\(\int \frac {(A+B x) (c+d x)^4}{(a+b x^2)^{5/2}} \, dx\) [187]

Optimal result

Integrand size = 24, antiderivative size = 232 \[ \int \frac {(A+B x) (c+d x)^4}{\left (a+b x^2\right )^{5/2}} \, dx=-\frac {(c+d x)^3 (a (B c+A d)-(A b c-a B d) x)}{3 a b \left (a+b x^2\right )^{3/2}}-\frac {(c+d x) \left (a d \left (A b c^2+8 a B c d+3 a A d^2\right )-2 \left (2 a B d \left (b c^2-a d^2\right )+A b c \left (b c^2+2 a d^2\right )\right ) x\right )}{3 a^2 b^2 \sqrt {a+b x^2}}-\frac {d \left (4 a B d \left (b c^2-2 a d^2\right )+A b c \left (2 b c^2+5 a d^2\right )\right ) \sqrt {a+b x^2}}{3 a^2 b^3}+\frac {d^3 (4 B c+A d) \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{b^{5/2}} \] Output:

-1/3*(d*x+c)^3*(a*(A*d+B*c)-(A*b*c-B*a*d)*x)/a/b/(b*x^2+a)^(3/2)-1/3*(d*x+ 
c)*(a*d*(3*A*a*d^2+A*b*c^2+8*B*a*c*d)-2*(2*a*B*d*(-a*d^2+b*c^2)+A*b*c*(2*a 
*d^2+b*c^2))*x)/a^2/b^2/(b*x^2+a)^(1/2)-1/3*d*(4*a*B*d*(-2*a*d^2+b*c^2)+A* 
b*c*(5*a*d^2+2*b*c^2))*(b*x^2+a)^(1/2)/a^2/b^3+d^3*(A*d+4*B*c)*arctanh(b^( 
1/2)*x/(b*x^2+a)^(1/2))/b^(5/2)
 

Mathematica [A] (verified)

Time = 1.83 (sec) , antiderivative size = 236, normalized size of antiderivative = 1.02 \[ \int \frac {(A+B x) (c+d x)^4}{\left (a+b x^2\right )^{5/2}} \, dx=\frac {8 a^4 B d^4+2 A b^4 c^4 x^3-a^3 b d^2 \left (A d (8 c+3 d x)+12 B \left (c^2+c d x-d^2 x^2\right )\right )+a b^3 c^2 x \left (4 B c d x^2+3 A \left (c^2+2 d^2 x^2\right )\right )-a^2 b^2 \left (4 A d \left (c^3+3 c d^2 x^2+d^3 x^3\right )+B \left (c^4+18 c^2 d^2 x^2+16 c d^3 x^3-3 d^4 x^4\right )\right )-3 a^2 \sqrt {b} d^3 (4 B c+A d) \left (a+b x^2\right )^{3/2} \log \left (-\sqrt {b} x+\sqrt {a+b x^2}\right )}{3 a^2 b^3 \left (a+b x^2\right )^{3/2}} \] Input:

Integrate[((A + B*x)*(c + d*x)^4)/(a + b*x^2)^(5/2),x]
 

Output:

(8*a^4*B*d^4 + 2*A*b^4*c^4*x^3 - a^3*b*d^2*(A*d*(8*c + 3*d*x) + 12*B*(c^2 
+ c*d*x - d^2*x^2)) + a*b^3*c^2*x*(4*B*c*d*x^2 + 3*A*(c^2 + 2*d^2*x^2)) - 
a^2*b^2*(4*A*d*(c^3 + 3*c*d^2*x^2 + d^3*x^3) + B*(c^4 + 18*c^2*d^2*x^2 + 1 
6*c*d^3*x^3 - 3*d^4*x^4)) - 3*a^2*Sqrt[b]*d^3*(4*B*c + A*d)*(a + b*x^2)^(3 
/2)*Log[-(Sqrt[b]*x) + Sqrt[a + b*x^2]])/(3*a^2*b^3*(a + b*x^2)^(3/2))
 

Rubi [A] (verified)

Time = 0.45 (sec) , antiderivative size = 248, normalized size of antiderivative = 1.07, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {684, 684, 27, 455, 224, 219}

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.

