3.24.88 \(\int (1-2 x)^{5/2} (2+3 x)^3 \sqrt {3+5 x} \, dx\) [2388]

Optimal. Leaf size=172 \[ \frac {339629939 \sqrt {1-2 x} \sqrt {3+5 x}}{256000000}+\frac {30875449 (1-2 x)^{3/2} \sqrt {3+5 x}}{76800000}+\frac {2806859 (1-2 x)^{5/2} \sqrt {3+5 x}}{19200000}-\frac {255169 (1-2 x)^{7/2} \sqrt {3+5 x}}{640000}-\frac {3}{70} (1-2 x)^{7/2} (2+3 x)^2 (3+5 x)^{3/2}-\frac {3 (1-2 x)^{7/2} (3+5 x)^{3/2} (33857+26700 x)}{280000}+\frac {3735929329 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{256000000 \sqrt {10}} \]

[Out]

-3/70*(1-2*x)^(7/2)*(2+3*x)^2*(3+5*x)^(3/2)-3/280000*(1-2*x)^(7/2)*(3+5*x)^(3/2)*(33857+26700*x)[phone]/25
60000000*arcsin(1/11*22^(1/2)*(3+5*x)^(1/2))*10^(1/2)[phone]/76800000*(1-2*x)^(3/2)*(3+5*x)^(1/2)+2806859/19
200000*(1-2*x)^(5/2)*(3+5*x)^(1/2)-255169/640000*(1-2*x)^(7/2)*(3+5*x)^(1/2)[phone]/256000000*(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 {3735929329 \text {ArcSin}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{256000000 \sqrt {10}}-\frac {3}{70} (3 x+2)^2 (5 x+3)^{3/2} (1-2 x)^{7/2}-\frac {3 (5 x+3)^{3/2} (26700 x+33857) (1-2 x)^{7/2}}{280000}-\frac {255169 \sqrt {5 x+3} (1-2 x)^{7/2}}{640000}+\frac {2806859 \sqrt {5 x+3} (1-2 x)^{5/2}}{19200000}+\frac {30875449 \sqrt {5 x+3} (1-2 x)^{3/2}}{76800000}+\frac {339629939 \sqrt {5 x+3} \sqrt {1-2 x}}{256000000} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(339629939*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/256000000 + (30875449*(1 - 2*x)^(3/2)*Sqrt[3 + 5*x])/76800000 + (28068
59*(1 - 2*x)^(5/2)*Sqrt[3 + 5*x])/19200000 - (255169*(1 - 2*x)^(7/2)*Sqrt[3 + 5*x])/640000 - (3*(1 - 2*x)^(7/2
)*(2 + 3*x)^2*(3 + 5*x)^(3/2))/70 - (3*(1 - 2*x)^(7/2)*(3 + 5*x)^(3/2)*(33857 + 26700*x))/280000 + (3735929329
*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(256000000*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 (1-2 x)^{5/2} (2+3 x)^3 \sqrt {3+5 x} \, dx &=-\frac {3}{70} (1-2 x)^{7/2} (2+3 x)^2 (3+5 x)^{3/2}-\frac {1}{70} \int \left (-253-\frac {801 x}{2}\right ) (1-2 x)^{5/2} (2+3 x) \sqrt {3+5 x} \, dx\\ &=-\frac {3}{70} (1-2 x)^{7/2} (2+3 x)^2 (3+5 x)^{3/2}-\frac {3 (1-2 x)^{7/2} (3+5 x)^{3/2} (33857+26700 x)}{280000}+\frac {255169 \int (1-2 x)^{5/2} \sqrt {3+5 x} \, dx}{80000}\\ &=-\frac {255169 (1-2 x)^{7/2} \sqrt {3+5 x}}{640000}-\frac {3}{70} (1-2 x)^{7/2} (2+3 x)^2 (3+5 x)^{3/2}-\frac {3 (1-2 x)^{7/2} (3+5 x)^{3/2} (33857+26700 x)}{280000}+\frac {2806859 \int \frac {(1-2 x)^{5/2}}{\sqrt {3+5 x}} \, dx}{1280000}\\ &=\frac {2806859 (1-2 x)^{5/2} \sqrt {3+5 x}}{19200000}-\frac {255169 (1-2 x)^{7/2} \sqrt {3+5 x}}{640000}-\frac {3}{70} (1-2 