\(\int \frac {(c+d x^2)^3}{x^{9/2} (a+b x^2)} \, dx\) [770]

Optimal result

Integrand size = 24, antiderivative size = 217 \[ \int \frac {\left (c+d x^2\right )^3}{x^{9/2} \left (a+b x^2\right )} \, dx=-\frac {2 c^3}{7 a x^{7/2}}+\frac {2 c^2 (b c-3 a d)}{3 a^2 x^{3/2}}+\frac {2 d^3 \sqrt {x}}{b}-\frac {(b c-a d)^3 \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} a^{11/4} b^{5/4}}+\frac {(b c-a d)^3 \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} a^{11/4} b^{5/4}}+\frac {(b c-a d)^3 \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}{\sqrt {a}+\sqrt {b} x}\right )}{\sqrt {2} a^{11/4} b^{5/4}} \] Output:

-2/7*c^3/a/x^(7/2)+2/3*c^2*(-3*a*d+b*c)/a^2/x^(3/2)+2*d^3*x^(1/2)/b-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^(11/4)/b^(5 
/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^( 
11/4)/b^(5/4)+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^(11/4)/b^(5/4)
 

Mathematica [A] (verified)

Time = 0.23 (sec) , antiderivative size = 177, normalized size of antiderivative = 0.82 \[ \int \frac {\left (c+d x^2\right )^3}{x^{9/2} \left (a+b x^2\right )} \, dx=\frac {\frac {4 a^{3/4} \sqrt [4]{b} \left (7 b^2 c^3 x^2+21 a^2 d^3 x^4-3 a b c^2 \left (c+7 d x^2\right )\right )}{x^{7/2}}+21 \sqrt {2} (-b c+a d)^3 \arctan \left (\frac {\sqrt {a}-\sqrt {b} x}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}\right )+21 \sqrt {2} (b c-a d)^3 \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}{\sqrt {a}+\sqrt {b} x}\right )}{42 a^{11/4} b^{5/4}} \] Input:

Integrate[(c + d*x^2)^3/(x^(9/2)*(a + b*x^2)),x]
 

Output:

((4*a^(3/4)*b^(1/4)*(7*b^2*c^3*x^2 + 21*a^2*d^3*x^4 - 3*a*b*c^2*(c + 7*d*x 
^2)))/x^(7/2) + 21*Sqrt[2]*(-(b*c) + a*d)^3*ArcTan[(Sqrt[a] - Sqrt[b]*x)/( 
Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])] + 21*Sqrt[2]*(b*c - a*d)^3*ArcTanh[(Sqrt 
[2]*a^(1/4)*b^(1/4)*Sqrt[x])/(Sqrt[a] + Sqrt[b]*x)])/(42*a^(11/4)*b^(5/4))
 

Rubi [A] (verified)

Time = 0.43 (sec) , antiderivative size = 289, normalized size of antiderivative = 1.33, 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.

\(\displaystyle \int \frac {\left (c+d x^2\right )^3}{x^{9/2} \left (a+b x^2\right )} \, dx\)

\(\Big \downarrow \) 368

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

\(\Big \downarrow \) 961

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

\(\Big \downarrow \) 2009

\(\displaystyle 2 \left (-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right ) (b c-a d)^3}{2 \sqrt {2} a^{11/4} b^{5/4}}+\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right ) (b c-a d)^3}{2 \sqrt {2} a^{11/4} b^{5/4}}-\frac {(b c-a d)^3 \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{4 \sqrt {2} a^{11/4} b^{5/4}}+\frac {(b c-a d)^3 \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{4 \sqrt {2} a^{11/4} b^{5/4}}+\frac {c^2 (b c-3 a d)}{3 a^2 x^{3/2}}-\frac {c^3}{7 a x^{7/2}}+\frac {d^3 \sqrt {x}}{b}\right )\)

Input:

Int[(c + d*x^2)^3/(x^(9/2)*(a + b*x^2)),x]
 

Output:

2*(-1/7*c^3/(a*x^(7/2)) + (c^2*(b*c - 3*a*d))/(3*a^2*x^(3/2)) + (d^3*Sqrt[ 
x])/b - ((b*c - a*d)^3*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/(2*S 
qrt[2]*a^(11/4)*b^(5/4)) + ((b*c - a*d)^3*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt 
[x])/a^(1/4)])/(2*Sqrt[2]*a^(11/4)*b^(5/4)) - ((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^(11/4)*b^(5/4) 
) + ((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^(11/4)*b^(5/4)))
 

