\(\int \frac {(c+d x^2)^3}{x^{13/2} (a+b x^2)} \, dx\) [772]

Optimal result

Integrand size = 24, antiderivative size = 240 \[ \int \frac {\left (c+d x^2\right )^3}{x^{13/2} \left (a+b x^2\right )} \, dx=-\frac {2 c^3}{11 a x^{11/2}}+\frac {2 c^2 (b c-3 a d)}{7 a^2 x^{7/2}}-\frac {2 c \left (b^2 c^2-3 a b c d+3 a^2 d^2\right )}{3 a^3 x^{3/2}}+\frac {(b c-a d)^3 \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} a^{15/4} \sqrt [4]{b}}-\frac {(b c-a d)^3 \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} a^{15/4} \sqrt [4]{b}}-\frac {(b c-a d)^3 \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}{\sqrt {a}+\sqrt {b} x}\right )}{\sqrt {2} a^{15/4} \sqrt [4]{b}} \] Output:

-2/11*c^3/a/x^(11/2)+2/7*c^2*(-3*a*d+b*c)/a^2/x^(7/2)-2/3*c*(3*a^2*d^2-3*a 
*b*c*d+b^2*c^2)/a^3/x^(3/2)+1/2*(-a*d+b*c)^3*arctan(1-2^(1/2)*b^(1/4)*x^(1 
/2)/a^(1/4))*2^(1/2)/a^(15/4)/b^(1/4)-1/2*(-a*d+b*c)^3*arctan(1+2^(1/2)*b^ 
(1/4)*x^(1/2)/a^(1/4))*2^(1/2)/a^(15/4)/b^(1/4)-1/2*(-a*d+b*c)^3*arctanh(2 
^(1/2)*a^(1/4)*b^(1/4)*x^(1/2)/(a^(1/2)+b^(1/2)*x))*2^(1/2)/a^(15/4)/b^(1/ 
4)
 

Mathematica [A] (verified)

Time = 0.25 (sec) , antiderivative size = 194, normalized size of antiderivative = 0.81 \[ \int \frac {\left (c+d x^2\right )^3}{x^{13/2} \left (a+b x^2\right )} \, dx=\frac {-\frac {4 a^{3/4} c \left (77 b^2 c^2 x^4-33 a b c x^2 \left (c+7 d x^2\right )+3 a^2 \left (7 c^2+33 c d x^2+77 d^2 x^4\right )\right )}{x^{11/2}}+\frac {231 \sqrt {2} (b c-a d)^3 \arctan \left (\frac {\sqrt {a}-\sqrt {b} x}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}\right )}{\sqrt [4]{b}}+\frac {231 \sqrt {2} (-b c+a d)^3 \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}{\sqrt {a}+\sqrt {b} x}\right )}{\sqrt [4]{b}}}{462 a^{15/4}} \] Input:

Integrate[(c + d*x^2)^3/(x^(13/2)*(a + b*x^2)),x]
 

Output:

((-4*a^(3/4)*c*(77*b^2*c^2*x^4 - 33*a*b*c*x^2*(c + 7*d*x^2) + 3*a^2*(7*c^2 
 + 33*c*d*x^2 + 77*d^2*x^4)))/x^(11/2) + (231*Sqrt[2]*(b*c - a*d)^3*ArcTan 
[(Sqrt[a] - Sqrt[b]*x)/(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])])/b^(1/4) + (231* 
Sqrt[2]*(-(b*c) + a*d)^3*ArcTanh[(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])/(Sqrt[a 
] + Sqrt[b]*x)])/b^(1/4))/(462*a^(15/4))
 

Rubi [A] (verified)

Time = 0.45 (sec) , antiderivative size = 312, normalized size of antiderivative = 1.30, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {368, 961, 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.

