3.23.59 \(\int \frac {(1-2 x)^{5/2} \sqrt {3+5 x}}{(2+3 x)^7} \, dx\)

Optimal. Leaf size=209 \[ -\frac {\sqrt {5 x+3} (1-2 x)^{5/2}}{18 (3 x+2)^6}+\frac {\sqrt {5 x+3} (1-2 x)^{3/2}}{12 (3 x+2)^5}+\frac {2770202075 \sqrt {5 x+3} \sqrt {1-2 x}}{14224896 (3 x+2)}+\frac {26486645 \sqrt {5 x+3} \sqrt {1-2 x}}{1016064 (3 x+2)^2}+\frac {151621 \sqrt {5 x+3} \sqrt {1-2 x}}{36288 (3 x+2)^3}+\frac {647 \sqrt {5 x+3} \sqrt {1-2 x}}{864 (3 x+2)^4}-\frac {391280725 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{175616 \sqrt {7}} \]

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Rubi [A]  time = 0.08, antiderivative size = 209, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 6, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {97, 149, 151, 12, 93, 204} \begin {gather*} -\frac {\sqrt {5 x+3} (1-2 x)^{5/2}}{18 (3 x+2)^6}+\frac {\sqrt {5 x+3} (1-2 x)^{3/2}}{12 (3 x+2)^5}+\frac {2770202075 \sqrt {5 x+3} \sqrt {1-2 x}}{14224896 (3 x+2)}+\frac {26486645 \sqrt {5 x+3} \sqrt {1-2 x}}{1016064 (3 x+2)^2}+\frac {151621 \sqrt {5 x+3} \sqrt {1-2 x}}{36288 (3 x+2)^3}+\frac {647 \sqrt {5 x+3} \sqrt {1-2 x}}{864 (3 x+2)^4}-\frac {391280725 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{175616 \sqrt {7}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((1 - 2*x)^(5/2)*Sqrt[3 + 5*x])/(2 + 3*x)^7,x]

[Out]

-((1 - 2*x)^(5/2)*Sqrt[3 + 5*x])/(18*(2 + 3*x)^6) + ((1 - 2*x)^(3/2)*Sqrt[3 + 5*x])/(12*(2 + 3*x)^5) + (647*Sq
rt[1 - 2*x]*Sqrt[3 + 5*x])/(864*(2 + 3*x)^4) + (151621*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(36288*(2 + 3*x)^3) + (264
86645*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(1016064*(2 + 3*x)^2) + (2770202075*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(14224896*
(2 + 3*x)) - (391280725*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(175616*Sqrt[7])

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 93

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 97

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 149

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

Rule 151

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)), x_Symb
ol] :> Simp[((b*g - a*h)*(a + b*x)^(m + 1)*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/((m + 1)*(b*c - a*d)*(b*e - a*
f)), x] + Dist[1/((m + 1)*(b*c - a*d)*(b*e - a*f)), Int[(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*x)^p*Simp[(a*d*f*
g - b*(d*e + c*f)*g + b*c*e*h)*(m + 1) - (b*g - a*h)*(d*e*(n + 1) + c*f*(p + 1)) - d*f*(b*g - a*h)*(m + n + p
+ 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x] && LtQ[m, -1] && IntegerQ[m]

