3.1.41 \(\int \frac {x^3}{a+b \sec (c+d \sqrt {x})} \, dx\) [41]

3.1.41.1 Optimal result

Integrand size = 20, antiderivative size = 1041 \[ \int \frac {x^3}{a+b \sec \left (c+d \sqrt {x}\right )} \, dx=\frac {x^4}{4 a}+\frac {2 i b x^{7/2} \log \left (1+\frac {a e^{i \left (c+d \sqrt {x}\right )}}{b-\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d}-\frac {2 i b x^{7/2} \log \left (1+\frac {a e^{i \left (c+d \sqrt {x}\right )}}{b+\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d}+\frac {14 b x^3 \operatorname {PolyLog}\left (2,-\frac {a e^{i \left (c+d \sqrt {x}\right )}}{b-\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d^2}-\frac {14 b x^3 \operatorname {PolyLog}\left (2,-\frac {a e^{i \left (c+d \sqrt {x}\right )}}{b+\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d^2}+\frac {84 i b x^{5/2} \operatorname {PolyLog}\left (3,-\frac {a e^{i \left (c+d \sqrt {x}\right )}}{b-\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d^3}-\frac {84 i b x^{5/2} \operatorname {PolyLog}\left (3,-\frac {a e^{i \left (c+d \sqrt {x}\right )}}{b+\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d^3}-\frac {420 b x^2 \operatorname {PolyLog}\left (4,-\frac {a e^{i \left (c+d \sqrt {x}\right )}}{b-\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d^4}+\frac {420 b x^2 \operatorname {PolyLog}\left (4,-\frac {a e^{i \left (c+d \sqrt {x}\right )}}{b+\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d^4}-\frac {1680 i b x^{3/2} \operatorname {PolyLog}\left (5,-\frac {a e^{i \left (c+d \sqrt {x}\right )}}{b-\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d^5}+\frac {1680 i b x^{3/2} \operatorname {PolyLog}\left (5,-\frac {a e^{i \left (c+d \sqrt {x}\right )}}{b+\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d^5}+\frac {5040 b x \operatorname {PolyLog}\left (6,-\frac {a e^{i \left (c+d \sqrt {x}\right )}}{b-\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d^6}-\frac {5040 b x \operatorname {PolyLog}\left (6,-\frac {a e^{i \left (c+d \sqrt {x}\right )}}{b+\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d^6}+\frac {10080 i b \sqrt {x} \operatorname {PolyLog}\left (7,-\frac {a e^{i \left (c+d \sqrt {x}\right )}}{b-\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d^7}-\frac {10080 i b \sqrt {x} \operatorname {PolyLog}\left (7,-\frac {a e^{i \left (c+d \sqrt {x}\right )}}{b+\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d^7}-\frac {10080 b \operatorname {PolyLog}\left (8,-\frac {a e^{i \left (c+d \sqrt {x}\right )}}{b-\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d^8}+\frac {10080 b \operatorname {PolyLog}\left (8,-\frac {a e^{i \left (c+d \sqrt {x}\right )}}{b+\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d^8} \]