\(\displaystyle \int \frac {\left (c+d x^2\right )^3}{x^{13/2} \left (a+b x^2\right )} \, dx\)

\(\Big \downarrow \) 368

\(\displaystyle 2 \int \frac {\left (d x^2+c\right )^3}{x^6 \left (b x^2+a\right )}d\sqrt {x}\)

\(\Big \downarrow \) 961

\(\displaystyle 2 \int \left (\frac {c^3}{a x^6}+\frac {(3 a d-b c) c^2}{a^2 x^4}+\frac {\left (b^2 c^2-3 a b d c+3 a^2 d^2\right ) c}{a^3 x^2}+\frac {(a d-b c)^3}{a^3 \left (b x^2+a\right )}\right )d\sqrt {x}\)

\(\Big \downarrow \) 2009

\(\displaystyle 2 \left (\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right ) (b c-a d)^3}{2 \sqrt {2} a^{15/4} \sqrt [4]{b}}-\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right ) (b c-a d)^3}{2 \sqrt {2} a^{15/4} \sqrt [4]{b}}+\frac {(b c-a d)^3 \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{4 \sqrt {2} a^{15/4} \sqrt [4]{b}}-\frac {(b c-a d)^3 \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{4 \sqrt {2} a^{15/4} \sqrt [4]{b}}+\frac {c^2 (b c-3 a d)}{7 a^2 x^{7/2}}-\frac {c \left (3 a^2 d^2-3 a b c d+b^2 c^2\right )}{3 a^3 x^{3/2}}-\frac {c^3}{11 a x^{11/2}}\right )\)

Input:

Int[(c + d*x^2)^3/(x^(13/2)*(a + b*x^2)),x]
 

Output:

2*(-1/11*c^3/(a*x^(11/2)) + (c^2*(b*c - 3*a*d))/(7*a^2*x^(7/2)) - (c*(b^2* 
c^2 - 3*a*b*c*d + 3*a^2*d^2))/(3*a^3*x^(3/2)) + ((b*c - a*d)^3*ArcTan[1 - 
(Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/(2*Sqrt[2]*a^(15/4)*b^(1/4)) - ((b*c - 
 a*d)^3*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/(2*Sqrt[2]*a^(15/4) 
*b^(1/4)) + ((b*c - a*d)^3*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + 
 Sqrt[b]*x])/(4*Sqrt[2]*a^(15/4)*b^(1/4)) - ((b*c - a*d)^3*Log[Sqrt[a] + S 
qrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(4*Sqrt[2]*a^(15/4)*b^(1/4)))
 

Defintions of rubi rules used

rule 368

Int[((e_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_) 
, x_Symbol] :> With[{k = Denominator[m]}, Simp[k/e   Subst[Int[x^(k*(m + 1) 
 - 1)*(a + b*(x^(k*2)/e^2))^p*(c + d*(x^(k*2)/e^2))^q, x], x, (e*x)^(1/k)], 
 x]] /; FreeQ[{a, b, c, d, e, p, q}, x] && NeQ[b*c - a*d, 0] && FractionQ[m 
] && IntegerQ[p]
 

rule 961

Int[(((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_))/((c_) + (d_.)*(x_)^( 
n_)), x_Symbol] :> Int[ExpandIntegrand[(e*x)^m*((a + b*x^n)^p/(c + d*x^n)), 
 x], x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] 
&& IGtQ[p, 0] && (IntegerQ[m] || IGtQ[2*(m + 1), 0] ||  !RationalQ[m])
 

rule 2009

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

Maple [A] (verified)

Time = 0.50 (sec) , antiderivative size = 205, normalized size of antiderivative = 0.85

method	result	size
derivativedivides	\(-\frac {2 c^{3}}{11 a \,x^{\frac {11}{2}}}-\frac {2 c \left (3 a^{2} d^{2}-3 a b c d +b^{2} c^{2}\right )}{3 a^{3} x^{\frac {3}{2}}}-\frac {2 c^{2} \left (3 a d -b c \right )}{7 a^{2} x^{\frac {7}{2}}}+\frac {\left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) \left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x +\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}{x -\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{4 a^{4}}\)	\(205\)
default	\(-\frac {2 c^{3}}{11 a \,x^{\frac {11}{2}}}-\frac {2 c \left (3 a^{2} d^{2}-3 a b c d +b^{2} c^{2}\right )}{3 a^{3} x^{\frac {3}{2}}}-\frac {2 c^{2} \left (3 a d -b c \right )}{7 a^{2} x^{\frac {7}{2}}}+\frac {\left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) \left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x +\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}{x -\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{4 a^{4}}\)	\(205\)
risch	\(-\frac {2 \left (231 a^{2} d^{2} x^{4}-231 a b c d \,x^{4}+77 b^{2} c^{2} x^{4}+99 a^{2} c d \,x^{2}-33 a b \,c^{2} x^{2}+21 a^{2} c^{2}\right ) c}{231 a^{3} x^{\frac {11}{2}}}+\frac {\left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) \left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x +\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}{x -\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{4 a^{4}}\)	\(212\)

