 x} \sqrt {10 x + 6} \cdot \left (20 x + 1\right )}{3872} - \frac {\sqrt {5} \sqrt {1 - 2 x} \sqrt {10 x + 6} \cdot \left (12100 x - 2000 \left (1 - 2 x\right )^{3} + 6600 \left (1 - 2 x\right )^{2} - 4719\right )}{1874048} + \frac {5 \operatorname {asin}{\left (\frac {\sqrt {55} \sqrt {1 - 2 x}}{11} \right )}}{128}\right )}{625} & \text {for}\: \sqrt {1 - 2 x} > - \frac {\sqrt {55}}{5} \wedge \sqrt {1 - 2 x} < \frac {\sqrt {55}}{5} \end {cases}\right )}{32} + \frac {17541 \sqrt {2} \left (\begin {cases} \frac {161051 \sqrt {5} \cdot \left (\frac {5 \sqrt {5} \left (1 - 2 x\right )^{\frac {5}{2}} \left (10 x + 6\right )^{\frac {5}{2}}}{322102} - \frac {5 \sqrt {5} \left (1 - 2 x\right )^{\frac {3}{2}} \left (10 x + 6\right )^{\frac {3}{2}}}{7986} - \frac {\sqrt {5} \sqrt {1 - 2 x} \sqrt {10 x + 6} \cdot \left (20 x + 1\right )}{7744} - \frac {3 \sqrt {5} \sqrt {1 - 2 x} \sqrt {10 x + 6} \cdot \left (12100 x - 2000 \left (1 - 2 x\right )^{3} + 6600 \left (1 - 2 x\right )^{2} - 4719\right )}{3748096} + \frac {7 \operatorname {asin}{\left (\frac {\sqrt {55} \sqrt {1 - 2 x}}{11} \right )}}{256}\right )}{3125} & \text {for}\: \sqrt {1 - 2 x} > - \frac {\sqrt {55}}{5} \wedge \sqrt {1 - 2 x} < \frac {\sqrt {55}}{5} \end {cases}\right )}{32} - \frac {7695 \sqrt {2} \left (\begin {cases} \frac {1771561 \sqrt {5} \cdot \left (\frac {5 \sqrt {5} \left (1 - 2 x\right )^{\frac {5}{2}} \left (10 x + 6\right )^{\frac {5}{2}}}{161051} + \frac {5 \sqrt {5} \left (1 - 2 x\right )^{\frac {3}{2}} \left (10 x + 6\right )^{\frac {3}{2}} \left (20 x + 1\right )^{3}}{170069856} - \frac {5 \sqrt {5} \left (1 - 2 x\right )^{\frac {3}{2}} \left (10 x + 6\right )^{\frac {3}{2}}}{7986} - \frac {\sqrt {5} \sqrt {1 - 2 x} \sqrt {10 x + 6} \cdot \left (20 x + 1\right )}{15488} - \frac {13 \sqrt {5} \sqrt {1 - 2 x} \sqrt {10 x + 6} \cdot \left (12100 x - 2000 \left (1 - 2 x\right )^{3} + 6600 \left (1 - 2 x\right )^{2} - 4719\right )}{14992384} + \frac {21 \operatorname {asin}{\left (\frac {\sqrt {55} \sqrt {1 - 2 x}}{11} \right )}}{1024}\right )}{15625} & \text {for}\: \sqrt {1 - 2 x} > - \frac {\sqrt {55}}{5} \wedge \sqrt {1 - 2 x} < \frac {\sqrt {55}}{5} \end {cases}\right )}{64} + \frac {675 \sqrt {2} \left (\begin {cases} \frac {19487171 \sqrt {5} \left (- \frac {125 \sqrt {5} \left (1 - 2 x\right )^{\frac {7}{2}} \left (10 x + 6\right )^{\frac {7}{2}}}{272820394} + \frac {15 \sqrt {5} \left (1 - 2 x\right )^{\frac {5}{2}} \left (10 x + 6\right )^{\frac {5}{2}}}{322102} + \frac {25 \sqrt {5} \left (1 - 2 x\right )^{\frac {3}{2}} \left (10 x + 6\right )^{\frac {3}{2}} \left (20 x + 1\right )^{3}}{340139712} - \frac {5 \sqrt {5} \left (1 - 2 x\right )^{\frac {3}{2}} \left (10 x + 6\right )^{\frac {3}{2}}}{7986} - \frac {\sqrt {5} \sqrt {1 - 2 x} \sqrt {10 x + 6} \cdot \left (20 x + 1\right )}{30976} - \frac {25 \sqrt {5} \sqrt {1 - 2 x} \sqrt {10 x + 6} \cdot \left (12100 x - 2000 \left (1 - 2 x\right )^{3} + 6600 \left (1 - 2 x\right )^{2} - 4719\right )}{29984768} + \frac {33 \operatorname {asin}{\left (\frac {\sqrt {55} \sqrt {1 - 2 x}}{11} \right )}}{2048}\right )}{78125} & \text {for}\: \sqrt {1 - 2 x} > - \frac {\sqrt {55}}{5} \wedge \sqrt {1 - 2 x} < \frac {\sqrt {55}}{5} \end {cases}\right )}{64} \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]

-41503*sqrt(2)*Piecewise((121*sqrt(5)*(-sqrt(5)*sqrt(1 - 2*x)*sqrt(10*x + 6)*(20*x + 1)/121 + asin(sqrt(55)*sq
rt(1 - 2*x)/11))/200, (sqrt(1 - 2*x) > -sqrt(55)/5) & (sqrt(1 - 2*x) < sqrt(55)/5)))/64 + 91091*sqrt(2)*Piecew
ise((1331*sqrt(5)*(-5*sqrt(5)*(1 - 2*x)**(3/2)*(10*x + 6)**(3/2)/7986 - sqrt(5)*sqrt(1 - 2*x)*sqrt(10*x + 6)*(
20*x + 1)/1936 + asin(sqrt(55)*sqrt(1 - 2*x)/11)/16)/125, (sqrt(1 - 2*x) > -sqrt(55)/5) & (sqrt(1 - 2*x) < sqr
t(55)/5)))/64 - 39977*sqrt(2)*Piecewise((14641*sqrt(5)*(-5*sqrt(5)*(1 - 2*x)**(3/2)*(10*x + 6)**(3/2)/7986 - s
qrt(5)*sqrt(1 - 2*x)*sqrt(10*x + 6)*(20*x + 1)/3872 - sqrt(5)*sqrt(1 - 2*x)*sqrt(10*x + 6)*(12100*x - 2000*(1
- 2*x)**3 + 6600*(1 - 2*x)**2 - 4719)/1874048 + 5*asin(sqrt(55)*sqrt(1 - 2*x)/11)/128)/625, (sqrt(1 - 2*x) > -
sqrt(55)/5) & (sqrt(1 - 2*x) < sqrt(55)/5)))/32 + 17541*sqrt(2)*Piecewise((161051*sqrt(5)*(5*sqrt(5)*(1 - 2*x)
**(5/2)*(10*x + 6)**(5/2)/322102 - 5*sqrt(5)*(1 - 2*x)**(3/2)*(10*x + 6)**(3/2)/7986 - sqrt(5)*sqrt(1 - 2*x)*s
qrt(10*x + 6)*(20*x + 1)/7744 - 3*sqrt(5)*sqrt(1 - 2*x)*sqrt(10*x + 6)*(12100*x - 2000*(1 - 2*x)**3 + 6600*(1
- 2*x)**2 - 4719)/3748096 + 7*asin(sqrt(55)*sqrt(1 - 2*x)/11)/256)/3125, (sqrt(1 - 2*x) > -sqrt(55)/5) & (sqrt
(1 - 2*x) < sqrt(55)/5)))/32 - 7695*sqrt(2)*Piecewise((1771561*sqrt(5)*(5*sqrt(5)*(1 - 2*x)**(5/2)*(10*x + 6)*
*(5/2)/161051 + 5*sqrt(5)*(1 - 2*x)**(3/2)*(10*x + 6)**(3/2)*(20*x + 1)**3/170069856 - 5*sqrt(5)*(1 - 2*x)**(3
/2)*(10*x + 6)**(3/2)/7986 - sqrt(5)*sqrt(1 - 2*x)*sqrt(10*x + 6)*(20*x + 1)/15488 - 13*sqrt(5)*sqrt(1 - 2*x)*
sqrt(10*x + 6)*(12100*x - 2000*(1 - 2*x)**3 + 6600*(1 - 2*x)**2 - 4719)/14992384 + 21*asin(sqrt(55)*sqrt(1 - 2
*x)/11)/1024)/15625, (sqrt(1 - 2*x) > -sqrt(55)/5) & (sqrt(1 - 2*x) < sqrt(55)/5)))/64 + 675*sqrt(2)*Piecewise
((19487171*sqrt(5)*(-125*sqrt(5)*(1 - 2*x)**(7/2)*(10*x + 6)**(7/2)/272820394 + 15*sqrt(5)*(1 - 2*x)**(5/2)*(1
0*x + 6)**(5/2)/322102 + 25*sqrt(5)*(1 - 2*x)**(3/2)*(10*x + 6)**(3/2)*(20*x + 1)**3/340139712 - 5*sqrt(5)*(1
- 2*x)**(3/2)*(10*x + 6)**(3/2)/7986 - sqrt(5)*sqrt(1 - 2*x)*sqrt(10*x + 6)*(20*x + 1)/30976 - 25*sqrt(5)*sqrt
(1 - 2*x)*sqrt(10*x + 6)*(12100*x - 2000*(1 - 2*x)**3 + 6600*(1 - 2*x)**2 - 4719)/29984768 + 33*asin(sqrt(55)*
sqrt(1 - 2*x)/11)/2048)/78125, (sqrt(1 - 2*x) > -sqrt(55)/5) & (sqrt(1 - 2*x) < sqrt(55)/5)))/64

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 446 vs. \(2 (127) = 254\).