Rule 204

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

Rubi steps

\begin {align*} \int \frac {(1-2 x)^{5/2} \sqrt {3+5 x}}{(2+3 x)^7} \, dx &=-\frac {(1-2 x)^{5/2} \sqrt {3+5 x}}{18 (2+3 x)^6}+\frac {1}{18} \int \frac {\left (-\frac {25}{2}-30 x\right ) (1-2 x)^{3/2}}{(2+3 x)^6 \sqrt {3+5 x}} \, dx\\ &=-\frac {(1-2 x)^{5/2} \sqrt {3+5 x}}{18 (2+3 x)^6}+\frac {(1-2 x)^{3/2} \sqrt {3+5 x}}{12 (2+3 x)^5}-\frac {1}{270} \int \frac {\sqrt {1-2 x} \left (-\frac {2235}{4}+375 x\right )}{(2+3 x)^5 \sqrt {3+5 x}} \, dx\\ &=-\frac {(1-2 x)^{5/2} \sqrt {3+5 x}}{18 (2+3 x)^6}+\frac {(1-2 x)^{3/2} \sqrt {3+5 x}}{12 (2+3 x)^5}+\frac {647 \sqrt {1-2 x} \sqrt {3+5 x}}{864 (2+3 x)^4}+\frac {\int \frac {\frac {385905}{8}-\frac {139575 x}{2}}{\sqrt {1-2 x} (2+3 x)^4 \sqrt {3+5 x}} \, dx}{3240}\\ &=-\frac {(1-2 x)^{5/2} \sqrt {3+5 x}}{18 (2+3 x)^6}+\frac {(1-2 x)^{3/2} \sqrt {3+5 x}}{12 (2+3 x)^5}+\frac {647 \sqrt {1-2 x} \sqrt {3+5 x}}{864 (2+3 x)^4}+\frac {151621 \sqrt {1-2 x} \sqrt {3+5 x}}{36288 (2+3 x)^3}+\frac {\int \frac {\frac {71784825}{16}-\frac {11371575 x}{2}}{\sqrt {1-2 x} (2+3 x)^3 \sqrt {3+5 x}} \, dx}{68040}\\ &=-\frac {(1-2 x)^{5/2} \sqrt {3+5 x}}{18 (2+3 x)^6}+\frac {(1-2 x)^{3/2} \sqrt {3+5 x}}{12 (2+3 x)^5}+\frac {647 \sqrt {1-2 x} \sqrt {3+5 x}}{864 (2+3 x)^4}+\frac {151621 \sqrt {1-2 x} \sqrt {3+5 x}}{36288 (2+3 x)^3}+\frac {26486645 \sqrt {1-2 x} \sqrt {3+5 x}}{1016064 (2+3 x)^2}+\frac {\int \frac {\frac {8553681375}{32}-\frac {1986498375 x}{8}}{\sqrt {1-2 x} (2+3 x)^2 \sqrt {3+5 x}} \, dx}{952560}\\ &=-\frac {(1-2 x)^{5/2} \sqrt {3+5 x}}{18 (2+3 x)^6}+\frac {(1-2 x)^{3/2} \sqrt {3+5 x}}{12 (2+3 x)^5}+\frac {647 \sqrt {1-2 x} \sqrt {3+5 x}}{864 (2+3 x)^4}+\frac {151621 \sqrt {1-2 x} \sqrt {3+5 x}}{36288 (2+3 x)^3}+\frac {26486645 \sqrt {1-2 x} \sqrt {3+5 x}}{1016064 (2+3 x)^2}+\frac {2770202075 \sqrt {1-2 x} \sqrt {3+5 x}}{14224896 (2+3 x)}+\frac {\int \frac {475406080875}{64 \sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}} \, dx}{6667920}\\ &=-\frac {(1-2 x)^{5/2} \sqrt {3+5 x}}{18 (2+3 x)^6}+\frac {(1-2 x)^{3/2} \sqrt {3+5 x}}{12 (2+3 x)^5}+\frac {647 \sqrt {1-2 x} \sqrt {3+5 x}}{864 (2+3 x)^4}+\frac {151621 \sqrt {1-2 x} \sqrt {3+5 x}}{36288 (2+3 x)^3}+\frac {26486645 \sqrt {1-2 x} \sqrt {3+5 x}}{1016064 (2+3 x)^2}+\frac {2770202075 \sqrt {1-2 x} \sqrt {3+5 x}}{14224896 (2+3 x)}+\frac {391280725 \int \frac {1}{\sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}} \, dx}{351232}\\ &=-\frac {(1-2 x)^{5/2} \sqrt {3+5 x}}{18 (2+3 x)^6}+\frac {(1-2 x)^{3/2} \sqrt {3+5 x}}{12 (2+3 x)^5}+\frac {647 \sqrt {1-2 x} \sqrt {3+5 x}}{864 (2+3 x)^4}+\frac {151621 \sqrt {1-2 x} \sqrt {3+5 x}}{36288 (2+3 x)^3}+\frac {26486645 \sqrt {1-2 x} \sqrt {3+5 x}}{1016064 (2+3 x)^2}+\frac {2770202075 \sqrt {1-2 x} \sqrt {3+5 x}}{14224896 (2+3 x)}+\frac {391280725 \operatorname {Subst}\left (\int \frac {1}{-7-x^2} \, dx,x,\frac {\sqrt {1-2 x}}{\sqrt {3+5 x}}\right )}{175616}\\ &=-\frac {(1-2 x)^{5/2} \sqrt {3+5 x}}{18 (2+3 x)^6}+\frac {(1-2 x)^{3/2} \sqrt {3+5 x}}{12 (2+3 x)^5}+\frac {647 \sqrt {1-2 x} \sqrt {3+5 x}}{864 (2+3 x)^4}+\frac {151621 \sqrt {1-2 x} \sqrt {3+5 x}}{36288 (2+3 x)^3}+\frac {26486645 \sqrt {1-2 x} \sqrt {3+5 x}}{1016064 (2+3 x)^2}+\frac {2770202075 \sqrt {1-2 x} \sqrt {3+5 x}}{14224896 (2+3 x)}-\frac {391280725 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{175616 \sqrt {7}}\\ \end {align*}