Input:

int((d*x^2+c)^3/x^(13/2)/(b*x^2+a),x,method=_RETURNVERBOSE)
 

Output:

-2/11*c^3/a/x^(11/2)-2/3*c*(3*a^2*d^2-3*a*b*c*d+b^2*c^2)/a^3/x^(3/2)-2/7*c 
^2*(3*a*d-b*c)/a^2/x^(7/2)+1/4*(a^3*d^3-3*a^2*b*c*d^2+3*a*b^2*c^2*d-b^3*c^ 
3)/a^4*(a/b)^(1/4)*2^(1/2)*(ln((x+(a/b)^(1/4)*x^(1/2)*2^(1/2)+(a/b)^(1/2)) 
/(x-(a/b)^(1/4)*x^(1/2)*2^(1/2)+(a/b)^(1/2)))+2*arctan(2^(1/2)/(a/b)^(1/4) 
*x^(1/2)+1)+2*arctan(2^(1/2)/(a/b)^(1/4)*x^(1/2)-1))
 

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.12 (sec) , antiderivative size = 1665, normalized size of antiderivative = 6.94 \[ \int \frac {\left (c+d x^2\right )^3}{x^{13/2} \left (a+b x^2\right )} \, dx=\text {Too large to display} \] Input:

integrate((d*x^2+c)^3/x^(13/2)/(b*x^2+a),x, algorithm="fricas")
 

Output:

1/462*(231*a^3*x^6*(-(b^12*c^12 - 12*a*b^11*c^11*d + 66*a^2*b^10*c^10*d^2 
- 220*a^3*b^9*c^9*d^3 + 495*a^4*b^8*c^8*d^4 - 792*a^5*b^7*c^7*d^5 + 924*a^ 
6*b^6*c^6*d^6 - 792*a^7*b^5*c^5*d^7 + 495*a^8*b^4*c^4*d^8 - 220*a^9*b^3*c^ 
3*d^9 + 66*a^10*b^2*c^2*d^10 - 12*a^11*b*c*d^11 + a^12*d^12)/(a^15*b))^(1/ 
4)*log(a^4*(-(b^12*c^12 - 12*a*b^11*c^11*d + 66*a^2*b^10*c^10*d^2 - 220*a^ 
3*b^9*c^9*d^3 + 495*a^4*b^8*c^8*d^4 - 792*a^5*b^7*c^7*d^5 + 924*a^6*b^6*c^ 
6*d^6 - 792*a^7*b^5*c^5*d^7 + 495*a^8*b^4*c^4*d^8 - 220*a^9*b^3*c^3*d^9 + 
66*a^10*b^2*c^2*d^10 - 12*a^11*b*c*d^11 + a^12*d^12)/(a^15*b))^(1/4) - (b^ 
3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3)*sqrt(x)) + 231*I*a^3*x^6* 
(-(b^12*c^12 - 12*a*b^11*c^11*d + 66*a^2*b^10*c^10*d^2 - 220*a^3*b^9*c^9*d 
^3 + 495*a^4*b^8*c^8*d^4 - 792*a^5*b^7*c^7*d^5 + 924*a^6*b^6*c^6*d^6 - 792 
*a^7*b^5*c^5*d^7 + 495*a^8*b^4*c^4*d^8 - 220*a^9*b^3*c^3*d^9 + 66*a^10*b^2 
*c^2*d^10 - 12*a^11*b*c*d^11 + a^12*d^12)/(a^15*b))^(1/4)*log(I*a^4*(-(b^1 
2*c^12 - 12*a*b^11*c^11*d + 66*a^2*b^10*c^10*d^2 - 220*a^3*b^9*c^9*d^3 + 4 
95*a^4*b^8*c^8*d^4 - 792*a^5*b^7*c^7*d^5 + 924*a^6*b^6*c^6*d^6 - 792*a^7*b 
^5*c^5*d^7 + 495*a^8*b^4*c^4*d^8 - 220*a^9*b^3*c^3*d^9 + 66*a^10*b^2*c^2*d 
^10 - 12*a^11*b*c*d^11 + a^12*d^12)/(a^15*b))^(1/4) - (b^3*c^3 - 3*a*b^2*c 
^2*d + 3*a^2*b*c*d^2 - a^3*d^3)*sqrt(x)) - 231*I*a^3*x^6*(-(b^12*c^12 - 12 
*a*b^11*c^11*d + 66*a^2*b^10*c^10*d^2 - 220*a^3*b^9*c^9*d^3 + 495*a^4*b^8* 
c^8*d^4 - 792*a^5*b^7*c^7*d^5 + 924*a^6*b^6*c^6*d^6 - 792*a^7*b^5*c^5*d...
 