\(\displaystyle \int \frac {(A+B x) (c+d x)^4}{\left (a+b x^2\right )^{5/2}} \, dx\)

\(\Big \downarrow \) 684

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

\(\Big \downarrow \) 684

\(\displaystyle \frac {\frac {\int \frac {d \left (3 a^2 d^2 (4 B c+A d)-\left (4 a B d \left (b c^2-2 a d^2\right )+A b c \left (2 b c^2+5 a d^2\right )\right ) x\right )}{\sqrt {b x^2+a}}dx}{a b}-\frac {(c+d x) \left (a d \left (3 a A d^2+8 a B c d+A b c^2\right )-2 x \left (A b c \left (2 a d^2+b c^2\right )+2 a B d \left (b c^2-a d^2\right )\right )\right )}{a b \sqrt {a+b x^2}}}{3 a b}-\frac {(c+d x)^3 (a (A d+B c)-x (A b c-a B d))}{3 a b \left (a+b x^2\right )^{3/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {d \int \frac {3 a^2 d^2 (4 B c+A d)-\left (4 a B d \left (b c^2-2 a d^2\right )+A b c \left (2 b c^2+5 a d^2\right )\right ) x}{\sqrt {b x^2+a}}dx}{a b}-\frac {(c+d x) \left (a d \left (3 a A d^2+8 a B c d+A b c^2\right )-2 x \left (A b c \left (2 a d^2+b c^2\right )+2 a B d \left (b c^2-a d^2\right )\right )\right )}{a b \sqrt {a+b x^2}}}{3 a b}-\frac {(c+d x)^3 (a (A d+B c)-x (A b c-a B d))}{3 a b \left (a+b x^2\right )^{3/2}}\)

\(\Big \downarrow \) 455

\(\displaystyle \frac {\frac {d \left (3 a^2 d^2 (A d+4 B c) \int \frac {1}{\sqrt {b x^2+a}}dx-\frac {\sqrt {a+b x^2} \left (A b c \left (5 a d^2+2 b c^2\right )+4 a B d \left (b c^2-2 a d^2\right )\right )}{b}\right )}{a b}-\frac {(c+d x) \left (a d \left (3 a A d^2+8 a B c d+A b c^2\right )-2 x \left (A b c \left (2 a d^2+b c^2\right )+2 a B d \left (b c^2-a d^2\right )\right )\right )}{a b \sqrt {a+b x^2}}}{3 a b}-\frac {(c+d x)^3 (a (A d+B c)-x (A b c-a B d))}{3 a b \left (a+b x^2\right )^{3/2}}\)

\(\Big \downarrow \) 224

\(\displaystyle \frac {\frac {d \left (3 a^2 d^2 (A d+4 B c) \int \frac {1}{1-\frac {b x^2}{b x^2+a}}d\frac {x}{\sqrt {b x^2+a}}-\frac {\sqrt {a+b x^2} \left (A b c \left (5 a d^2+2 b c^2\right )+4 a B d \left (b c^2-2 a d^2\right )\right )}{b}\right )}{a b}-\frac {(c+d x) \left (a d \left (3 a A d^2+8 a B c d+A b c^2\right )-2 x \left (A b c \left (2 a d^2+b c^2\right )+2 a B d \left (b c^2-a d^2\right )\right )\right )}{a b \sqrt {a+b x^2}}}{3 a b}-\frac {(c+d x)^3 (a (A d+B c)-x (A b c-a B d))}{3 a b \left (a+b x^2\right )^{3/2}}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {\frac {d \left (\frac {3 a^2 d^2 \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right ) (A d+4 B c)}{\sqrt {b}}-\frac {\sqrt {a+b x^2} \left (A b c \left (5 a d^2+2 b c^2\right )+4 a B d \left (b c^2-2 a d^2\right )\right )}{b}\right )}{a b}-\frac {(c+d x) \left (a d \left (3 a A d^2+8 a B c d+A b c^2\right )-2 x \left (A b c \left (2 a d^2+b c^2\right )+2 a B d \left (b c^2-a d^2\right )\right )\right )}{a b \sqrt {a+b x^2}}}{3 a b}-\frac {(c+d x)^3 (a (A d+B c)-x (A b c-a B d))}{3 a b \left (a+b x^2\right )^{3/2}}\)

Input:

Int[((A + B*x)*(c + d*x)^4)/(a + b*x^2)^(5/2),x]
 

Output:

-1/3*((c + d*x)^3*(a*(B*c + A*d) - (A*b*c - a*B*d)*x))/(a*b*(a + b*x^2)^(3 
/2)) + (-(((c + d*x)*(a*d*(A*b*c^2 + 8*a*B*c*d + 3*a*A*d^2) - 2*(2*a*B*d*( 
b*c^2 - a*d^2) + A*b*c*(b*c^2 + 2*a*d^2))*x))/(a*b*Sqrt[a + b*x^2])) + (d* 
(-(((4*a*B*d*(b*c^2 - 2*a*d^2) + A*b*c*(2*b*c^2 + 5*a*d^2))*Sqrt[a + b*x^2 
])/b) + (3*a^2*d^2*(4*B*c + A*d)*ArcTanh[(Sqrt[b]*x)/Sqrt[a + b*x^2]])/Sqr 
t[b]))/(a*b))/(3*a*b)
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 219

