3.2347 \(\int \frac {(1-2 x)^{3/2} (3+5 x)^{3/2}}{(2+3 x)^7} \, dx\)

Optimal. Leaf size=209 \[ \frac {37 \sqrt {1-2 x} (5 x+3)^{3/2}}{180 (3 x+2)^5}-\frac {(1-2 x)^{3/2} (5 x+3)^{3/2}}{18 (3 x+2)^6}+\frac {137752591 \sqrt {1-2 x} \sqrt {5 x+3}}{14224896 (3 x+2)}+\frac {1316353 \sqrt {1-2 x} \sqrt {5 x+3}}{1016064 (3 x+2)^2}+\frac {37333 \sqrt {1-2 x} \sqrt {5 x+3}}{181440 (3 x+2)^3}-\frac {7591 \sqrt {1-2 x} \sqrt {5 x+3}}{30240 (3 x+2)^4}-\frac {19457889 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{175616 \sqrt {7}} \]

[Out]

-1/18*(1-2*x)^(3/2)*(3+5*x)^(3/2)/(2+3*x)^6-19457889/1229312*arctan(1/7*(1-2*x)^(1/2)*7^(1/2)/(3+5*x)^(1/2))*7
^(1/2)+37/180*(3+5*x)^(3/2)*(1-2*x)^(1/2)/(2+3*x)^5-7591/30240*(1-2*x)^(1/2)*(3+5*x)^(1/2)/(2+3*x)^4+37333/181
440*(1-2*x)^(1/2)*(3+5*x)^(1/2)/(2+3*x)^3+1316353/1016064*(1-2*x)^(1/2)*(3+5*x)^(1/2)/(2+3*x)^2+137752591/1422
4896*(1-2*x)^(1/2)*(3+5*x)^(1/2)/(2+3*x)