Sympy [F(-1)]

Timed out. \[ \int \frac {\left (c+d x^2\right )^3}{x^{13/2} \left (a+b x^2\right )} \, dx=\text {Timed out} \] Input:

integrate((d*x**2+c)**3/x**(13/2)/(b*x**2+a),x)
 

Output:

Timed out
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 389 vs. \(2 (183) = 366\).

Time = 0.15 (sec) , antiderivative size = 389, normalized size of antiderivative = 1.62 \[ \int \frac {\left (c+d x^2\right )^3}{x^{13/2} \left (a+b x^2\right )} \, dx=-\frac {\frac {2 \, \sqrt {2} {\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} + 2 \, \sqrt {b} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b}}}\right )}{\sqrt {a} \sqrt {\sqrt {a} \sqrt {b}}} + \frac {2 \, \sqrt {2} {\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} - 2 \, \sqrt {b} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b}}}\right )}{\sqrt {a} \sqrt {\sqrt {a} \sqrt {b}}} + \frac {\sqrt {2} {\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} \log \left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} \sqrt {x} + \sqrt {b} x + \sqrt {a}\right )}{a^{\frac {3}{4}} b^{\frac {1}{4}}} - \frac {\sqrt {2} {\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} \log \left (-\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} \sqrt {x} + \sqrt {b} x + \sqrt {a}\right )}{a^{\frac {3}{4}} b^{\frac {1}{4}}}}{4 \, a^{3}} - \frac {2 \, {\left (21 \, a^{2} c^{3} + 77 \, {\left (b^{2} c^{3} - 3 \, a b c^{2} d + 3 \, a^{2} c d^{2}\right )} x^{4} - 33 \, {\left (a b c^{3} - 3 \, a^{2} c^{2} d\right )} x^{2}\right )}}{231 \, a^{3} x^{\frac {11}{2}}} \] Input:

integrate((d*x^2+c)^3/x^(13/2)/(b*x^2+a),x, algorithm="maxima")
 

Output:

-1/4*(2*sqrt(2)*(b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3)*arctan 
(1/2*sqrt(2)*(sqrt(2)*a^(1/4)*b^(1/4) + 2*sqrt(b)*sqrt(x))/sqrt(sqrt(a)*sq 
rt(b)))/(sqrt(a)*sqrt(sqrt(a)*sqrt(b))) + 2*sqrt(2)*(b^3*c^3 - 3*a*b^2*c^2 
*d + 3*a^2*b*c*d^2 - a^3*d^3)*arctan(-1/2*sqrt(2)*(sqrt(2)*a^(1/4)*b^(1/4) 
 - 2*sqrt(b)*sqrt(x))/sqrt(sqrt(a)*sqrt(b)))/(sqrt(a)*sqrt(sqrt(a)*sqrt(b) 
)) + sqrt(2)*(b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3)*log(sqrt( 
2)*a^(1/4)*b^(1/4)*sqrt(x) + sqrt(b)*x + sqrt(a))/(a^(3/4)*b^(1/4)) - sqrt 
(2)*(b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3)*log(-sqrt(2)*a^(1/ 
4)*b^(1/4)*sqrt(x) + sqrt(b)*x + sqrt(a))/(a^(3/4)*b^(1/4)))/a^3 - 2/231*( 
21*a^2*c^3 + 77*(b^2*c^3 - 3*a*b*c^2*d + 3*a^2*c*d^2)*x^4 - 33*(a*b*c^3 - 
3*a^2*c^2*d)*x^2)/(a^3*x^(11/2))
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 483 vs. \(2 (183) = 366\).