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 = 0.52 (sec) , antiderivative size = 188, normalized size of antiderivative = 0.87

method	result	size
derivativedivides	\(\frac {2 d^{3} \sqrt {x}}{b}+\frac {\left (-a^{3} d^{3}+3 a^{2} b c \,d^{2}-3 a \,b^{2} c^{2} d +b^{3} c^{3}\right ) \left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x +\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}{x -\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{4 a^{3} b}-\frac {2 c^{3}}{7 a \,x^{\frac {7}{2}}}-\frac {2 c^{2} \left (3 a d -b c \right )}{3 a^{2} x^{\frac {3}{2}}}\)	\(188\)
default	\(\frac {2 d^{3} \sqrt {x}}{b}+\frac {\left (-a^{3} d^{3}+3 a^{2} b c \,d^{2}-3 a \,b^{2} c^{2} d +b^{3} c^{3}\right ) \left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x +\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}{x -\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{4 a^{3} b}-\frac {2 c^{3}}{7 a \,x^{\frac {7}{2}}}-\frac {2 c^{2} \left (3 a d -b c \right )}{3 a^{2} x^{\frac {3}{2}}}\)	\(188\)
risch	\(\frac {2 a^{2} d^{3} x^{4}-2 a b \,c^{2} d \,x^{2}+\frac {2}{3} b^{2} c^{3} x^{2}-\frac {2}{7} a b \,c^{3}}{b \,x^{\frac {7}{2}} a^{2}}-\frac {\left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) \left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x +\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}{x -\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{4 b \,a^{3}}\)	\(198\)

Input:

int((d*x^2+c)^3/x^(9/2)/(b*x^2+a),x,method=_RETURNVERBOSE)
 

Output:

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

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.11 (sec) , antiderivative size = 1656, normalized size of antiderivative = 7.63 \[ \int \frac {\left (c+d x^2\right )^3}{x^{9/2} \left (a+b x^2\right )} \, dx=\text {Too large to display} \] Input:

integrate((d*x^2+c)^3/x^(9/2)/(b*x^2+a),x, algorithm="fricas")
 

Output:

-1/42*(21*a^2*b*x^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^11*b^5))^ 
(1/4)*log(a^3*b*(-(b^12*c^12 - 12*a*b^11*c^11*d + 66*a^2*b^10*c^10*d^2 - 2 
20*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^11*b^5))^(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)) + 21*I*a^ 
2*b*x^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^11*b^5))^(1/4)*log(I* 
a^3*b*(-(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^11*b^5))^(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)) - 21*I*a^2*b*x^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 - 79...
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 607 vs. \(2 (206) = 412\).

Time = 58.71 (sec) , antiderivative size = 607, normalized size of antiderivative = 2.80 \[ \int \frac {\left (c+d x^2\right )^3}{x^{9/2} \left (a+b x^2\right )} \, dx =\text {Too large to display} \] Input:

integrate((d*x**2+c)**3/x**(9/2)/(b*x**2+a),x)
 

Output:

Piecewise((zoo*(-2*c**3/(11*x**(11/2)) - 6*c**2*d/(7*x**(7/2)) - 2*c*d**2/ 
x**(3/2) + 2*d**3*sqrt(x)), Eq(a, 0) & Eq(b, 0)), ((-2*c**3/(11*x**(11/2)) 
 - 6*c**2*d/(7*x**(7/2)) - 2*c*d**2/x**(3/2) + 2*d**3*sqrt(x))/b, Eq(a, 0) 
), ((-2*c**3/(7*x**(7/2)) - 2*c**2*d/x**(3/2) + 6*c*d**2*sqrt(x) + 2*d**3* 
x**(5/2)/5)/a, Eq(b, 0)), (2*d**3*sqrt(x)/b + d**3*(-a/b)**(1/4)*log(sqrt( 
x) - (-a/b)**(1/4))/(2*b) - d**3*(-a/b)**(1/4)*log(sqrt(x) + (-a/b)**(1/4) 
)/(2*b) - d**3*(-a/b)**(1/4)*atan(sqrt(x)/(-a/b)**(1/4))/b - 2*c**3/(7*a*x 
**(7/2)) - 2*c**2*d/(a*x**(3/2)) - 3*c*d**2*(-a/b)**(1/4)*log(sqrt(x) - (- 
a/b)**(1/4))/(2*a) + 3*c*d**2*(-a/b)**(1/4)*log(sqrt(x) + (-a/b)**(1/4))/( 
2*a) + 3*c*d**2*(-a/b)**(1/4)*atan(sqrt(x)/(-a/b)**(1/4))/a + 2*b*c**3/(3* 
a**2*x**(3/2)) + 3*b*c**2*d*(-a/b)**(1/4)*log(sqrt(x) - (-a/b)**(1/4))/(2* 
a**2) - 3*b*c**2*d*(-a/b)**(1/4)*log(sqrt(x) + (-a/b)**(1/4))/(2*a**2) - 3 
*b*c**2*d*(-a/b)**(1/4)*atan(sqrt(x)/(-a/b)**(1/4))/a**2 - b**2*c**3*(-a/b 
)**(1/4)*log(sqrt(x) - (-a/b)**(1/4))/(2*a**3) + b**2*c**3*(-a/b)**(1/4)*l 
og(sqrt(x) + (-a/b)**(1/4))/(2*a**3) + b**2*c**3*(-a/b)**(1/4)*atan(sqrt(x 
)/(-a/b)**(1/4))/a**3, True))
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 368 vs. \(2 (163) = 326\).

Time = 0.15 (sec) , antiderivative size = 368, normalized size of antiderivative = 1.70 \[ \int \frac {\left (c+d x^2\right )^3}{x^{9/2} \left (a+b x^2\right )} \, dx=\frac {2 \, d^{3} \sqrt {x}}{b} + \frac {\frac {2 \, \sqrt {2} {\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} + 2 \, \sqrt {b} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b}}}\right )}{\sqrt {a} \sqrt {\sqrt {a} \sqrt {b}}} + \frac {2 \, \sqrt {2} {\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} - 2 \, \sqrt {b} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b}}}\right )}{\sqrt {a} \sqrt {\sqrt {a} \sqrt {b}}} + \frac {\sqrt {2} {\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} \log \left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} \sqrt {x} + \sqrt {b} x + \sqrt {a}\right )}{a^{\frac {3}{4}} b^{\frac {1}{4}}} - \frac {\sqrt {2} {\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} \log \left (-\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} \sqrt {x} + \sqrt {b} x + \sqrt {a}\right )}{a^{\frac {3}{4}} b^{\frac {1}{4}}}}{4 \, a^{2} b} - \frac {2 \, {\left (3 \, a c^{3} - 7 \, {\left (b c^{3} - 3 \, a c^{2} d\right )} x^{2}\right )}}{21 \, a^{2} x^{\frac {7}{2}}} \] Input:

integrate((d*x^2+c)^3/x^(9/2)/(b*x^2+a),x, algorithm="maxima")
 

Output:

2*d^3*sqrt(x)/b + 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)*sqrt(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)*sqr 
t(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)*l 
og(-sqrt(2)*a^(1/4)*b^(1/4)*sqrt(x) + sqrt(b)*x + sqrt(a))/(a^(3/4)*b^(1/4 
)))/(a^2*b) - 2/21*(3*a*c^3 - 7*(b*c^3 - 3*a*c^2*d)*x^2)/(a^2*x^(7/2))
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 455 vs. \(2 (163) = 326\).