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* 
ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt 
Q[a, 0] || LtQ[b, 0])
 

rule 224

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], 
x, x/Sqrt[a + b*x^2]] /; FreeQ[{a, b}, x] &&  !GtQ[a, 0]
 

rule 455

Int[((c_) + (d_.)*(x_))*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[d*(( 
a + b*x^2)^(p + 1)/(2*b*(p + 1))), x] + Simp[c   Int[(a + b*x^2)^p, x], x] 
/; FreeQ[{a, b, c, d, p}, x] &&  !LeQ[p, -1]
 

rule 684

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p 
_.), x_Symbol] :> Simp[(d + e*x)^(m - 1)*(a + c*x^2)^(p + 1)*((a*(e*f + d*g 
) - (c*d*f - a*e*g)*x)/(2*a*c*(p + 1))), x] - Simp[1/(2*a*c*(p + 1))   Int[ 
(d + e*x)^(m - 2)*(a + c*x^2)^(p + 1)*Simp[a*e*(e*f*(m - 1) + d*g*m) - c*d^ 
2*f*(2*p + 3) + e*(a*e*g*m - c*d*f*(m + 2*p + 2))*x, x], x], x] /; FreeQ[{a 
, c, d, e, f, g}, x] && LtQ[p, -1] && GtQ[m, 1] && (EqQ[d, 0] || (EqQ[m, 2] 
 && EqQ[p, -3] && RationalQ[a, c, d, e, f, g]) ||  !ILtQ[m + 2*p + 3, 0])
 

Maple [A] (verified)

Time = 1.33 (sec) , antiderivative size = 310, normalized size of antiderivative = 1.34