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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} \[ \frac {37 \sqrt {1-2 x} (5 x+3)^{3/2}}{180 (3 x+2)^5}-\frac {(1-2 x)^{3/2} (5 x+3)^{3/2}}{18 (3 x+2)^6}+\frac {137752591 \sqrt {1-2 x} \sqrt {5 x+3}}{14224896 (3 x+2)}+\frac {1316353 \sqrt {1-2 x} \sqrt {5 x+3}}{1016064 (3 x+2)^2}+\frac {37333 \sqrt {1-2 x} \sqrt {5 x+3}}{181440 (3 x+2)^3}-\frac {7591 \sqrt {1-2 x} \sqrt {5 x+3}}{30240 (3 x+2)^4}-\frac {19457889 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{175616 \sqrt {7}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-7591*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(30240*(2 + 3*x)^4) + (37333*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(181440*(2 + 3*x
)^3) + (1316353*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(1016064*(2 + 3*x)^2) + (137752591*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(
14224896*(2 + 3*x)) - ((1 - 2*x)^(3/2)*(3 + 5*x)^(3/2))/(18*(2 + 3*x)^6) + (37*Sqrt[1 - 2*x]*(3 + 5*x)^(3/2))/
(180*(2 + 3*x)^5) - (19457889*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)^{3/2} (3+5 x)^{3/2}}{(2+3 x)^7} \, dx &=-\frac {(1-2 x)^{3/2} (3+5 x)^{3/2}}{18 (2+3 x)^6}+\frac {1}{18} \int \frac {\left (-\frac {3}{2}-30 x\right ) \sqrt {1-2 x} \sqrt {3+5 x}}{(2+3 x)^6} \, dx\\ &=-\frac {(1-2 x)^{3/2} (3+5 x)^{3/2}}{18 (2+3 x)^6}+\frac {37 \sqrt {1-2 x} (3+5 x)^{3/2}}{180 (2+3 x)^5}-\frac {1}{270} \int \frac {\sqrt {3+5 x} \left (-\frac {3951}{4}+1365 x\right )}{\sqrt {1-2 x} (2+3 x)^5} \, dx\\ &=-\frac {7591 \sqrt {1-2 x} \sqrt {3+5 x}}{30240 (2+3 x)^4}-\frac {(1-2 x)^{3/2} (3+5 x)^{3/2}}{18 (2+3 x)^6}+\frac {37 \sqrt {1-2 x} (3+5 x)^{3/2}}{180 (2+3 x)^5}-\frac {\int \frac {-\frac {153051}{8}+\frac {40605 x}{2}}{\sqrt {1-2 x} (2+3 x)^4 \sqrt {3+5 x}} \, dx}{22680}\\ &=-\frac {7591 \sqrt {1-2 x} \sqrt {3+5 x}}{30240 (2+3 x)^4}+\frac {37333 \sqrt {1-2 x} \sqrt {3+5 x}}{181440 (2+3 x)^3}-\frac {(1-2 x)^{3/2} (3+5 x)^{3/2}}{18 (2+3 x)^6}+\frac {37 \sqrt {1-2 x} (3+5 x)^{3/2}}{180 (2+3 x)^5}-\frac {\int \frac {-\frac {25165875}{16}+\frac {3919965 x}{2}}{\sqrt {1-2 x} (2+3 x)^3 \sqrt {3+5 x}} \, dx}{476280}\\ &=-\frac {7591 \sqrt {1-2 x} \sqrt {3+5 x}}{30240 (2+3 x)^4}+\frac {37333 \sqrt {1-2 x} \sqrt {3+5 x}}{181440 (2+3 x)^3}+\frac {1316353 \sqrt {1-2 x} \sqrt {3+5 x}}{1016064 (2+3 x)^2}-\frac {(1-2 x)^{3/2} (3+5 x)^{3/2}}{18 (2+3 x)^6}+\frac {37 \sqrt {1-2 x} (3+5 x)^{3/2}}{180 (2+3 x)^5}-\frac {\int \frac {-\frac {2978446485}{32}+\frac {691085325 x}{8}}{\sqrt {1-2 x} (2+3 x)^2 \sqrt {3+5 x}} \, dx}{6667920}\\ &=-\frac {7591 \sqrt {1-2 x} \sqrt {3+5 x}}{30240 (2+3 x)^4}+\frac {37333 \sqrt {1-2 x} \sqrt {3+5 x}}{181440 (2+3 x)^3}+\frac {1316353 \sqrt {1-2 x} \sqrt {3+5 x}}{1016064 (2+3 x)^2}+\frac {137752591 \sqrt {1-2 x} \sqrt {3+5 x}}{14224896 (2+3 x)}-\frac {(1-2 x)^{3/2} (3+5 x)^{3/2}}{18 (2+3 x)^6}+\frac {37 \sqrt {1-2 x} (3+5 x)^{3/2}}{180 (2+3 x)^5}-\frac {\int -\frac {165489345945}{64 \sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}} \, dx}{46675440}\\ &=-\frac {7591 \sqrt {1-2 x} \sqrt {3+5 x}}{30240 (2+3 x)^4}+\frac {37333 \sqrt {1-2 x} \sqrt {3+5 x}}{181440 (2+3 x)^3}+\frac {1316353 \sqrt {1-2 x} \sqrt {3+5 x}}{1016064 (2+3 x)^2}+\frac {137752591 \sqrt {1-2 x} \sqrt {3+5 x}}{14224896 (2+3 x)}-\frac {(1-2 x)^{3/2} (3+5 x)^{3/2}}{18 (2+3 x)^6}+\frac {37 \sqrt {1-2 x} (3+5 x)^{3/2}}{180 (2+3 x)^5}+\frac {19457889 \int \frac {1}{\sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}} \, dx}{351232}\\ &=-\frac {7591 \sqrt {1-2 x} \sqrt {3+5 x}}{30240 (2+3 x)^4}+\frac {37333 \sqrt {1-2 x} \sqrt {3+5 x}}{181440 (2+3 x)^3}+\frac {1316353 \sqrt {1-2 x} \sqrt {3+5 x}}{1016064 (2+3 x)^2}+\frac {137752591 \sqrt {1-2 x} \sqrt {3+5 x}}{14224896 (2+3 x)}-\frac {(1-2 x)^{3/2} (3+5 x)^{3/2}}{18 (2+3 x)^6}+\frac {37 \sqrt {1-2 x} (3+5 x)^{3/2}}{180 (2+3 x)^5}+\frac {19457889 \operatorname {Subst}\left (\int \frac {1}{-7-x^2} \, dx,x,\frac {\sqrt {1-2 x}}{\sqrt {3+5 x}}\right )}{175616}\\ &=-\frac {7591 \sqrt {1-2 x} \sqrt {3+5 x}}{30240 (2+3 x)^4}+\frac {37333 \sqrt {1-2 x} \sqrt {3+5 x}}{181440 (2+3 x)^3}+\frac {1316353 \sqrt {1-2 x} \sqrt {3+5 x}}{1016064 (2+3 x)^2}+\frac {137752591 \sqrt {1-2 x} \sqrt {3+5 x}}{14224896 (2+3 x)}-\frac {(1-2 x)^{3/2} (3+5 x)^{3/2}}{18 (2+3 x)^6}+\frac {37 \sqrt {1-2 x} (3+5 x)^{3/2}}{180 (2+3 x)^5}-\frac {19457889 \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 = 138, normalized size = 0.66 \[ \frac {1}{280} \left (\frac {2215 \left (\frac {7 \sqrt {1-2 x} \sqrt {5 x+3} \left (100159 x^3+213240 x^2+145940 x+32400\right )}{(3 x+2)^4}-43923 \sqrt {7} \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )\right )}{21952}+\frac {74 (1-2 x)^{5/2} (5 x+3)^{5/2}}{(3 x+2)^5}+\frac {20 (1-2 x)^{5/2} (5 x+3)^{5/2}}{(3 x+2)^6}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