output

1/4*x^4/a-2*I*b*x^(7/2)*ln(1+a*exp(I*(c+d*x^(1/2)))/(b+(-a^2+b^2)^(1/2)))/ 
a/d/(-a^2+b^2)^(1/2)+84*I*b*x^(5/2)*polylog(3,-a*exp(I*(c+d*x^(1/2)))/(b-( 
-a^2+b^2)^(1/2)))/a/d^3/(-a^2+b^2)^(1/2)+14*b*x^3*polylog(2,-a*exp(I*(c+d* 
x^(1/2)))/(b-(-a^2+b^2)^(1/2)))/a/d^2/(-a^2+b^2)^(1/2)-14*b*x^3*polylog(2, 
-a*exp(I*(c+d*x^(1/2)))/(b+(-a^2+b^2)^(1/2)))/a/d^2/(-a^2+b^2)^(1/2)-10080 
*I*b*polylog(7,-a*exp(I*(c+d*x^(1/2)))/(b+(-a^2+b^2)^(1/2)))*x^(1/2)/a/d^7 
/(-a^2+b^2)^(1/2)+1680*I*b*x^(3/2)*polylog(5,-a*exp(I*(c+d*x^(1/2)))/(b+(- 
a^2+b^2)^(1/2)))/a/d^5/(-a^2+b^2)^(1/2)-420*b*x^2*polylog(4,-a*exp(I*(c+d* 
x^(1/2)))/(b-(-a^2+b^2)^(1/2)))/a/d^4/(-a^2+b^2)^(1/2)+420*b*x^2*polylog(4 
,-a*exp(I*(c+d*x^(1/2)))/(b+(-a^2+b^2)^(1/2)))/a/d^4/(-a^2+b^2)^(1/2)+1008 
0*I*b*polylog(7,-a*exp(I*(c+d*x^(1/2)))/(b-(-a^2+b^2)^(1/2)))*x^(1/2)/a/d^ 
7/(-a^2+b^2)^(1/2)+2*I*b*x^(7/2)*ln(1+a*exp(I*(c+d*x^(1/2)))/(b-(-a^2+b^2) 
^(1/2)))/a/d/(-a^2+b^2)^(1/2)+5040*b*x*polylog(6,-a*exp(I*(c+d*x^(1/2)))/( 
b-(-a^2+b^2)^(1/2)))/a/d^6/(-a^2+b^2)^(1/2)-5040*b*x*polylog(6,-a*exp(I*(c 
+d*x^(1/2)))/(b+(-a^2+b^2)^(1/2)))/a/d^6/(-a^2+b^2)^(1/2)-10080*b*polylog( 
8,-a*exp(I*(c+d*x^(1/2)))/(b-(-a^2+b^2)^(1/2)))/a/d^8/(-a^2+b^2)^(1/2)+100 
80*b*polylog(8,-a*exp(I*(c+d*x^(1/2)))/(b+(-a^2+b^2)^(1/2)))/a/d^8/(-a^2+b 
^2)^(1/2)-84*I*b*x^(5/2)*polylog(3,-a*exp(I*(c+d*x^(1/2)))/(b+(-a^2+b^2)^( 
1/2)))/a/d^3/(-a^2+b^2)^(1/2)-1680*I*b*x^(3/2)*polylog(5,-a*exp(I*(c+d*x^( 
1/2)))/(b-(-a^2+b^2)^(1/2)))/a/d^5/(-a^2+b^2)^(1/2)
 

3.1.41.2 Mathematica [A] (verified)

Time = 1.66 (sec) , antiderivative size = 802, normalized size of antiderivative = 0.77 \[ \int \frac {x^3}{a+b \sec \left (c+d \sqrt {x}\right )} \, dx=\frac {\sqrt {-a^2+b^2} d^8 x^4+8 i b d^7 x^{7/2} \log \left (1-\frac {a e^{i \left (c+d \sqrt {x}\right )}}{-b+\sqrt {-a^2+b^2}}\right )-8 i b d^7 x^{7/2} \log \left (1+\frac {a e^{i \left (c+d \sqrt {x}\right )}}{b+\sqrt {-a^2+b^2}}\right )+56 b d^6 x^3 \operatorname {PolyLog}\left (2,\frac {a e^{i \left (c+d \sqrt {x}\right )}}{-b+\sqrt {-a^2+b^2}}\right )-56 b d^6 x^3 \operatorname {PolyLog}\left (2,-\frac {a e^{i \left (c+d \sqrt {x}\right )}}{b+\sqrt {-a^2+b^2}}\right )+336 i b d^5 x^{5/2} \operatorname {PolyLog}\left (3,\frac {a e^{i \left (c+d \sqrt {x}\right )}}{-b+\sqrt {-a^2+b^2}}\right )-336 i b d^5 x^{5/2} \operatorname {PolyLog}\left (3,-\frac {a e^{i \left (c+d \sqrt {x}\right )}}{b+\sqrt {-a^2+b^2}}\right )-1680 b d^4 x^2 \operatorname {PolyLog}\left (4,\frac {a e^{i \left (c+d \sqrt {x}\right )}}{-b+\sqrt {-a^2+b^2}}\right )+1680 b d^4 x^2 \operatorname {PolyLog}\left (4,-\frac {a e^{i \left (c+d \sqrt {x}\right )}}{b+\sqrt {-a^2+b^2}}\right )-6720 i b d^3 x^{3/2} \operatorname {PolyLog}\left (5,\frac {a e^{i \left (c+d \sqrt {x}\right )}}{-b+\sqrt {-a^2+b^2}}\right )+6720 i b d^3 x^{3/2} \operatorname {PolyLog}\left (5,-\frac {a e^{i \left (c+d \sqrt {x}\right )}}{b+\sqrt {-a^2+b^2}}\right )+20160 b d^2 x \operatorname {PolyLog}\left (6,\frac {a e^{i \left (c+d \sqrt {x}\right )}}{-b+\sqrt {-a^2+b^2}}\right )-20160 b d^2 x \operatorname {PolyLog}\left (6,-\frac {a e^{i \left (c+d \sqrt {x}\right )}}{b+\sqrt {-a^2+b^2}}\right )+40320 i b d \sqrt {x} \operatorname {PolyLog}\left (7,\frac {a e^{i \left (c+d \sqrt {x}\right )}}{-b+\sqrt {-a^2+b^2}}\right )-40320 i b d \sqrt {x} \operatorname {PolyLog}\left (7,-\frac {a e^{i \left (c+d \sqrt {x}\right )}}{b+\sqrt {-a^2+b^2}}\right )-40320 b \operatorname {PolyLog}\left (8,\frac {a e^{i \left (c+d \sqrt {x}\right )}}{-b+\sqrt {-a^2+b^2}}\right )+40320 b \operatorname {PolyLog}\left (8,-\frac {a e^{i \left (c+d \sqrt {x}\right )}}{b+\sqrt {-a^2+b^2}}\right )}{4 a \sqrt {-a^2+b^2} d^8} \]