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Mathematica [A]  time = 0.15, size = 191, normalized size = 0.91 \begin {gather*} \frac {1}{392} \left (\frac {130 (5 x+3)^{3/2} (1-2 x)^{7/2}}{(3 x+2)^5}+\frac {28 (5 x+3)^{3/2} (1-2 x)^{7/2}}{(3 x+2)^6}+\frac {5345 \left (2352 (5 x+3)^{3/2} (1-2 x)^{5/2}+55 (3 x+2) \left (392 (1-2 x)^{3/2} (5 x+3)^{3/2}+33 (3 x+2) \left (7 \sqrt {1-2 x} \sqrt {5 x+3} (37 x+20)-121 \sqrt {7} (3 x+2)^2 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )\right )\right )\right )}{9408 (3 x+2)^4}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((1 - 2*x)^(5/2)*Sqrt[3 + 5*x])/(2 + 3*x)^7,x]

[Out]

((28*(1 - 2*x)^(7/2)*(3 + 5*x)^(3/2))/(2 + 3*x)^6 + (130*(1 - 2*x)^(7/2)*(3 + 5*x)^(3/2))/(2 + 3*x)^5 + (5345*
(2352*(1 - 2*x)^(5/2)*(3 + 5*x)^(3/2) + 55*(2 + 3*x)*(392*(1 - 2*x)^(3/2)*(3 + 5*x)^(3/2) + 33*(2 + 3*x)*(7*Sq
rt[1 - 2*x]*Sqrt[3 + 5*x]*(20 + 37*x) - 121*Sqrt[7]*(2 + 3*x)^2*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])
)))/(9408*(2 + 3*x)^4))/392

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IntegrateAlgebraic [A]  time = 0.48, size = 154, normalized size = 0.74 \begin {gather*} -\frac {14641 \sqrt {1-2 x} \left (\frac {80175 (1-2 x)^5}{(5 x+3)^5}-\frac {5600525 (1-2 x)^4}{(5 x+3)^4}-\frac {62995674 (1-2 x)^3}{(5 x+3)^3}-\frac {363000330 (1-2 x)^2}{(5 x+3)^2}-\frac {1090834325 (1-2 x)}{5 x+3}-1347501225\right )}{526848 \sqrt {5 x+3} \left (\frac {1-2 x}{5 x+3}+7\right )^6}-\frac {391280725 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{175616 \sqrt {7}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[((1 - 2*x)^(5/2)*Sqrt[3 + 5*x])/(2 + 3*x)^7,x]

[Out]