((20*(1 - 2*x)^(5/2)*(3 + 5*x)^(5/2))/(2 + 3*x)^6 + (74*(1 - 2*x)^(5/2)*(3 + 5*x)^(5/2))/(2 + 3*x)^5 + (2215*(
(7*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(32400 + 145940*x + 213240*x^2 + 100159*x^3))/(2 + 3*x)^4 - 43923*Sqrt[7]*ArcTa
n[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])]))/21952)/280

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fricas [A]  time = 0.85, size = 146, normalized size = 0.70 \[ -\frac {97289445 \, \sqrt {7} {\left (729 \, x^{6} + 2916 \, x^{5} + 4860 \, x^{4} + 4320 \, x^{3} + 2160 \, x^{2} + 576 \, x + 64\right )} \arctan \left (\frac {\sqrt {7} {\left (37 \, x + 20\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{14 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) - 14 \, {\left (2066288865 \, x^{5} + 6979774260 \, x^{4} + 9434103472 \, x^{3} + 6379024416 \, x^{2} + 2157325040 \, x + 291805632\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{12293120 \, {\left (729 \, x^{6} + 2916 \, x^{5} + 4860 \, x^{4} + 4320 \, x^{3} + 2160 \, x^{2} + 576 \, x + 64\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-2*x)^(3/2)*(3+5*x)^(3/2)/(2+3*x)^7,x, algorithm="fricas")

[Out]

-1/12293120*(97289445*sqrt(7)*(729*x^6 + 2916*x^5 + 4860*x^4 + 4320*x^3 + 2160*x^2 + 576*x + 64)*arctan(1/14*s
qrt(7)*(37*x + 20)*sqrt(5*x + 3)*sqrt(-2*x + 1)/(10*x^2 + x - 3)) - 14*(2066288865*x^5 + 6979774260*x^4 + 9434
103472*x^3 + 6379024416*x^2 + 2157325040*x + 291805632)*sqrt(5*x + 3)*sqrt(-2*x + 1))/(729*x^6 + 2916*x^5 + 48
60*x^4 + 4320*x^3 + 2160*x^2 + 576*x + 64)

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giac [B]  time = 4.02, size = 484, normalized size = 2.32 \[ \frac {19457889}{24586240} \, \sqrt {70} \sqrt {10} {\left (\pi + 2 \, \arctan \left (-\frac {\sqrt {70} \sqrt {5 \, x + 3} {\left (\frac {{\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}\right )\right )} - \frac {14641 \, \sqrt {10} {\left (1329 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{11} + 2108680 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{9} - 1434500480 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{7} - 382530534400 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{5} - 46289743360000 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{3} - \frac {2287257907200000 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}{\sqrt {5 \, x + 3}} + \frac {9149031628800000 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}}{87808 \, {\left ({\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{2} + 280\right )}^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-2*x)^(3/2)*(3+5*x)^(3/2)/(2+3*x)^7,x, algorithm="giac")

[Out]