method	result	size
default	\(A \,c^{4} \left (\frac {x}{3 a \left (b \,x^{2}+a \right )^{\frac {3}{2}}}+\frac {2 x}{3 a^{2} \sqrt {b \,x^{2}+a}}\right )+d^{3} \left (A d +4 B c \right ) \left (-\frac {x^{3}}{3 b \left (b \,x^{2}+a \right )^{\frac {3}{2}}}+\frac {-\frac {x}{b \sqrt {b \,x^{2}+a}}+\frac {\ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{b^{\frac {3}{2}}}}{b}\right )+2 c \,d^{2} \left (2 A d +3 B c \right ) \left (-\frac {x^{2}}{b \left (b \,x^{2}+a \right )^{\frac {3}{2}}}-\frac {2 a}{3 b^{2} \left (b \,x^{2}+a \right )^{\frac {3}{2}}}\right )+2 c^{2} d \left (3 A d +2 B c \right ) \left (-\frac {x}{2 b \left (b \,x^{2}+a \right )^{\frac {3}{2}}}+\frac {a \left (\frac {x}{3 a \left (b \,x^{2}+a \right )^{\frac {3}{2}}}+\frac {2 x}{3 a^{2} \sqrt {b \,x^{2}+a}}\right )}{2 b}\right )-\frac {c^{3} \left (4 A d +B c \right )}{3 b \left (b \,x^{2}+a \right )^{\frac {3}{2}}}+B \,d^{4} \left (\frac {x^{4}}{b \left (b \,x^{2}+a \right )^{\frac {3}{2}}}-\frac {4 a \left (-\frac {x^{2}}{b \left (b \,x^{2}+a \right )^{\frac {3}{2}}}-\frac {2 a}{3 b^{2} \left (b \,x^{2}+a \right )^{\frac {3}{2}}}\right )}{b}\right )\)	\(310\)
risch	\(\frac {B \,d^{4} \sqrt {b \,x^{2}+a}}{b^{3}}+\frac {\frac {d^{3} \left (A d +4 B c \right ) \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{\sqrt {b}}-\frac {\left (4 A \sqrt {-a b}\, a b c \,d^{3}-4 A \sqrt {-a b}\, b^{2} c^{3} d +A \,a^{2} b \,d^{4}-6 A a \,b^{2} c^{2} d^{2}+A \,b^{3} c^{4}-B \sqrt {-a b}\, a^{2} d^{4}+6 B \sqrt {-a b}\, a b \,c^{2} d^{2}-B \sqrt {-a b}\, b^{2} c^{4}+4 B \,a^{2} b c \,d^{3}-4 B a \,b^{2} c^{3} d \right ) \left (\frac {\sqrt {\left (x +\frac {\sqrt {-a b}}{b}\right )^{2} b -2 \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right )}}{3 \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right )^{2}}-\frac {\sqrt {\left (x +\frac {\sqrt {-a b}}{b}\right )^{2} b -2 \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right )}}{3 a \left (x +\frac {\sqrt {-a b}}{b}\right )}\right )}{4 a \,b^{2}}+\frac {\left (4 A \sqrt {-a b}\, a b c \,d^{3}-4 A \sqrt {-a b}\, b^{2} c^{3} d -A \,a^{2} b \,d^{4}+6 A a \,b^{2} c^{2} d^{2}-A \,b^{3} c^{4}-B \sqrt {-a b}\, a^{2} d^{4}+6 B \sqrt {-a b}\, a b \,c^{2} d^{2}-B \sqrt {-a b}\, b^{2} c^{4}-4 B \,a^{2} b c \,d^{3}+4 B a \,b^{2} c^{3} d \right ) \left (-\frac {\sqrt {\left (x -\frac {\sqrt {-a b}}{b}\right )^{2} b +2 \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right )}}{3 \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right )^{2}}-\frac {\sqrt {\left (x -\frac {\sqrt {-a b}}{b}\right )^{2} b +2 \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right )}}{3 a \left (x -\frac {\sqrt {-a b}}{b}\right )}\right )}{4 a \,b^{2}}+\frac {\left (-3 A \,a^{2} b \,d^{4}+6 A a \,b^{2} c^{2} d^{2}+8 A \sqrt {-a b}\, a b c \,d^{3}+A \,b^{3} c^{4}-12 B \,a^{2} b c \,d^{3}+4 B a \,b^{2} c^{3} d -4 B \sqrt {-a b}\, a^{2} d^{4}+12 B \sqrt {-a b}\, a b \,c^{2} d^{2}\right ) \sqrt {\left (x -\frac {\sqrt {-a b}}{b}\right )^{2} b +2 \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right )}}{4 a^{2} b^{2} \left (x -\frac {\sqrt {-a b}}{b}\right )}-\frac {\left (8 A \sqrt {-a b}\, a b c \,d^{3}+3 A \,a^{2} b \,d^{4}-6 A a \,b^{2} c^{2} d^{2}-A \,b^{3} c^{4}-4 B \sqrt {-a b}\, a^{2} d^{4}+12 B \sqrt {-a b}\, a b \,c^{2} d^{2}+12 B \,a^{2} b c \,d^{3}-4 B a \,b^{2} c^{3} d \right ) \sqrt {\left (x +\frac {\sqrt {-a b}}{b}\right )^{2} b -2 \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right )}}{4 a^{2} b^{2} \left (x +\frac {\sqrt {-a b}}{b}\right )}}{b^{2}}\)	\(909\)

Input:

int((B*x+A)*(d*x+c)^4/(b*x^2+a)^(5/2),x,method=_RETURNVERBOSE)
 

Output:

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

Fricas [A] (verification not implemented)