Time = 0.14 (sec) , antiderivative size = 483, normalized size of antiderivative = 2.01 \[ \int \frac {\left (c+d x^2\right )^3}{x^{13/2} \left (a+b x^2\right )} \, dx=-\frac {\sqrt {2} {\left (\left (a b^{3}\right )^{\frac {1}{4}} b^{3} c^{3} - 3 \, \left (a b^{3}\right )^{\frac {1}{4}} a b^{2} c^{2} d + 3 \, \left (a b^{3}\right )^{\frac {1}{4}} a^{2} b c d^{2} - \left (a b^{3}\right )^{\frac {1}{4}} a^{3} d^{3}\right )} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a}{b}\right )^{\frac {1}{4}} + 2 \, \sqrt {x}\right )}}{2 \, \left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{2 \, a^{4} b} - \frac {\sqrt {2} {\left (\left (a b^{3}\right )^{\frac {1}{4}} b^{3} c^{3} - 3 \, \left (a b^{3}\right )^{\frac {1}{4}} a b^{2} c^{2} d + 3 \, \left (a b^{3}\right )^{\frac {1}{4}} a^{2} b c d^{2} - \left (a b^{3}\right )^{\frac {1}{4}} a^{3} d^{3}\right )} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a}{b}\right )^{\frac {1}{4}} - 2 \, \sqrt {x}\right )}}{2 \, \left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{2 \, a^{4} b} - \frac {\sqrt {2} {\left (\left (a b^{3}\right )^{\frac {1}{4}} b^{3} c^{3} - 3 \, \left (a b^{3}\right )^{\frac {1}{4}} a b^{2} c^{2} d + 3 \, \left (a b^{3}\right )^{\frac {1}{4}} a^{2} b c d^{2} - \left (a b^{3}\right )^{\frac {1}{4}} a^{3} d^{3}\right )} \log \left (\sqrt {2} \sqrt {x} \left (\frac {a}{b}\right )^{\frac {1}{4}} + x + \sqrt {\frac {a}{b}}\right )}{4 \, a^{4} b} + \frac {\sqrt {2} {\left (\left (a b^{3}\right )^{\frac {1}{4}} b^{3} c^{3} - 3 \, \left (a b^{3}\right )^{\frac {1}{4}} a b^{2} c^{2} d + 3 \, \left (a b^{3}\right )^{\frac {1}{4}} a^{2} b c d^{2} - \left (a b^{3}\right )^{\frac {1}{4}} a^{3} d^{3}\right )} \log \left (-\sqrt {2} \sqrt {x} \left (\frac {a}{b}\right )^{\frac {1}{4}} + x + \sqrt {\frac {a}{b}}\right )}{4 \, a^{4} b} - \frac {2 \, {\left (77 \, b^{2} c^{3} x^{4} - 231 \, a b c^{2} d x^{4} + 231 \, a^{2} c d^{2} x^{4} - 33 \, a b c^{3} x^{2} + 99 \, a^{2} c^{2} d x^{2} + 21 \, a^{2} c^{3}\right )}}{231 \, a^{3} x^{\frac {11}{2}}} \] Input:

integrate((d*x^2+c)^3/x^(13/2)/(b*x^2+a),x, algorithm="giac")
 

Output:

-1/2*sqrt(2)*((a*b^3)^(1/4)*b^3*c^3 - 3*(a*b^3)^(1/4)*a*b^2*c^2*d + 3*(a*b 
^3)^(1/4)*a^2*b*c*d^2 - (a*b^3)^(1/4)*a^3*d^3)*arctan(1/2*sqrt(2)*(sqrt(2) 
*(a/b)^(1/4) + 2*sqrt(x))/(a/b)^(1/4))/(a^4*b) - 1/2*sqrt(2)*((a*b^3)^(1/4 
)*b^3*c^3 - 3*(a*b^3)^(1/4)*a*b^2*c^2*d + 3*(a*b^3)^(1/4)*a^2*b*c*d^2 - (a 
*b^3)^(1/4)*a^3*d^3)*arctan(-1/2*sqrt(2)*(sqrt(2)*(a/b)^(1/4) - 2*sqrt(x)) 
/(a/b)^(1/4))/(a^4*b) - 1/4*sqrt(2)*((a*b^3)^(1/4)*b^3*c^3 - 3*(a*b^3)^(1/ 
4)*a*b^2*c^2*d + 3*(a*b^3)^(1/4)*a^2*b*c*d^2 - (a*b^3)^(1/4)*a^3*d^3)*log( 
sqrt(2)*sqrt(x)*(a/b)^(1/4) + x + sqrt(a/b))/(a^4*b) + 1/4*sqrt(2)*((a*b^3 
)^(1/4)*b^3*c^3 - 3*(a*b^3)^(1/4)*a*b^2*c^2*d + 3*(a*b^3)^(1/4)*a^2*b*c*d^ 
2 - (a*b^3)^(1/4)*a^3*d^3)*log(-sqrt(2)*sqrt(x)*(a/b)^(1/4) + x + sqrt(a/b 
))/(a^4*b) - 2/231*(77*b^2*c^3*x^4 - 231*a*b*c^2*d*x^4 + 231*a^2*c*d^2*x^4 
 - 33*a*b*c^3*x^2 + 99*a^2*c^2*d*x^2 + 21*a^2*c^3)/(a^3*x^(11/2))
 