input

Integrate[x^3/(a + b*Sec[c + d*Sqrt[x]]),x]
 

output

(Sqrt[-a^2 + b^2]*d^8*x^4 + (8*I)*b*d^7*x^(7/2)*Log[1 - (a*E^(I*(c + d*Sqr 
t[x])))/(-b + Sqrt[-a^2 + b^2])] - (8*I)*b*d^7*x^(7/2)*Log[1 + (a*E^(I*(c 
+ d*Sqrt[x])))/(b + Sqrt[-a^2 + b^2])] + 56*b*d^6*x^3*PolyLog[2, (a*E^(I*( 
c + d*Sqrt[x])))/(-b + Sqrt[-a^2 + b^2])] - 56*b*d^6*x^3*PolyLog[2, -((a*E 
^(I*(c + d*Sqrt[x])))/(b + Sqrt[-a^2 + b^2]))] + (336*I)*b*d^5*x^(5/2)*Pol 
yLog[3, (a*E^(I*(c + d*Sqrt[x])))/(-b + Sqrt[-a^2 + b^2])] - (336*I)*b*d^5 
*x^(5/2)*PolyLog[3, -((a*E^(I*(c + d*Sqrt[x])))/(b + Sqrt[-a^2 + b^2]))] - 
 1680*b*d^4*x^2*PolyLog[4, (a*E^(I*(c + d*Sqrt[x])))/(-b + Sqrt[-a^2 + b^2 
])] + 1680*b*d^4*x^2*PolyLog[4, -((a*E^(I*(c + d*Sqrt[x])))/(b + Sqrt[-a^2 
 + b^2]))] - (6720*I)*b*d^3*x^(3/2)*PolyLog[5, (a*E^(I*(c + d*Sqrt[x])))/( 
-b + Sqrt[-a^2 + b^2])] + (6720*I)*b*d^3*x^(3/2)*PolyLog[5, -((a*E^(I*(c + 
 d*Sqrt[x])))/(b + Sqrt[-a^2 + b^2]))] + 20160*b*d^2*x*PolyLog[6, (a*E^(I* 
(c + d*Sqrt[x])))/(-b + Sqrt[-a^2 + b^2])] - 20160*b*d^2*x*PolyLog[6, -((a 
*E^(I*(c + d*Sqrt[x])))/(b + Sqrt[-a^2 + b^2]))] + (40320*I)*b*d*Sqrt[x]*P 
olyLog[7, (a*E^(I*(c + d*Sqrt[x])))/(-b + Sqrt[-a^2 + b^2])] - (40320*I)*b 
*d*Sqrt[x]*PolyLog[7, -((a*E^(I*(c + d*Sqrt[x])))/(b + Sqrt[-a^2 + b^2]))] 
 - 40320*b*PolyLog[8, (a*E^(I*(c + d*Sqrt[x])))/(-b + Sqrt[-a^2 + b^2])] + 
 40320*b*PolyLog[8, -((a*E^(I*(c + d*Sqrt[x])))/(b + Sqrt[-a^2 + b^2]))])/ 
(4*a*Sqrt[-a^2 + b^2]*d^8)
 