Time = 0.14 (sec) , antiderivative size = 724, normalized size of antiderivative = 3.12 \[ \int \frac {(A+B x) (c+d x)^4}{\left (a+b x^2\right )^{5/2}} \, dx=\left [\frac {3 \, {\left (4 \, B a^{4} c d^{3} + A a^{4} d^{4} + {\left (4 \, B a^{2} b^{2} c d^{3} + A a^{2} b^{2} d^{4}\right )} x^{4} + 2 \, {\left (4 \, B a^{3} b c d^{3} + A a^{3} b d^{4}\right )} x^{2}\right )} \sqrt {b} \log \left (-2 \, b x^{2} - 2 \, \sqrt {b x^{2} + a} \sqrt {b} x - a\right ) + 2 \, {\left (3 \, B a^{2} b^{2} d^{4} x^{4} - B a^{2} b^{2} c^{4} - 4 \, A a^{2} b^{2} c^{3} d - 12 \, B a^{3} b c^{2} d^{2} - 8 \, A a^{3} b c d^{3} + 8 \, B a^{4} d^{4} + 2 \, {\left (A b^{4} c^{4} + 2 \, B a b^{3} c^{3} d + 3 \, A a b^{3} c^{2} d^{2} - 8 \, B a^{2} b^{2} c d^{3} - 2 \, A a^{2} b^{2} d^{4}\right )} x^{3} - 6 \, {\left (3 \, B a^{2} b^{2} c^{2} d^{2} + 2 \, A a^{2} b^{2} c d^{3} - 2 \, B a^{3} b d^{4}\right )} x^{2} + 3 \, {\left (A a b^{3} c^{4} - 4 \, B a^{3} b c d^{3} - A a^{3} b d^{4}\right )} x\right )} \sqrt {b x^{2} + a}}{6 \, {\left (a^{2} b^{5} x^{4} + 2 \, a^{3} b^{4} x^{2} + a^{4} b^{3}\right )}}, -\frac {3 \, {\left (4 \, B a^{4} c d^{3} + A a^{4} d^{4} + {\left (4 \, B a^{2} b^{2} c d^{3} + A a^{2} b^{2} d^{4}\right )} x^{4} + 2 \, {\left (4 \, B a^{3} b c d^{3} + A a^{3} b d^{4}\right )} x^{2}\right )} \sqrt {-b} \arctan \left (\frac {\sqrt {-b} x}{\sqrt {b x^{2} + a}}\right ) - {\left (3 \, B a^{2} b^{2} d^{4} x^{4} - B a^{2} b^{2} c^{4} - 4 \, A a^{2} b^{2} c^{3} d - 12 \, B a^{3} b c^{2} d^{2} - 8 \, A a^{3} b c d^{3} + 8 \, B a^{4} d^{4} + 2 \, {\left (A b^{4} c^{4} + 2 \, B a b^{3} c^{3} d + 3 \, A a b^{3} c^{2} d^{2} - 8 \, B a^{2} b^{2} c d^{3} - 2 \, A a^{2} b^{2} d^{4}\right )} x^{3} - 6 \, {\left (3 \, B a^{2} b^{2} c^{2} d^{2} + 2 \, A a^{2} b^{2} c d^{3} - 2 \, B a^{3} b d^{4}\right )} x^{2} + 3 \, {\left (A a b^{3} c^{4} - 4 \, B a^{3} b c d^{3} - A a^{3} b d^{4}\right )} x\right )} \sqrt {b x^{2} + a}}{3 \, {\left (a^{2} b^{5} x^{4} + 2 \, a^{3} b^{4} x^{2} + a^{4} b^{3}\right )}}\right ] \] Input:

integrate((B*x+A)*(d*x+c)^4/(b*x^2+a)^(5/2),x, algorithm="fricas")
 

Output:

[1/6*(3*(4*B*a^4*c*d^3 + A*a^4*d^4 + (4*B*a^2*b^2*c*d^3 + A*a^2*b^2*d^4)*x 
^4 + 2*(4*B*a^3*b*c*d^3 + A*a^3*b*d^4)*x^2)*sqrt(b)*log(-2*b*x^2 - 2*sqrt( 
b*x^2 + a)*sqrt(b)*x - a) + 2*(3*B*a^2*b^2*d^4*x^4 - B*a^2*b^2*c^4 - 4*A*a 
^2*b^2*c^3*d - 12*B*a^3*b*c^2*d^2 - 8*A*a^3*b*c*d^3 + 8*B*a^4*d^4 + 2*(A*b 
^4*c^4 + 2*B*a*b^3*c^3*d + 3*A*a*b^3*c^2*d^2 - 8*B*a^2*b^2*c*d^3 - 2*A*a^2 
*b^2*d^4)*x^3 - 6*(3*B*a^2*b^2*c^2*d^2 + 2*A*a^2*b^2*c*d^3 - 2*B*a^3*b*d^4 
)*x^2 + 3*(A*a*b^3*c^4 - 4*B*a^3*b*c*d^3 - A*a^3*b*d^4)*x)*sqrt(b*x^2 + a) 
)/(a^2*b^5*x^4 + 2*a^3*b^4*x^2 + a^4*b^3), -1/3*(3*(4*B*a^4*c*d^3 + A*a^4* 
d^4 + (4*B*a^2*b^2*c*d^3 + A*a^2*b^2*d^4)*x^4 + 2*(4*B*a^3*b*c*d^3 + A*a^3 
*b*d^4)*x^2)*sqrt(-b)*arctan(sqrt(-b)*x/sqrt(b*x^2 + a)) - (3*B*a^2*b^2*d^ 
4*x^4 - B*a^2*b^2*c^4 - 4*A*a^2*b^2*c^3*d - 12*B*a^3*b*c^2*d^2 - 8*A*a^3*b 
*c*d^3 + 8*B*a^4*d^4 + 2*(A*b^4*c^4 + 2*B*a*b^3*c^3*d + 3*A*a*b^3*c^2*d^2 
- 8*B*a^2*b^2*c*d^3 - 2*A*a^2*b^2*d^4)*x^3 - 6*(3*B*a^2*b^2*c^2*d^2 + 2*A* 
a^2*b^2*c*d^3 - 2*B*a^3*b*d^4)*x^2 + 3*(A*a*b^3*c^4 - 4*B*a^3*b*c*d^3 - A* 
a^3*b*d^4)*x)*sqrt(b*x^2 + a))/(a^2*b^5*x^4 + 2*a^3*b^4*x^2 + a^4*b^3)]
 