(-14641*Sqrt[1 - 2*x]*(-1347501225 + (80175*(1 - 2*x)^5)/(3 + 5*x)^5 - (5600525*(1 - 2*x)^4)/(3 + 5*x)^4 - (62
995674*(1 - 2*x)^3)/(3 + 5*x)^3 - (363000330*(1 - 2*x)^2)/(3 + 5*x)^2 - (1090834325*(1 - 2*x))/(3 + 5*x)))/(52
6848*Sqrt[3 + 5*x]*(7 + (1 - 2*x)/(3 + 5*x))^6) - (391280725*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(1
75616*Sqrt[7])

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fricas [A]  time = 1.65, size = 146, normalized size = 0.70 \begin {gather*} -\frac {1173842175 \, \sqrt {7} {\left (729 \, x^{6} + 2916 \, x^{5} + 4860 \, x^{4} + 4320 \, x^{3} + 2160 \, x^{2} + 576 \, x + 64\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 ) - 14 \, {\left (24931818675 \, x^{5} + 84218501340 \, x^{4} + 113834022672 \, x^{3} + 76960600672 \, x^{2} + 26026519504 \, x + 3522190656\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{7375872 \, {\left (729 \, x^{6} + 2916 \, x^{5} + 4860 \, x^{4} + 4320 \, x^{3} + 2160 \, x^{2} + 576 \, x + 64\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)^7,x, algorithm="fricas")

[Out]

-1/7375872*(1173842175*sqrt(7)*(729*x^6 + 2916*x^5 + 4860*x^4 + 4320*x^3 + 2160*x^2 + 576*x + 64)*arctan(1/14*
sqrt(7)*(37*x + 20)*sqrt(5*x + 3)*sqrt(-2*x + 1)/(10*x^2 + x - 3)) - 14*(24931818675*x^5 + 84218501340*x^4 + 1
13834022672*x^3 + 76960600672*x^2 + 26026519504*x + 3522190656)*sqrt(5*x + 3)*sqrt(-2*x + 1))/(729*x^6 + 2916*
x^5 + 4860*x^4 + 4320*x^3 + 2160*x^2 + 576*x + 64)

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giac [B]  time = 4.14, size = 484, normalized size = 2.32 \begin {gather*} \frac {78256145}{4917248} \, \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 {366025 \, \sqrt {10} {\left (3207 \, {\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 )}^{11} - 8960840 \, {\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 )}^{9} - 4031723136 \, {\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 )}^{7} - 929280844800 \, {\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 )}^{5} - 111701434880000 \, {\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 )}^{3} - \frac {5519365017600000 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}{\sqrt {5 \, x + 3}} + \frac {22077460070400000 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}}{263424 \, {\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 )}^{6}} \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)^7,x, algorithm="giac")

[Out]

78256145/4917248*sqrt(70)*sqrt(10)*(pi + 2*arctan(-1/140*sqrt(70)*sqrt(5*x + 3)*((sqrt(2)*sqrt(-10*x + 5) - sq
rt(22))^2/(5*x + 3) - 4)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22)))) - 366025/263424*sqrt(10)*(3207*((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)))^11 - 8960840*((sq
rt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) - 4*sqrt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22)))^9 - 40
31723136*((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)))^7 - 929280844800*((sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) - 4*sqrt(5*x + 3)/(sqrt(2)*sqrt(-1
0*x + 5) - sqrt(22)))^5 - 111701434880000*((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)))^3 - 5519365017600000*(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3
) + 22077460070400000*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)^6

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maple [B]  time = 0.01, size = 346, normalized size = 1.66 \begin {gather*} \frac {\sqrt {-2 x +1}\, \sqrt {5 x +3}\, \left (855730945575 \sqrt {7}\, x^{6} \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )[phone] \sqrt {7}\, x^{5} \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )[phone] \sqrt {-10 x^{2}-x +3}\, x^{5}[phone] \sqrt {7}\, x^{4} \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )[phone] \sqrt {-10 x^{2}-x +3}\, x^{4}[phone] \sqrt {7}\, x^{3} \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )[phone] \sqrt {-10 x^{2}-x +3}\, x^{3}[phone] \sqrt {7}\, x^{2} \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )[phone] \sqrt {-10 x^{2}-x +3}\, x^{2}[phone] \sqrt {7}\, x \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )[phone] \sqrt {-10 x^{2}-x +3}\, x [phone] \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )[phone] \sqrt {-10 x^{2}-x +3}\right )}{7375872 \sqrt {-10 x^{2}-x +3}\, \left (3 x +2\right )^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-2*x+1)^(5/2)*(5*x+3)^(1/2)/(3*x+2)^7,x)