3.1.41.3 Rubi [A] (verified)

Time = 1.72 (sec) , antiderivative size = 1043, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {4692, 3042, 4679, 2009}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \frac {x^3}{a+b \sec \left (c+d \sqrt {x}\right )} \, dx\)

\(\Big \downarrow \) 4692

\(\displaystyle 2 \int \frac {x^{7/2}}{a+b \sec \left (c+d \sqrt {x}\right )}d\sqrt {x}\)

\(\Big \downarrow \) 3042

\(\displaystyle 2 \int \frac {x^{7/2}}{a+b \csc \left (c+d \sqrt {x}+\frac {\pi }{2}\right )}d\sqrt {x}\)

\(\Big \downarrow \) 4679

\(\displaystyle 2 \int \left (\frac {x^{7/2}}{a}-\frac {b x^{7/2}}{a \left (b+a \cos \left (c+d \sqrt {x}\right )\right )}\right )d\sqrt {x}\)

\(\Big \downarrow \) 2009

\(\displaystyle 2 \left (\frac {x^4}{8 a}+\frac {i b \log \left (\frac {e^{i \left (c+d \sqrt {x}\right )} a}{b-\sqrt {b^2-a^2}}+1\right ) x^{7/2}}{a \sqrt {b^2-a^2} d}-\frac {i b \log \left (\frac {e^{i \left (c+d \sqrt {x}\right )} a}{b+\sqrt {b^2-a^2}}+1\right ) x^{7/2}}{a \sqrt {b^2-a^2} d}+\frac {7 b \operatorname {PolyLog}\left (2,-\frac {a e^{i \left (c+d \sqrt {x}\right )}}{b-\sqrt {b^2-a^2}}\right ) x^3}{a \sqrt {b^2-a^2} d^2}-\frac {7 b \operatorname {PolyLog}\left (2,-\frac {a e^{i \left (c+d \sqrt {x}\right )}}{b+\sqrt {b^2-a^2}}\right ) x^3}{a \sqrt {b^2-a^2} d^2}+\frac {42 i b \operatorname {PolyLog}\left (3,-\frac {a e^{i \left (c+d \sqrt {x}\right )}}{b-\sqrt {b^2-a^2}}\right ) x^{5/2}}{a \sqrt {b^2-a^2} d^3}-\frac {42 i b \operatorname {PolyLog}\left (3,-\frac {a e^{i \left (c+d \sqrt {x}\right )}}{b+\sqrt {b^2-a^2}}\right ) x^{5/2}}{a \sqrt {b^2-a^2} d^3}-\frac {210 b \operatorname {PolyLog}\left (4,-\frac {a e^{i \left (c+d \sqrt {x}\right )}}{b-\sqrt {b^2-a^2}}\right ) x^2}{a \sqrt {b^2-a^2} d^4}+\frac {210 b \operatorname {PolyLog}\left (4,-\frac {a e^{i \left (c+d \sqrt {x}\right )}}{b+\sqrt {b^2-a^2}}\right ) x^2}{a \sqrt {b^2-a^2} d^4}-\frac {840 i b \operatorname {PolyLog}\left (5,-\frac {a e^{i \left (c+d \sqrt {x}\right )}}{b-\sqrt {b^2-a^2}}\right ) x^{3/2}}{a \sqrt {b^2-a^2} d^5}+\frac {840 i b \operatorname {PolyLog}\left (5,-\frac {a e^{i \left (c+d \sqrt {x}\right )}}{b+\sqrt {b^2-a^2}}\right ) x^{3/2}}{a \sqrt {b^2-a^2} d^5}+\frac {2520 b \operatorname {PolyLog}\left (6,-\frac {a e^{i \left (c+d \sqrt {x}\right )}}{b-\sqrt {b^2-a^2}}\right ) x}{a \sqrt {b^2-a^2} d^6}-\frac {2520 b \operatorname {PolyLog}\left (6,-\frac {a e^{i \left (c+d \sqrt {x}\right )}}{b+\sqrt {b^2-a^2}}\right ) x}{a \sqrt {b^2-a^2} d^6}+\frac {5040 i b \operatorname {PolyLog}\left (7,-\frac {a e^{i \left (c+d \sqrt {x}\right )}}{b-\sqrt {b^2-a^2}}\right ) \sqrt {x}}{a \sqrt {b^2-a^2} d^7}-\frac {5040 i b \operatorname {PolyLog}\left (7,-\frac {a e^{i \left (c+d \sqrt {x}\right )}}{b+\sqrt {b^2-a^2}}\right ) \sqrt {x}}{a \sqrt {b^2-a^2} d^7}-\frac {5040 b \operatorname {PolyLog}\left (8,-\frac {a e^{i \left (c+d \sqrt {x}\right )}}{b-\sqrt {b^2-a^2}}\right )}{a \sqrt {b^2-a^2} d^8}+\frac {5040 b \operatorname {PolyLog}\left (8,-\frac {a e^{i \left (c+d \sqrt {x}\right )}}{b+\sqrt {b^2-a^2}}\right )}{a \sqrt {b^2-a^2} d^8}\right )\)

