3.2398 \(\int \frac{(1-2 x)^{5/2} \sqrt{3+5 x}}{(2+3 x)^7} \, dx\)

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

[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])

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Rubi [A]  time = 0.0793283, antiderivative size = 209, normalized size of antiderivative = 1., 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} \[ -\frac{\sqrt{5 x+3} (1-2 x)^{5/2}}{18 (3 x+2)^6}+\frac{\sqrt{5 x+3} (1-2 x)^{3/2}}{12 (3 x+2)^5}+\frac{2770202075 \sqrt{5 x+3} \sqrt{1-2 x}}{14224896 (3 x+2)}+\frac{26486645 \sqrt{5 x+3} \sqrt{1-2 x}}{1016064 (3 x+2)^2}+\frac{151621 \sqrt{5 x+3} \sqrt{1-2 x}}{36288 (3 x+2)^3}+\frac{647 \sqrt{5 x+3} \sqrt{1-2 x}}{864 (3 x+2)^4}-\frac{391280725 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{175616 \sqrt{7}} \]

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 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 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 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*}

Mathematica [A]  time = 0.143802, size = 191, normalized size = 0.91 \[ \frac{1}{392} \left (\frac{130 (5 x+3)^{3/2} (1-2 x)^{7/2}}{(3 x+2)^5}+\frac{28 (5 x+3)^{3/2} (1-2 x)^{7/2}}{(3 x+2)^6}+\frac{5345 \left (2352 (5 x+3)^{3/2} (1-2 x)^{5/2}+55 (3 x+2) \left (392 (1-2 x)^{3/2} (5 x+3)^{3/2}+33 (3 x+2) \left (7 \sqrt{1-2 x} \sqrt{5 x+3} (37 x+20)-121 \sqrt{7} (3 x+2)^2 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )\right )\right )\right )}{9408 (3 x+2)^4}\right ) \]

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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Maple [B]  time = 0.012, size = 346, normalized size = 1.7 \begin{align*}{\frac{1}{7375872\, \left ( 2+3\,x \right ) ^{6}}\sqrt{1-2\,x}\sqrt{3+5\,x} \left ( 855730945575\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{6}[phone]\,\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) \sqrt{7}{x}^{5}[phone]\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{4}[phone]\,{x}^{5}\sqrt{-10\,{x}^{2}-x+3}[phone]\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{3}[phone]\,{x}^{4}\sqrt{-10\,{x}^{2}-x+3}[phone]\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{2}[phone]\,{x}^{3}\sqrt{-10\,{x}^{2}-x+3}[phone]\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) x+1077448409408\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}[phone]\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) [phone]\,x\sqrt{-10\,{x}^{2}-x+3}[phone]\,\sqrt{-10\,{x}^{2}-x+3} \right ){\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}}} \end{align*}

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)^7,x)

[Out]

1/7375872*(1-2*x)^(1/2)*(3+5*x)^(1/2)*(855730945575*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))
*x^6+3422923782300*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*7^(1/2)*x^5+5704872970500*7^(1/2)*arctan
(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^4+349045461450*x^5*(-10*x^2-x+3)^(1/2)[phone]*7^(1/2)*ar
ctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^3+1179059018760*x^4*(-10*x^2-x+3)^(1/2)[phone]*7^(1/
2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^2+1593676317408*x^3*(-10*x^2-x+3)^(1/2)[phone]*7
^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x+1077448409408*x^2*(-10*x^2-x+3)^(1/2)[phone]*
7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))[phone]*x*(-10*x^2-x+3)^(1/2)[phone]*(-10
*x^2-x+3)^(1/2))/(-10*x^2-x+3)^(1/2)/(2+3*x)^6

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Maxima [A]  time = 3.27936, size = 329, normalized size = 1.57 \begin{align*} \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{align*}

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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Fricas [A]  time = 1.5527, size = 516, normalized size = 2.47 \begin{align*} -\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{align*}

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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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

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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Giac [B]  time = 3.67731, size = 676, normalized size = 3.23 \begin{align*} \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 \,{\left (3207 \, \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 )}^{11} - 8960840 \, \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 )}^{9} - 4031723136 \, \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 )}^{7} - 929280844800 \, \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 )}^{5} - 111701434880000 \, \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 )}^{3} - 5519365017600000 \, \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 )}\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{align*}

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*(3207*sqrt(10)*((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*sqrt
(10)*((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)
))^9 - 4031723136*sqrt(10)*((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(10)*((sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) - 4*sqr
t(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22)))^5 - 111701434880000*sqrt(10)*((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(10)*((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))))/(((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)))^2 + 28
0)^6