Time = 0.14 (sec) , antiderivative size = 455, normalized size of antiderivative = 2.10 \[ \int \frac {\left (c+d x^2\right )^3}{x^{9/2} \left (a+b x^2\right )} \, dx=\frac {2 \, d^{3} \sqrt {x}}{b} + \frac {\sqrt {2} {\left (\left (a b^{3}\right )^{\frac {1}{4}} b^{3} c^{3} - 3 \, \left (a b^{3}\right )^{\frac {1}{4}} a b^{2} c^{2} d + 3 \, \left (a b^{3}\right )^{\frac {1}{4}} a^{2} b c d^{2} - \left (a b^{3}\right )^{\frac {1}{4}} a^{3} d^{3}\right )} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a}{b}\right )^{\frac {1}{4}} + 2 \, \sqrt {x}\right )}}{2 \, \left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{2 \, a^{3} b^{2}} + \frac {\sqrt {2} {\left (\left (a b^{3}\right )^{\frac {1}{4}} b^{3} c^{3} - 3 \, \left (a b^{3}\right )^{\frac {1}{4}} a b^{2} c^{2} d + 3 \, \left (a b^{3}\right )^{\frac {1}{4}} a^{2} b c d^{2} - \left (a b^{3}\right )^{\frac {1}{4}} a^{3} d^{3}\right )} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a}{b}\right )^{\frac {1}{4}} - 2 \, \sqrt {x}\right )}}{2 \, \left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{2 \, a^{3} b^{2}} + \frac {\sqrt {2} {\left (\left (a b^{3}\right )^{\frac {1}{4}} b^{3} c^{3} - 3 \, \left (a b^{3}\right )^{\frac {1}{4}} a b^{2} c^{2} d + 3 \, \left (a b^{3}\right )^{\frac {1}{4}} a^{2} b c d^{2} - \left (a b^{3}\right )^{\frac {1}{4}} a^{3} d^{3}\right )} \log \left (\sqrt {2} \sqrt {x} \left (\frac {a}{b}\right )^{\frac {1}{4}} + x + \sqrt {\frac {a}{b}}\right )}{4 \, a^{3} b^{2}} - \frac {\sqrt {2} {\left (\left (a b^{3}\right )^{\frac {1}{4}} b^{3} c^{3} - 3 \, \left (a b^{3}\right )^{\frac {1}{4}} a b^{2} c^{2} d + 3 \, \left (a b^{3}\right )^{\frac {1}{4}} a^{2} b c d^{2} - \left (a b^{3}\right )^{\frac {1}{4}} a^{3} d^{3}\right )} \log \left (-\sqrt {2} \sqrt {x} \left (\frac {a}{b}\right )^{\frac {1}{4}} + x + \sqrt {\frac {a}{b}}\right )}{4 \, a^{3} b^{2}} + \frac {2 \, {\left (7 \, b c^{3} x^{2} - 21 \, a c^{2} d x^{2} - 3 \, a c^{3}\right )}}{21 \, a^{2} x^{\frac {7}{2}}} \] Input:

integrate((d*x^2+c)^3/x^(9/2)/(b*x^2+a),x, algorithm="giac")
 

Output:

2*d^3*sqrt(x)/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^3*b^2) + 1/2*sq 
rt(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^3*b^2) + 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^3*b^2 
) - 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^3*b^2) + 2/21*(7*b*c^3*x^2 - 21*a*c^2*d*x^2 - 
 3*a*c^3)/(a^2*x^(7/2))
 

Mupad [B] (verification not implemented)

Time = 0.84 (sec) , antiderivative size = 1564, normalized size of antiderivative = 7.21 \[ \int \frac {\left (c+d x^2\right )^3}{x^{9/2} \left (a+b x^2\right )} \, dx=\text {Too large to display} \] Input:

int((c + d*x^2)^3/(x^(9/2)*(a + b*x^2)),x)
 

Output:

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

Reduce [B] (verification not implemented)

Time = 0.25 (sec) , antiderivative size = 742, normalized size of antiderivative = 3.42 \[ \int \frac {\left (c+d x^2\right )^3}{x^{9/2} \left (a+b x^2\right )} \, dx =\text {Too large to display} \] Input:

int((d*x^2+c)^3/x^(9/2)/(b*x^2+a),x)
 

Output:

(42*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**3 - 126*sqrt(x) 
*b**(3/4)*a**(1/4)*sqrt(2)*atan((b**(1/4)*a**(1/4)*sqrt(2) - 2*sqrt(x)*sqr 
t(b))/(b**(1/4)*a**(1/4)*sqrt(2)))*a**2*b*c*d**2*x**3 + 126*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**3 - 42*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**3 - 42*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)*sq 
rt(2)))*a**3*d**3*x**3 + 126*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** 
2*b*c*d**2*x**3 - 126*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**3 + 42*sqrt(x)*b**(3/4)*a**(1/4)*sqrt(2)*atan((b**(1/4)*a**(1/4)*sq 
rt(2) + 2*sqrt(x)*sqrt(b))/(b**(1/4)*a**(1/4)*sqrt(2)))*b**3*c**3*x**3 + 2 
1*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**3 - 63*sqrt(x)*b**(3/4)*a**(1/4)*sq 
rt(2)*log( - sqrt(x)*b**(1/4)*a**(1/4)*sqrt(2) + sqrt(a) + sqrt(b)*x)*a**2 
*b*c*d**2*x**3 + 63*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*x**3 - 21*sqr...