input

Int[x^3/(a + b*Sec[c + d*Sqrt[x]]),x]
 

output

2*(x^4/(8*a) + (I*b*x^(7/2)*Log[1 + (a*E^(I*(c + d*Sqrt[x])))/(b - Sqrt[-a 
^2 + b^2])])/(a*Sqrt[-a^2 + b^2]*d) - (I*b*x^(7/2)*Log[1 + (a*E^(I*(c + d* 
Sqrt[x])))/(b + Sqrt[-a^2 + b^2])])/(a*Sqrt[-a^2 + b^2]*d) + (7*b*x^3*Poly 
Log[2, -((a*E^(I*(c + d*Sqrt[x])))/(b - Sqrt[-a^2 + b^2]))])/(a*Sqrt[-a^2 
+ b^2]*d^2) - (7*b*x^3*PolyLog[2, -((a*E^(I*(c + d*Sqrt[x])))/(b + Sqrt[-a 
^2 + b^2]))])/(a*Sqrt[-a^2 + b^2]*d^2) + ((42*I)*b*x^(5/2)*PolyLog[3, -((a 
*E^(I*(c + d*Sqrt[x])))/(b - Sqrt[-a^2 + b^2]))])/(a*Sqrt[-a^2 + b^2]*d^3) 
 - ((42*I)*b*x^(5/2)*PolyLog[3, -((a*E^(I*(c + d*Sqrt[x])))/(b + Sqrt[-a^2 
 + b^2]))])/(a*Sqrt[-a^2 + b^2]*d^3) - (210*b*x^2*PolyLog[4, -((a*E^(I*(c 
+ d*Sqrt[x])))/(b - Sqrt[-a^2 + b^2]))])/(a*Sqrt[-a^2 + b^2]*d^4) + (210*b 
*x^2*PolyLog[4, -((a*E^(I*(c + d*Sqrt[x])))/(b + Sqrt[-a^2 + b^2]))])/(a*S 
qrt[-a^2 + b^2]*d^4) - ((840*I)*b*x^(3/2)*PolyLog[5, -((a*E^(I*(c + d*Sqrt 
[x])))/(b - Sqrt[-a^2 + b^2]))])/(a*Sqrt[-a^2 + b^2]*d^5) + ((840*I)*b*x^( 
3/2)*PolyLog[5, -((a*E^(I*(c + d*Sqrt[x])))/(b + Sqrt[-a^2 + b^2]))])/(a*S 
qrt[-a^2 + b^2]*d^5) + (2520*b*x*PolyLog[6, -((a*E^(I*(c + d*Sqrt[x])))/(b 
 - Sqrt[-a^2 + b^2]))])/(a*Sqrt[-a^2 + b^2]*d^6) - (2520*b*x*PolyLog[6, -( 
(a*E^(I*(c + d*Sqrt[x])))/(b + Sqrt[-a^2 + b^2]))])/(a*Sqrt[-a^2 + b^2]*d^ 
6) + ((5040*I)*b*Sqrt[x]*PolyLog[7, -((a*E^(I*(c + d*Sqrt[x])))/(b - Sqrt[ 
-a^2 + b^2]))])/(a*Sqrt[-a^2 + b^2]*d^7) - ((5040*I)*b*Sqrt[x]*PolyLog[7, 
-((a*E^(I*(c + d*Sqrt[x])))/(b + Sqrt[-a^2 + b^2]))])/(a*Sqrt[-a^2 + b^...
 