Sympy [F]

\[ \int \frac {(A+B x) (c+d x)^4}{\left (a+b x^2\right )^{5/2}} \, dx=\int \frac {\left (A + B x\right ) \left (c + d x\right )^{4}}{\left (a + b x^{2}\right )^{\frac {5}{2}}}\, dx \] Input:

integrate((B*x+A)*(d*x+c)**4/(b*x**2+a)**(5/2),x)
 

Output:

Integral((A + B*x)*(c + d*x)**4/(a + b*x**2)**(5/2), x)
 

Maxima [A] (verification not implemented)

Time = 0.05 (sec) , antiderivative size = 375, normalized size of antiderivative = 1.62 \[ \int \frac {(A+B x) (c+d x)^4}{\left (a+b x^2\right )^{5/2}} \, dx=\frac {B d^{4} x^{4}}{{\left (b x^{2} + a\right )}^{\frac {3}{2}} b} + \frac {4 \, B a d^{4} x^{2}}{{\left (b x^{2} + a\right )}^{\frac {3}{2}} b^{2}} + \frac {2 \, A c^{4} x}{3 \, \sqrt {b x^{2} + a} a^{2}} + \frac {A c^{4} x}{3 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} a} - \frac {1}{3} \, {\left (4 \, B c d^{3} + A d^{4}\right )} x {\left (\frac {3 \, x^{2}}{{\left (b x^{2} + a\right )}^{\frac {3}{2}} b} + \frac {2 \, a}{{\left (b x^{2} + a\right )}^{\frac {3}{2}} b^{2}}\right )} - \frac {B c^{4}}{3 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} b} - \frac {4 \, A c^{3} d}{3 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} b} + \frac {8 \, B a^{2} d^{4}}{3 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} b^{3}} - \frac {2 \, {\left (3 \, B c^{2} d^{2} + 2 \, A c d^{3}\right )} x^{2}}{{\left (b x^{2} + a\right )}^{\frac {3}{2}} b} - \frac {{\left (4 \, B c d^{3} + A d^{4}\right )} x}{3 \, \sqrt {b x^{2} + a} b^{2}} - \frac {2 \, {\left (2 \, B c^{3} d + 3 \, A c^{2} d^{2}\right )} x}{3 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} b} + \frac {2 \, {\left (2 \, B c^{3} d + 3 \, A c^{2} d^{2}\right )} x}{3 \, \sqrt {b x^{2} + a} a b} + \frac {{\left (4 \, B c d^{3} + A d^{4}\right )} \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{b^{\frac {5}{2}}} - \frac {4 \, {\left (3 \, B c^{2} d^{2} + 2 \, A c d^{3}\right )} a}{3 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} b^{2}} \] Input:

integrate((B*x+A)*(d*x+c)^4/(b*x^2+a)^(5/2),x, algorithm="maxima")
 

Output:

B*d^4*x^4/((b*x^2 + a)^(3/2)*b) + 4*B*a*d^4*x^2/((b*x^2 + a)^(3/2)*b^2) + 
2/3*A*c^4*x/(sqrt(b*x^2 + a)*a^2) + 1/3*A*c^4*x/((b*x^2 + a)^(3/2)*a) - 1/ 
3*(4*B*c*d^3 + A*d^4)*x*(3*x^2/((b*x^2 + a)^(3/2)*b) + 2*a/((b*x^2 + a)^(3 
/2)*b^2)) - 1/3*B*c^4/((b*x^2 + a)^(3/2)*b) - 4/3*A*c^3*d/((b*x^2 + a)^(3/ 
2)*b) + 8/3*B*a^2*d^4/((b*x^2 + a)^(3/2)*b^3) - 2*(3*B*c^2*d^2 + 2*A*c*d^3 
)*x^2/((b*x^2 + a)^(3/2)*b) - 1/3*(4*B*c*d^3 + A*d^4)*x/(sqrt(b*x^2 + a)*b 
^2) - 2/3*(2*B*c^3*d + 3*A*c^2*d^2)*x/((b*x^2 + a)^(3/2)*b) + 2/3*(2*B*c^3 
*d + 3*A*c^2*d^2)*x/(sqrt(b*x^2 + a)*a*b) + (4*B*c*d^3 + A*d^4)*arcsinh(b* 
x/sqrt(a*b))/b^(5/2) - 4/3*(3*B*c^2*d^2 + 2*A*c*d^3)*a/((b*x^2 + a)^(3/2)* 
b^2)
 