x)^{7/2} (2+3 x)^2 (3+5 x)^{3/2}-\frac {3 (1-2 x)^{7/2} (3+5 x)^{3/2} (33857+26700 x)}{280000}+\frac {30875449 \int \frac {(1-2 x)^{3/2}}{\sqrt {3+5 x}} \, dx}{7680000}\\ &=\frac {30875449 (1-2 x)^{3/2} \sqrt {3+5 x}}{76800000}+\frac {2806859 (1-2 x)^{5/2} \sqrt {3+5 x}}{19200000}-\frac {255169 (1-2 x)^{7/2} \sqrt {3+5 x}}{640000}-\frac {3}{70} (1-2 x)^{7/2} (2+3 x)^2 (3+5 x)^{3/2}-\frac {3 (1-2 x)^{7/2} (3+5 x)^{3/2} (33857+26700 x)}{280000}+\frac {339629939 \int \frac {\sqrt {1-2 x}}{\sqrt {3+5 x}} \, dx}{51200000}\\ &=\frac {339629939 \sqrt {1-2 x} \sqrt {3+5 x}}{256000000}+\frac {30875449 (1-2 x)^{3/2} \sqrt {3+5 x}}{76800000}+\frac {2806859 (1-2 x)^{5/2} \sqrt {3+5 x}}{19200000}-\frac {255169 (1-2 x)^{7/2} \sqrt {3+5 x}}{640000}-\frac {3}{70} (1-2 x)^{7/2} (2+3 x)^2 (3+5 x)^{3/2}-\frac {3 (1-2 x)^{7/2} (3+5 x)^{3/2} (33857+26700 x)}{280000}+\frac {3735929329 \int \frac {1}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx}{512000000}\\ &=\frac {339629939 \sqrt {1-2 x} \sqrt {3+5 x}}{256000000}+\frac {30875449 (1-2 x)^{3/2} \sqrt {3+5 x}}{76800000}+\frac {2806859 (1-2 x)^{5/2} \sqrt {3+5 x}}{19200000}-\frac {255169 (1-2 x)^{7/2} \sqrt {3+5 x}}{640000}-\frac {3}{70} (1-2 x)^{7/2} (2+3 x)^2 (3+5 x)^{3/2}-\frac {3 (1-2 x)^{7/2} (3+5 x)^{3/2} (33857+26700 x)}{280000}+\frac {3735929329 \text {Subst}\left (\int \frac {1}{\sqrt {11-2 x^2}} \, dx,x,\sqrt {3+5 x}\right )}{256000000 \sqrt {5}}\\ &=\frac {339629939 \sqrt {1-2 x} \sqrt {3+5 x}}{256000000}+\frac {30875449 (1-2 x)^{3/2} \sqrt {3+5 x}}{76800000}+\frac {2806859 (1-2 x)^{5/2} \sqrt {3+5 x}}{19200000}-\frac {255169 (1-2 x)^{7/2} \sqrt {3+5 x}}{640000}-\frac {3}{70} (1-2 x)^{7/2} (2+3 x)^2 (3+5 x)^{3/2}-\frac {3 (1-2 x)^{7/2} (3+5 x)^{3/2} (33857+26700 x)}{280000}+\frac {3735929329 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{256000000 \sqrt {10}}\\ \end {align*}

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Mathematica [A]
time = 0.28, size = 93, normalized size = 0.54 \begin {gather*} \frac {10 \sqrt {1-2 x} \left (-2037835593+86060710125 x+178815054340 x^2-205753776800 x^3-582885168000 x^4+30032640000 x^5+736128000000 x^6+414720000000 x^7\right )-78454515909 \sqrt {30+50 x} \tan ^{-1}\left (\frac {\sqrt {\frac {5}{2}-5 x}}{\sqrt {3+5 x}}\right )}{53760000000 \sqrt {3+5 x}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(10*Sqrt[1 - 2*x]*(-2037835593 + 86060710125*x + 178815054340*x^2 - 205753776800*x^3 - 582885168000*x^4 + 3003
2640000*x^5 + 736128000000*x^6 + 414720000000*x^7) - 78454515909*Sqrt[30 + 50*x]*ArcTan[Sqrt[5/2 - 5*x]/Sqrt[3
+ 5*x]])/(53760000000*Sqrt[3 + 5*x])