3.1.41.3.1 Defintions of rubi rules used

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 4679

Int[(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(n_.)*((c_.) + (d_.)*(x_))^(m_.) 
, x_Symbol] :> Int[ExpandIntegrand[(c + d*x)^m, 1/(Sin[e + f*x]^n/(b + a*Si 
n[e + f*x])^n), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && ILtQ[n, 0] && IGt 
Q[m, 0]
 

rule 4692

Int[(x_)^(m_.)*((a_.) + (b_.)*Sec[(c_.) + (d_.)*(x_)^(n_)])^(p_.), x_Symbol 
] :> Simp[1/n   Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a + b*Sec[c + d*x])^ 
p, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p}, x] && IGtQ[Simplify[(m + 
 1)/n], 0] && IntegerQ[p]
 

3.1.41.4 Maple [F]

\[\int \frac {x^{3}}{a +b \sec \left (c +d \sqrt {x}\right )}d x\]

input

int(x^3/(a+b*sec(c+d*x^(1/2))),x)
 

output

int(x^3/(a+b*sec(c+d*x^(1/2))),x)
 

3.1.41.5 Fricas [F]

\[ \int \frac {x^3}{a+b \sec \left (c+d \sqrt {x}\right )} \, dx=\int { \frac {x^{3}}{b \sec \left (d \sqrt {x} + c\right ) + a} \,d x } \]

input

integrate(x^3/(a+b*sec(c+d*x^(1/2))),x, algorithm="fricas")
 

output

integral(x^3/(b*sec(d*sqrt(x) + c) + a), x)
 

3.1.41.6 Sympy [F]

\[ \int \frac {x^3}{a+b \sec \left (c+d \sqrt {x}\right )} \, dx=\int \frac {x^{3}}{a + b \sec {\left (c + d \sqrt {x} \right )}}\, dx \]

input

integrate(x**3/(a+b*sec(c+d*x**(1/2))),x)
 

output

Integral(x**3/(a + b*sec(c + d*sqrt(x))), x)
 

3.1.41.7 Maxima [F(-2)]

Exception generated. \[ \int \frac {x^3}{a+b \sec \left (c+d \sqrt {x}\right )} \, dx=\text {Exception raised: ValueError} \]

input

integrate(x^3/(a+b*sec(c+d*x^(1/2))),x, algorithm="maxima")
 

output

Exception raised: ValueError >> Computation failed since Maxima requested 
additional constraints; using the 'assume' command before evaluation *may* 
 help (example of legal syntax is 'assume(4*a^2-4*b^2>0)', see `assume?` f 
or more de
 

3.1.41.8 Giac [F]

\[ \int \frac {x^3}{a+b \sec \left (c+d \sqrt {x}\right )} \, dx=\int { \frac {x^{3}}{b \sec \left (d \sqrt {x} + c\right ) + a} \,d x } \]

input

integrate(x^3/(a+b*sec(c+d*x^(1/2))),x, algorithm="giac")
 

output

integrate(x^3/(b*sec(d*sqrt(x) + c) + a), x)
 

3.1.41.9 Mupad [F(-1)]

Timed out. \[ \int \frac {x^3}{a+b \sec \left (c+d \sqrt {x}\right )} \, dx=\int \frac {x^3}{a+\frac {b}{\cos \left (c+d\,\sqrt {x}\right )}} \,d x \]

input

int(x^3/(a + b/cos(c + d*x^(1/2))),x)
 

output

int(x^3/(a + b/cos(c + d*x^(1/2))), x)