time = 1.35, size = 446, normalized size = 2.59 \begin {gather*} \frac {9}{14336000000} \, \sqrt {5} {\left (2 \, {\left (4 \, {\left (8 \, {\left (4 \, {\left (16 \, {\left (20 \, {\left (120 \, x - 443\right )} {\left (5 \, x + 3\right )} + 94933\right )} {\left (5 \, x + 3\right )} - 7838433\right )} {\left (5 \, x + 3\right )} + 98794353\right )} {\left (5 \, x + 3\right )} - 1568443065\right )} {\left (5 \, x + 3\right )} + 8438816295\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5} + 17534989395 \, \sqrt {2} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right )\right )} + \frac {171}{512000000} \, \sqrt {5} {\left (2 \, {\left (4 \, {\left (8 \, {\left (4 \, {\left (16 \, {\left (100 \, x - 311\right )} {\left (5 \, x + 3\right )} + 46071\right )} {\left (5 \, x + 3\right )} - 775911\right )} {\left (5 \, x + 3\right )} + 15385695\right )} {\left (5 \, x + 3\right )} - 99422145\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5} - 220189365 \, \sqrt {2} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right )\right )} + \frac {1353}{64000000} \, \sqrt {5} {\left (2 \, {\left (4 \, {\left (8 \, {\left (12 \, {\left (80 \, x - 203\right )} {\left (5 \, x + 3\right )} + 19073\right )} {\left (5 \, x + 3\right )} - 506185\right )} {\left (5 \, x + 3\right )} + 4031895\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5} + 10392195 \, \sqrt {2} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right )\right )} + \frac {17119}{9600000} \, \sqrt {5} {\left (2 \, {\left (4 \, {\left (8 \, {\left (60 \, x - 119\right )} {\left (5 \, x + 3\right )} + 6163\right )} {\left (5 \, x + 3\right )} - 66189\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5} - 184305 \, \sqrt {2} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right )\right )} + \frac {1353}{20000} \, \sqrt {5} {\left (2 \, {\left (4 \, {\left (40 \, x - 59\right )} {\left (5 \, x + 3\right )} + 1293\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5} + 4785 \, \sqrt {2} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right )\right )} + \frac {513}{500} \, \sqrt {5} {\left (2 \, {\left (20 \, x - 23\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5} - 143 \, \sqrt {2} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right )\right )} + \frac {108}{25} \, \sqrt {5} {\left (11 \, \sqrt {2} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right ) + 2 \, \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5}\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="giac")

[Out]

9/14336000000*sqrt(5)*(2*(4*(8*(4*(16*(20*(120*x - 443)*(5*x + 3) + 94933)*(5*x + 3) - 7838433)*(5*x + 3) + 98
794353)*(5*x + 3) - 1568443065)*(5*x + 3) + 8438816295)*sqrt(5*x + 3)*sqrt(-10*x + 5) + 17534989395*sqrt(2)*ar
csin(1/11*sqrt(22)*sqrt(5*x + 3))) + 171/512000000*sqrt(5)*(2*(4*(8*(4*(16*(100*x - 311)*(5*x + 3) + 46071)*(5
*x + 3) - 775911)*(5*x + 3) + 15385695)*(5*x + 3) - 99422145)*sqrt(5*x + 3)*sqrt(-10*x + 5) - 220189365*sqrt(2
)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3))) + 1353/64000000*sqrt(5)*(2*(4*(8*(12*(80*x - 203)*(5*x + 3) + 19073)*(5
*x + 3) - 506185)*(5*x + 3) + 4031895)*sqrt(5*x + 3)*sqrt(-10*x + 5) + 10392195*sqrt(2)*arcsin(1/11*sqrt(22)*s
qrt(5*x + 3))) + 17119/9600000*sqrt(5)*(2*(4*(8*(60*x - 119)*(5*x + 3) + 6163)*(5*x + 3) - 66189)*sqrt(5*x + 3
)*sqrt(-10*x + 5) - 184305*sqrt(2)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3))) + 1353/20000*sqrt(5)*(2*(4*(40*x - 59)
*(5*x + 3) + 1293)*sqrt(5*x + 3)*sqrt(-10*x + 5) + 4785*sqrt(2)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3))) + 513/500
*sqrt(5)*(2*(20*x - 23)*sqrt(5*x + 3)*sqrt(-10*x + 5) - 143*sqrt(2)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3))) + 108
/25*sqrt(5)*(11*sqrt(2)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3)) + 2*sqrt(5*x + 3)*sqrt(-10*x + 5))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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