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Maple [A]
time = 0.14, size = 155, normalized size = 0.90

method	result	size
risch	\(-\frac {\left (82944000000 x^{6}[phone] x^{5}-52468992000 x^{4}-85095638400 x^{3}[phone] x^{2}[phone] x -679278531\right ) \sqrt {3+5 x}\, \left (-1+2 x \right ) \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{5376000000 \sqrt {-\left (3+5 x \right ) \left (-1+2 x \right )}\, \sqrt {1-2 x}}+\frac {3735929329 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right ) \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{5120000000 \sqrt {1-2 x}\, \sqrt {3+5 x}}\)	\(118\)
default	\(\frac {\sqrt {1-2 x}\, \sqrt {3+5 x}\, \left (1658880000000 \sqrt {-10 x^{2}-x +3}\, x^{6}[phone] x^{5} \sqrt {-10 x^{2}-x +3}-1049379840000 x^{4} \sqrt {-10 x^{2}-x +3}-1701912768000 x^{3} \sqrt {-10 x^{2}-x +3}[phone] x^{2} \sqrt {-10 x^{2}-x +3}[phone] \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )[phone] x \sqrt {-10 x^{2}-x +3}-13585570620 \sqrt {-10 x^{2}-x +3}\right )}{107520000000 \sqrt {-10 x^{2}-x +3}}\)	\(155\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/107520000000*(1-2*x)^(1/2)*(3+5*x)^(1/2)*(1658880000000*(-10*x^2-x+3)^(1/2)*x^6+1949184000000*x^5*(-10*x^2-x
+3)^(1/2)-1049379840000*x^4*(-10*x^2-x+3)^(1/2)-1701912768000*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)[phone]*x*(-10*x^2-x+3)^(1/2)-13585570620*(-10*x^2
-x+3)^(1/2))/(-10*x^2-x+3)^(1/2)

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Maxima [A]
time = 0.82, size = 121, normalized size = 0.70 \begin {gather*} -\frac {54}{35} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x^{4} - \frac {1161}{700} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x^{3} + \frac {47529}{70000} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x^{2} + \frac {5697497}{5600000} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x - \frac {5531929}{67200000} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} + \frac {30875449}{12800000} \, \sqrt {-10 \, x^{2} - x + 3} x - \frac {3735929329}{5120000000} \, \sqrt {10} \arcsin \left (-\frac {20}{11} \, x - \frac {1}{11}\right ) + \frac {30875449}{256000000} \, \sqrt {-10 \, x^{2} - x + 3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-54/35*(-10*x^2 - x + 3)^(3/2)*x^4 - 1161/700*(-10*x^2 - x + 3)^(3/2)*x^3 + 47529/70000*(-10*x^2 - x + 3)^(3/2
)*x^2 + 5697497/5600000*(-10*x^2 - x + 3)^(3/2)*x - 5531929/67200000*(-10*x^2 - x + 3)^(3/2) + 30875449/128000
00*sqrt(-10*x^2 - x + 3)*x - 3735929329/5120000000*sqrt(10)*arcsin(-20/11*x - 1/11) + 30875449/256000000*sqrt(
-10*x^2 - x + 3)

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Fricas [A]
time = 0.68, size = 87, normalized size = 0.51 \begin {gather*} \frac {1}{5376000000} \, {\left (82944000000 \, x^{6} + 97459200000 \, x^{5} - 52468992000 \, x^{4} - 85095638400 \, x^{3} + 9906627680 \, x^{2} + 29819034260 \, x - 679278531\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1} - \frac {3735929329}{5120000000} \, \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((1-2*x)^(5/2)*(2+3*x)^3*(3+5*x)^(1/2),x, algorithm="fricas")

[Out]