Mupad [B] (verification not implemented)

Time = 0.40 (sec) , antiderivative size = 1580, normalized size of antiderivative = 6.58 \[ \int \frac {\left (c+d x^2\right )^3}{x^{13/2} \left (a+b x^2\right )} \, dx=\text {Too large to display} \] Input:

int((c + d*x^2)^3/(x^(13/2)*(a + b*x^2)),x)
 

Output:

- ((2*c^3)/(11*a) + (2*c^2*x^2*(3*a*d - b*c))/(7*a^2) + (2*c*x^4*(3*a^2*d^ 
2 + b^2*c^2 - 3*a*b*c*d))/(3*a^3))/x^(11/2) - (atan(((((x^(1/2)*(16*a^9*b^ 
9*c^6 + 16*a^15*b^3*d^6 - 96*a^10*b^8*c^5*d - 96*a^14*b^4*c*d^5 + 240*a^11 
*b^7*c^4*d^2 - 320*a^12*b^6*c^3*d^3 + 240*a^13*b^5*c^2*d^4))/2 - ((a*d - b 
*c)^3*(16*a^13*b^6*c^3 - 16*a^16*b^3*d^3 - 48*a^14*b^5*c^2*d + 48*a^15*b^4 
*c*d^2))/(2*(-a)^(15/4)*b^(1/4)))*(a*d - b*c)^3*1i)/((-a)^(15/4)*b^(1/4)) 
+ (((x^(1/2)*(16*a^9*b^9*c^6 + 16*a^15*b^3*d^6 - 96*a^10*b^8*c^5*d - 96*a^ 
14*b^4*c*d^5 + 240*a^11*b^7*c^4*d^2 - 320*a^12*b^6*c^3*d^3 + 240*a^13*b^5* 
c^2*d^4))/2 + ((a*d - b*c)^3*(16*a^13*b^6*c^3 - 16*a^16*b^3*d^3 - 48*a^14* 
b^5*c^2*d + 48*a^15*b^4*c*d^2))/(2*(-a)^(15/4)*b^(1/4)))*(a*d - b*c)^3*1i) 
/((-a)^(15/4)*b^(1/4)))/((((x^(1/2)*(16*a^9*b^9*c^6 + 16*a^15*b^3*d^6 - 96 
*a^10*b^8*c^5*d - 96*a^14*b^4*c*d^5 + 240*a^11*b^7*c^4*d^2 - 320*a^12*b^6* 
c^3*d^3 + 240*a^13*b^5*c^2*d^4))/2 - ((a*d - b*c)^3*(16*a^13*b^6*c^3 - 16* 
a^16*b^3*d^3 - 48*a^14*b^5*c^2*d + 48*a^15*b^4*c*d^2))/(2*(-a)^(15/4)*b^(1 
/4)))*(a*d - b*c)^3)/((-a)^(15/4)*b^(1/4)) - (((x^(1/2)*(16*a^9*b^9*c^6 + 
16*a^15*b^3*d^6 - 96*a^10*b^8*c^5*d - 96*a^14*b^4*c*d^5 + 240*a^11*b^7*c^4 
*d^2 - 320*a^12*b^6*c^3*d^3 + 240*a^13*b^5*c^2*d^4))/2 + ((a*d - b*c)^3*(1 
6*a^13*b^6*c^3 - 16*a^16*b^3*d^3 - 48*a^14*b^5*c^2*d + 48*a^15*b^4*c*d^2)) 
/(2*(-a)^(15/4)*b^(1/4)))*(a*d - b*c)^3)/((-a)^(15/4)*b^(1/4))))*(a*d - b* 
c)^3*1i)/((-a)^(15/4)*b^(1/4)) - (atan(((((x^(1/2)*(16*a^9*b^9*c^6 + 16...
 