Giac [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 299, normalized size of antiderivative = 1.29 \[ \int \frac {(A+B x) (c+d x)^4}{\left (a+b x^2\right )^{5/2}} \, dx=\frac {{\left ({\left ({\left (\frac {3 \, B d^{4} x}{b} + \frac {2 \, {\left (A b^{6} c^{4} + 2 \, B a b^{5} c^{3} d + 3 \, A a b^{5} c^{2} d^{2} - 8 \, B a^{2} b^{4} c d^{3} - 2 \, A a^{2} b^{4} d^{4}\right )}}{a^{2} b^{5}}\right )} x - \frac {6 \, {\left (3 \, B a^{2} b^{4} c^{2} d^{2} + 2 \, A a^{2} b^{4} c d^{3} - 2 \, B a^{3} b^{3} d^{4}\right )}}{a^{2} b^{5}}\right )} x + \frac {3 \, {\left (A a b^{5} c^{4} - 4 \, B a^{3} b^{3} c d^{3} - A a^{3} b^{3} d^{4}\right )}}{a^{2} b^{5}}\right )} x - \frac {B a^{2} b^{4} c^{4} + 4 \, A a^{2} b^{4} c^{3} d + 12 \, B a^{3} b^{3} c^{2} d^{2} + 8 \, A a^{3} b^{3} c d^{3} - 8 \, B a^{4} b^{2} d^{4}}{a^{2} b^{5}}}{3 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}}} - \frac {{\left (4 \, B c d^{3} + A d^{4}\right )} \log \left ({\left | -\sqrt {b} x + \sqrt {b x^{2} + a} \right |}\right )}{b^{\frac {5}{2}}} \] Input:

integrate((B*x+A)*(d*x+c)^4/(b*x^2+a)^(5/2),x, algorithm="giac")
 

Output:

1/3*((((3*B*d^4*x/b + 2*(A*b^6*c^4 + 2*B*a*b^5*c^3*d + 3*A*a*b^5*c^2*d^2 - 
 8*B*a^2*b^4*c*d^3 - 2*A*a^2*b^4*d^4)/(a^2*b^5))*x - 6*(3*B*a^2*b^4*c^2*d^ 
2 + 2*A*a^2*b^4*c*d^3 - 2*B*a^3*b^3*d^4)/(a^2*b^5))*x + 3*(A*a*b^5*c^4 - 4 
*B*a^3*b^3*c*d^3 - A*a^3*b^3*d^4)/(a^2*b^5))*x - (B*a^2*b^4*c^4 + 4*A*a^2* 
b^4*c^3*d + 12*B*a^3*b^3*c^2*d^2 + 8*A*a^3*b^3*c*d^3 - 8*B*a^4*b^2*d^4)/(a 
^2*b^5))/(b*x^2 + a)^(3/2) - (4*B*c*d^3 + A*d^4)*log(abs(-sqrt(b)*x + sqrt 
(b*x^2 + a)))/b^(5/2)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {(A+B x) (c+d x)^4}{\left (a+b x^2\right )^{5/2}} \, dx=\int \frac {\left (A+B\,x\right )\,{\left (c+d\,x\right )}^4}{{\left (b\,x^2+a\right )}^{5/2}} \,d x \] Input:

int(((A + B*x)*(c + d*x)^4)/(a + b*x^2)^(5/2),x)
 

Output:

int(((A + B*x)*(c + d*x)^4)/(a + b*x^2)^(5/2), x)
 

Reduce [B] (verification not implemented)