1/5376000000*(82944000000*x^6 + 97459200000*x^5 - 52468992000*x^4 - 85095638400*x^3 + 9906627680*x^2 + 2981903
4260*x - 679278531)*sqrt(5*x + 3)*sqrt(-2*x + 1) - 3735929329/5120000000*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 = 110.18, size = 1051, normalized size = 6.11 \begin {gather*} \frac {242 \sqrt {5} \left (\begin {cases} \frac {121 \sqrt {2} \left (- \frac {\sqrt {2} \sqrt {5 - 10 x} \left (- 20 x - 1\right ) \sqrt {5 x + 3}}{121} + \operatorname {asin}{\left (\frac {\sqrt {22} \sqrt {5 x + 3}}{11} \right )}\right )}{32} & \text {for}\: \sqrt {5 x + 3} > - \frac {\sqrt {22}}{2} \wedge \sqrt {5 x + 3} < \frac {\sqrt {22}}{2} \end {cases}\right )}{78125} + \frac {418 \sqrt {5} \left (\begin {cases} \frac {1331 \sqrt {2} \left (- \frac {\sqrt {2} \left (5 - 10 x\right )^{\frac {3}{2}} \left (5 x + 3\right )^{\frac {3}{2}}}{3993} - \frac {\sqrt {2} \sqrt {5 - 10 x} \left (- 20 x - 1\right ) \sqrt {5 x + 3}}{1936} + \frac {\operatorname {asin}{\left (\frac {\sqrt {22} \sqrt {5 x + 3}}{11} \right )}}{16}\right )}{8} & \text {for}\: \sqrt {5 x + 3} > - \frac {\sqrt {22}}{2} \wedge \sqrt {5 x + 3} < \frac {\sqrt {22}}{2} \end {cases}\right )}{15625} + \frac {46 \sqrt {5} \left (\begin {cases} \frac {14641 \sqrt {2} \left (- \frac {\sqrt {2} \left (5 - 10 x\right )^{\frac {3}{2}} \left (5 x + 3\right )^{\frac {3}{2}}}{3993} - \frac {\sqrt {2} \sqrt {5 - 10 x} \left (- 20 x - 1\right ) \sqrt {5 x + 3}}{3872} - \frac {\sqrt {2} \sqrt {5 - 10 x} \sqrt {5 x + 3} \left (- 12100 x - 128 \left (5 x + 3\right )^{3} + 1056 \left (5 x + 3\right )^{2} - 5929\right )}{1874048} + \frac {5 \operatorname {asin}{\left (\frac {\sqrt {22} \sqrt {5 x + 3}}{11} \right )}}{128}\right )}{16} & \text {for}\: \sqrt {5 x + 3} > - \frac {\sqrt {22}}{2} \wedge \sqrt {5 x + 3} < \frac {\sqrt {22}}{2} \end {cases}\right )}{625} + \frac {846 \sqrt {5} \left (\begin {cases} \frac {161051 \sqrt {2} \cdot \left (\frac {2 \sqrt {2} \left (5 - 10 x\right )^{\frac {5}{2}} \left (5 x + 3\right )^{\frac {5}{2}}}{805255} - \frac {\sqrt {2} \left (5 - 10 x\right )^{\frac {3}{2}} \left (5 x + 3\right )^{\frac {3}{2}}}{3993} - \frac {\sqrt {2} \sqrt {5 - 10 x} \left (- 20 x - 1\right ) \sqrt {5 x + 3}}{7744} - \frac {3 \sqrt {2} \sqrt {5 - 10 x} \sqrt {5 x + 3} \left (- 12100 x - 128 \left (5 x + 3\right )^{3} + 1056 \left (5 x + 3\right )^{2} - 5929\right )}{3748096} + \frac {7 \operatorname {asin}{\left (\frac {\sqrt {22} \sqrt {5 x + 3}}{11} \right )}}{256}\right )}{32} & \text {for}\: \sqrt {5 x + 3} > - \frac {\sqrt {22}}{2} \wedge \sqrt {5 x + 3} < \frac {\sqrt {22}}{2} \end {cases}\right )}{15625} - \frac {432 \sqrt {5} \left (\begin {cases} \frac {1771561 \sqrt {2} \cdot \left (\frac {4 \sqrt {2} \left (5 - 10 x\right )^{\frac {5}{2}} \left (5 x + 3\right )^{\frac {5}{2}}}{805255} + \frac {\sqrt {2} \left (5 - 10 x\right )^{\frac {3}{2}} \left (- 20 x - 1\right )^{3} \left (5 x + 3\right )^{\frac {3}{2}}}{85034928} - \frac {\sqrt {2} \left (5 - 10 x\right )^{\frac {3}{2}} \left (5 x + 3\right )^{\frac {3}{2}}}{3993} - \frac {\sqrt {2} \sqrt {5 - 10 x} \left (- 20 x - 1\right ) \sqrt {5 x + 3}}{15488} - \frac {13 \sqrt {2} \sqrt {5 - 10 x} \sqrt {5 x + 3} \left (- 12100 x - 128 \left (5 x + 3\right )^{3} + 1056 \left (5 x + 3\right )^{2} - 5929\right )}{14992384} + \frac {21 \operatorname {asin}{\left (\frac {\sqrt {22} \sqrt {5 x + 3}}{11} \right )}}{1024}\right )}{64} & \text {for}\: \sqrt {5 x + 3} > - \frac {\sqrt {22}}{2} \wedge \sqrt {5 x + 3} < \frac {\sqrt {22}}{2} \end {cases}\right )}{15625} + \frac {216 \sqrt {5} \left (\begin {cases} \frac {19487171 \sqrt {2} \left (- \frac {4 \sqrt {2} \left (5 - 10 x\right )^{\frac {7}{2}} \left (5 x + 3\right )^{\frac {7}{2}}}{136410197} + \frac {6 \sqrt {2} \left (5 - 10 x\right )^{\frac {5}{2}} \left (5 x + 3\right )^{\frac {5}{2}}}{805255} + \frac {5 \sqrt {2} \left (5 - 10 x\right )^{\frac {3}{2}} \left (- 20 x - 1\right )^{3} \left (5 x + 3\right )^{\frac {3}{2}}}{170069856} - \frac {\sqrt {2} \left (5 - 10 x\right )^{\frac {3}{2}} \left (5 x + 3\right )^{\frac {3}{2}}}{3993} - \frac {\sqrt {2} \sqrt {5 - 10 x} \left (- 20 x - 1\right ) \sqrt {5 x + 3}}{30976} - \frac {25 \sqrt {2} \sqrt {5 - 10 x} \sqrt {5 x + 3} \left (- 12100 x - 128 \left (5 x + 3\right )^{3} + 1056 \left (5 x + 3\right )^{2} - 5929\right )}{29984768} + \frac {33 \operatorname {asin}{\left (\frac {\sqrt {22} \sqrt {5 x + 3}}{11} \right )}}{2048}\right )}{128} & \text {for}\: \sqrt {5 x + 3} > - \frac {\sqrt {22}}{2} \wedge \sqrt {5 x + 3} < \frac {\sqrt {22}}{2} \end {cases}\right )}{78125} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