Reduce [B] (verification not implemented)

Time = 0.20 (sec) , antiderivative size = 768, normalized size of antiderivative = 3.20 \[ \int \frac {\left (c+d x^2\right )^3}{x^{13/2} \left (a+b x^2\right )} \, dx =\text {Too large to display} \] Input:

int((d*x^2+c)^3/x^(13/2)/(b*x^2+a),x)
 

Output:

( - 462*sqrt(x)*b**(3/4)*a**(1/4)*sqrt(2)*atan((b**(1/4)*a**(1/4)*sqrt(2) 
- 2*sqrt(x)*sqrt(b))/(b**(1/4)*a**(1/4)*sqrt(2)))*a**3*d**3*x**5 + 1386*sq 
rt(x)*b**(3/4)*a**(1/4)*sqrt(2)*atan((b**(1/4)*a**(1/4)*sqrt(2) - 2*sqrt(x 
)*sqrt(b))/(b**(1/4)*a**(1/4)*sqrt(2)))*a**2*b*c*d**2*x**5 - 1386*sqrt(x)* 
b**(3/4)*a**(1/4)*sqrt(2)*atan((b**(1/4)*a**(1/4)*sqrt(2) - 2*sqrt(x)*sqrt 
(b))/(b**(1/4)*a**(1/4)*sqrt(2)))*a*b**2*c**2*d*x**5 + 462*sqrt(x)*b**(3/4 
)*a**(1/4)*sqrt(2)*atan((b**(1/4)*a**(1/4)*sqrt(2) - 2*sqrt(x)*sqrt(b))/(b 
**(1/4)*a**(1/4)*sqrt(2)))*b**3*c**3*x**5 + 462*sqrt(x)*b**(3/4)*a**(1/4)* 
sqrt(2)*atan((b**(1/4)*a**(1/4)*sqrt(2) + 2*sqrt(x)*sqrt(b))/(b**(1/4)*a** 
(1/4)*sqrt(2)))*a**3*d**3*x**5 - 1386*sqrt(x)*b**(3/4)*a**(1/4)*sqrt(2)*at 
an((b**(1/4)*a**(1/4)*sqrt(2) + 2*sqrt(x)*sqrt(b))/(b**(1/4)*a**(1/4)*sqrt 
(2)))*a**2*b*c*d**2*x**5 + 1386*sqrt(x)*b**(3/4)*a**(1/4)*sqrt(2)*atan((b* 
*(1/4)*a**(1/4)*sqrt(2) + 2*sqrt(x)*sqrt(b))/(b**(1/4)*a**(1/4)*sqrt(2)))* 
a*b**2*c**2*d*x**5 - 462*sqrt(x)*b**(3/4)*a**(1/4)*sqrt(2)*atan((b**(1/4)* 
a**(1/4)*sqrt(2) + 2*sqrt(x)*sqrt(b))/(b**(1/4)*a**(1/4)*sqrt(2)))*b**3*c* 
*3*x**5 - 231*sqrt(x)*b**(3/4)*a**(1/4)*sqrt(2)*log( - sqrt(x)*b**(1/4)*a* 
*(1/4)*sqrt(2) + sqrt(a) + sqrt(b)*x)*a**3*d**3*x**5 + 693*sqrt(x)*b**(3/4 
)*a**(1/4)*sqrt(2)*log( - sqrt(x)*b**(1/4)*a**(1/4)*sqrt(2) + sqrt(a) + sq 
rt(b)*x)*a**2*b*c*d**2*x**5 - 693*sqrt(x)*b**(3/4)*a**(1/4)*sqrt(2)*log( - 
 sqrt(x)*b**(1/4)*a**(1/4)*sqrt(2) + sqrt(a) + sqrt(b)*x)*a*b**2*c**2*d...