[Out]

1/7375872*(-2*x+1)^(1/2)*(5*x+3)^(1/2)*(855730945575*7^(1/2)*x^6*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(
1/2))[phone]*7^(1/2)*x^5*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))[phone]*7^(1/2)*x^4*a
rctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))[phone]*(-10*x^2-x+3)^(1/2)*x^5+5070998196000*7^(1/2)*x
^3*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))[phone]*(-10*x^2-x+3)^(1/2)*x^4+2535499098000*7^(1
/2)*x^2*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))[phone]*(-10*x^2-x+3)^(1/2)*x^3+676133092800*
7^(1/2)*x*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))[phone]*(-10*x^2-x+3)^(1/2)*x^2+75125899200
*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))[phone]*(-10*x^2-x+3)^(1/2)*x+49310669184*(-1
0*x^2-x+3)^(1/2))/(-10*x^2-x+3)^(1/2)/(3*x+2)^6

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maxima [A]  time = 1.28, size = 244, normalized size = 1.17 \begin {gather*} \frac {391280725}{2458624} \, \sqrt {7} \arcsin \left (\frac {37 \, x}{11 \, {\left | 3 \, x + 2 \right |}} + \frac {20}{11 \, {\left | 3 \, x + 2 \right |}}\right ) + \frac {16168625}{131712} \, \sqrt {-10 \, x^{2} - x + 3} + \frac {7 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}}{18 \, {\left (729 \, x^{6} + 2916 \, x^{5} + 4860 \, x^{4} + 4320 \, x^{3} + 2160 \, x^{2} + 576 \, x + 64\right )}} + \frac {19 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}}{12 \, {\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )}} + \frac {4673 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}}{672 \, {\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )}} + \frac {821945 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}}{28224 \, {\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}} + \frac {9701175 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}}{87808 \, {\left (9 \, x^{2} + 12 \, x + 4\right )}} - \frac {119647825 \, \sqrt {-10 \, x^{2} - x + 3}}{526848 \, {\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)^7,x, algorithm="maxima")

[Out]

391280725/2458624*sqrt(7)*arcsin(37/11*x/abs(3*x + 2) + 20/11/abs(3*x + 2)) + 16168625/131712*sqrt(-10*x^2 - x
+ 3) + 7/18*(-10*x^2 - x + 3)^(3/2)/(729*x^6 + 2916*x^5 + 4860*x^4 + 4320*x^3 + 2160*x^2 + 576*x + 64) + 19/1
2*(-10*x^2 - x + 3)^(3/2)/(243*x^5 + 810*x^4 + 1080*x^3 + 720*x^2 + 240*x + 32) + 4673/672*(-10*x^2 - x + 3)^(
3/2)/(81*x^4 + 216*x^3 + 216*x^2 + 96*x + 16) + 821945/28224*(-10*x^2 - x + 3)^(3/2)/(27*x^3 + 54*x^2 + 36*x +
8) + 9701175/87808*(-10*x^2 - x + 3)^(3/2)/(9*x^2 + 12*x + 4) - 119647825/526848*sqrt(-10*x^2 - x + 3)/(3*x +
2)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\left (1-2\,x\right )}^{5/2}\,\sqrt {5\,x+3}}{{\left (3\,x+2\right )}^7} \,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)^7,x)

[Out]

int(((1 - 2*x)^(5/2)*(5*x + 3)^(1/2))/(3*x + 2)^7, x)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \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)**7,x)

[Out]

Timed out

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