242*sqrt(5)*Piecewise((121*sqrt(2)*(-sqrt(2)*sqrt(5 - 10*x)*(-20*x - 1)*sqrt(5*x + 3)/121 + asin(sqrt(22)*sqrt
(5*x + 3)/11))/32, (sqrt(5*x + 3) > -sqrt(22)/2) & (sqrt(5*x + 3) < sqrt(22)/2)))/78125 + 418*sqrt(5)*Piecewis
e((1331*sqrt(2)*(-sqrt(2)*(5 - 10*x)**(3/2)*(5*x + 3)**(3/2)/3993 - sqrt(2)*sqrt(5 - 10*x)*(-20*x - 1)*sqrt(5*
x + 3)/1936 + asin(sqrt(22)*sqrt(5*x + 3)/11)/16)/8, (sqrt(5*x + 3) > -sqrt(22)/2) & (sqrt(5*x + 3) < sqrt(22)
/2)))/15625 + 46*sqrt(5)*Piecewise((14641*sqrt(2)*(-sqrt(2)*(5 - 10*x)**(3/2)*(5*x + 3)**(3/2)/3993 - sqrt(2)*
sqrt(5 - 10*x)*(-20*x - 1)*sqrt(5*x + 3)/3872 - sqrt(2)*sqrt(5 - 10*x)*sqrt(5*x + 3)*(-12100*x - 128*(5*x + 3)
**3 + 1056*(5*x + 3)**2 - 5929)/1874048 + 5*asin(sqrt(22)*sqrt(5*x + 3)/11)/128)/16, (sqrt(5*x + 3) > -sqrt(22
)/2) & (sqrt(5*x + 3) < sqrt(22)/2)))/625 + 846*sqrt(5)*Piecewise((161051*sqrt(2)*(2*sqrt(2)*(5 - 10*x)**(5/2)
*(5*x + 3)**(5/2)/805255 - sqrt(2)*(5 - 10*x)**(3/2)*(5*x + 3)**(3/2)/3993 - sqrt(2)*sqrt(5 - 10*x)*(-20*x - 1
)*sqrt(5*x + 3)/7744 - 3*sqrt(2)*sqrt(5 - 10*x)*sqrt(5*x + 3)*(-12100*x - 128*(5*x + 3)**3 + 1056*(5*x + 3)**2
- 5929)/3748096 + 7*asin(sqrt(22)*sqrt(5*x + 3)/11)/256)/32, (sqrt(5*x + 3) > -sqrt(22)/2) & (sqrt(5*x + 3) <
sqrt(22)/2)))/15625 - 432*sqrt(5)*Piecewise((1771561*sqrt(2)*(4*sqrt(2)*(5 - 10*x)**(5/2)*(5*x + 3)**(5/2)/80
5255 + sqrt(2)*(5 - 10*x)**(3/2)*(-20*x - 1)**3*(5*x + 3)**(3/2)/85034928 - sqrt(2)*(5 - 10*x)**(3/2)*(5*x + 3
)**(3/2)/3993 - sqrt(2)*sqrt(5 - 10*x)*(-20*x - 1)*sqrt(5*x + 3)/15488 - 13*sqrt(2)*sqrt(5 - 10*x)*sqrt(5*x +
3)*(-12100*x - 128*(5*x + 3)**3 + 1056*(5*x + 3)**2 - 5929)/14992384 + 21*asin(sqrt(22)*sqrt(5*x + 3)/11)/1024
)/64, (sqrt(5*x + 3) > -sqrt(22)/2) & (sqrt(5*x + 3) < sqrt(22)/2)))/15625 + 216*sqrt(5)*Piecewise((19487171*s
qrt(2)*(-4*sqrt(2)*(5 - 10*x)**(7/2)*(5*x + 3)**(7/2)/136410197 + 6*sqrt(2)*(5 - 10*x)**(5/2)*(5*x + 3)**(5/2)
/805255 + 5*sqrt(2)*(5 - 10*x)**(3/2)*(-20*x - 1)**3*(5*x + 3)**(3/2)/170069856 - sqrt(2)*(5 - 10*x)**(3/2)*(5
*x + 3)**(3/2)/3993 - sqrt(2)*sqrt(5 - 10*x)*(-20*x - 1)*sqrt(5*x + 3)/30976 - 25*sqrt(2)*sqrt(5 - 10*x)*sqrt(
5*x + 3)*(-12100*x - 128*(5*x + 3)**3 + 1056*(5*x + 3)**2 - 5929)/29984768 + 33*asin(sqrt(22)*sqrt(5*x + 3)/11
)/2048)/128, (sqrt(5*x + 3) > -sqrt(22)/2) & (sqrt(5*x + 3) < sqrt(22)/2)))/78125

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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 = 0.56, size = 446, normalized size = 2.59 \begin {gather*} \frac {9}{89600000000} \, \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 {9}{400000000} \, \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 {33}{320000000} \, \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 {17}{384000} \, \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 {77}{60000} \, \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 {13}{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 {12}{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((1-2*x)^(5/2)*(2+3*x)^3*(3+5*x)^(1/2),x, algorithm="giac")

[Out]

9/89600000000*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))) + 9/400000000*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))) + 33/320000000*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)*sqrt
(5*x + 3))) - 17/384000*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))) - 77/60000*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))) + 13/500*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))) + 12/25*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 {\left (1-2\,x\right )}^{5/2}\,{\left (3\,x+2\right )}^3\,\sqrt {5\,x+3} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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