19457889/24586240*sqrt(70)*sqrt(10)*(pi + 2*arctan(-1/140*sqrt(70)*sqrt(5*x + 3)*((sqrt(2)*sqrt(-10*x + 5) - s
qrt(22))^2/(5*x + 3) - 4)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22)))) - 14641/87808*sqrt(10)*(1329*((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 + 2108680*((sqr
t(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 - 143
4500480*((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 - 382530534400*((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)))^5 - 46289743360000*((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 - 2287257907200000*(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3)
+ 9149031628800000*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 \[ \frac {\sqrt {-2 x +1}\, \sqrt {5 x +3}\, \left (70924005405 \sqrt {7}\, x^{6} \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )[phone] \sqrt {7}\, x^{5} \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )[phone] \sqrt {-10 x^{2}-x +3}\, x^{5}[phone] \sqrt {7}\, x^{4} \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )[phone] \sqrt {-10 x^{2}-x +3}\, x^{4}[phone] \sqrt {7}\, x^{3} \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )[phone] \sqrt {-10 x^{2}-x +3}\, x^{3}[phone] \sqrt {7}\, x^{2} \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )[phone] \sqrt {-10 x^{2}-x +3}\, x^{2}[phone] \sqrt {7}\, x \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )[phone] \sqrt {-10 x^{2}-x +3}\, x [phone] \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )[phone] \sqrt {-10 x^{2}-x +3}\right )}{12293120 \sqrt {-10 x^{2}-x +3}\, \left (3 x +2\right )^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/12293120*(-2*x+1)^(1/2)*(5*x+3)^(1/2)*(70924005405*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*arc
tan(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+420290402400*7^(1/2)*x^3*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^4+210145201200*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+56038720320*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+6226524480*7^(1/2)*arc
tan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))[phone]*(-10*x^2-x+3)^(1/2)*x+4085278848*(-10*x^2-x+3)^(1/
2))/(-10*x^2-x+3)^(1/2)/(3*x+2)^6

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maxima [A]  time = 1.38, size = 273, normalized size = 1.31 \[ \frac {3652535}{921984} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} + \frac {{\left (-10 \, x^{2} - x + 3\right )}^{\frac {5}{2}}}{14 \, {\left (729 \, x^{6} + 2916 \, x^{5} + 4860 \, x^{4} + 4320 \, x^{3} + 2160 \, x^{2} + 576 \, x + 64\right )}} + \frac {37 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {5}{2}}}{140 \, {\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )}} + \frac {1329 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {5}{2}}}{1568 \, {\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )}} + \frac {49173 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {5}{2}}}{21952 \, {\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}} + \frac {2191521 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {5}{2}}}{614656 \, {\left (9 \, x^{2} + 12 \, x + 4\right )}} + \frac {29749665}{614656} \, \sqrt {-10 \, x^{2} - x + 3} x + \frac {19457889}{2458624} \, \sqrt {7} \arcsin \left (\frac {37 \, x}{11 \, {\left | 3 \, x + 2 \right |}} + \frac {20}{11 \, {\left | 3 \, x + 2 \right |}}\right ) - \frac {26211867}{1229312} \, \sqrt {-10 \, x^{2} - x + 3} + \frac {8670839 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}}{3687936 \, {\left (3 \, x + 2\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-2*x)^(3/2)*(3+5*x)^(3/2)/(2+3*x)^7,x, algorithm="maxima")

[Out]

3652535/921984*(-10*x^2 - x + 3)^(3/2) + 1/14*(-10*x^2 - x + 3)^(5/2)/(729*x^6 + 2916*x^5 + 4860*x^4 + 4320*x^
3 + 2160*x^2 + 576*x + 64) + 37/140*(-10*x^2 - x + 3)^(5/2)/(243*x^5 + 810*x^4 + 1080*x^3 + 720*x^2 + 240*x +
32) + 1329/1568*(-10*x^2 - x + 3)^(5/2)/(81*x^4 + 216*x^3 + 216*x^2 + 96*x + 16) + 49173/21952*(-10*x^2 - x +
3)^(5/2)/(27*x^3 + 54*x^2 + 36*x + 8) + 2191521/614656*(-10*x^2 - x + 3)^(5/2)/(9*x^2 + 12*x + 4) + 29749665/6
14656*sqrt(-10*x^2 - x + 3)*x + 19457889/2458624*sqrt(7)*arcsin(37/11*x/abs(3*x + 2) + 20/11/abs(3*x + 2)) - 2
6211867/1229312*sqrt(-10*x^2 - x + 3) + 8670839/3687936*(-10*x^2 - x + 3)^(3/2)/(3*x + 2)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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