Time = 0.26 (sec) , antiderivative size = 704, normalized size of antiderivative = 3.03 \[ \int \frac {(A+B x) (c+d x)^4}{\left (a+b x^2\right )^{5/2}} \, dx=\frac {-\sqrt {b \,x^{2}+a}\, a \,b^{3} c^{4}+8 \sqrt {b \,x^{2}+a}\, a^{3} b \,d^{4}+2 \sqrt {b \,x^{2}+a}\, b^{4} c^{4} x^{3}+3 \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}+\sqrt {b}\, x}{\sqrt {a}}\right ) a^{4} d^{4}-2 \sqrt {b}\, a^{2} b^{2} c^{4}-2 \sqrt {b}\, b^{4} c^{4} x^{4}-8 \sqrt {b \,x^{2}+a}\, a^{3} b c \,d^{3}-3 \sqrt {b \,x^{2}+a}\, a^{3} b \,d^{4} x -4 \sqrt {b \,x^{2}+a}\, a^{2} b^{2} c^{3} d -12 \sqrt {b \,x^{2}+a}\, a^{2} b^{2} c^{2} d^{2}-4 \sqrt {b \,x^{2}+a}\, a^{2} b^{2} d^{4} x^{3}+12 \sqrt {b \,x^{2}+a}\, a^{2} b^{2} d^{4} x^{2}+3 \sqrt {b \,x^{2}+a}\, a \,b^{3} c^{4} x +3 \sqrt {b \,x^{2}+a}\, a \,b^{3} d^{4} x^{4}+4 \sqrt {b \,x^{2}+a}\, b^{4} c^{3} d \,x^{3}+6 \sqrt {b}\, a^{3} b \,c^{2} d^{2}+4 \sqrt {b}\, a^{2} b^{2} c^{3} d -4 \sqrt {b}\, a \,b^{3} c^{4} x^{2}+4 \sqrt {b}\, b^{4} c^{3} d \,x^{4}+24 \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}+\sqrt {b}\, x}{\sqrt {a}}\right ) a^{2} b^{2} c \,d^{3} x^{2}+12 \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}+\sqrt {b}\, x}{\sqrt {a}}\right ) a \,b^{3} c \,d^{3} x^{4}-12 \sqrt {b \,x^{2}+a}\, a^{2} b^{2} c \,d^{3} x^{2}-12 \sqrt {b \,x^{2}+a}\, a^{2} b^{2} c \,d^{3} x +6 \sqrt {b \,x^{2}+a}\, a \,b^{3} c^{2} d^{2} x^{3}-18 \sqrt {b \,x^{2}+a}\, a \,b^{3} c^{2} d^{2} x^{2}-16 \sqrt {b \,x^{2}+a}\, a \,b^{3} c \,d^{3} x^{3}+12 \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}+\sqrt {b}\, x}{\sqrt {a}}\right ) a^{3} b c \,d^{3}+6 \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}+\sqrt {b}\, x}{\sqrt {a}}\right ) a^{3} b \,d^{4} x^{2}+3 \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}+\sqrt {b}\, x}{\sqrt {a}}\right ) a^{2} b^{2} d^{4} x^{4}+12 \sqrt {b}\, a^{2} b^{2} c^{2} d^{2} x^{2}+8 \sqrt {b}\, a \,b^{3} c^{3} d \,x^{2}+6 \sqrt {b}\, a \,b^{3} c^{2} d^{2} x^{4}}{3 a \,b^{3} \left (b^{2} x^{4}+2 a b \,x^{2}+a^{2}\right )} \] Input:

int((B*x+A)*(d*x+c)^4/(b*x^2+a)^(5/2),x)
 

Output:

( - 8*sqrt(a + b*x**2)*a**3*b*c*d**3 - 3*sqrt(a + b*x**2)*a**3*b*d**4*x + 
8*sqrt(a + b*x**2)*a**3*b*d**4 - 4*sqrt(a + b*x**2)*a**2*b**2*c**3*d - 12* 
sqrt(a + b*x**2)*a**2*b**2*c**2*d**2 - 12*sqrt(a + b*x**2)*a**2*b**2*c*d** 
3*x**2 - 12*sqrt(a + b*x**2)*a**2*b**2*c*d**3*x - 4*sqrt(a + b*x**2)*a**2* 
b**2*d**4*x**3 + 12*sqrt(a + b*x**2)*a**2*b**2*d**4*x**2 + 3*sqrt(a + b*x* 
*2)*a*b**3*c**4*x - sqrt(a + b*x**2)*a*b**3*c**4 + 6*sqrt(a + b*x**2)*a*b* 
*3*c**2*d**2*x**3 - 18*sqrt(a + b*x**2)*a*b**3*c**2*d**2*x**2 - 16*sqrt(a 
+ b*x**2)*a*b**3*c*d**3*x**3 + 3*sqrt(a + b*x**2)*a*b**3*d**4*x**4 + 2*sqr 
t(a + b*x**2)*b**4*c**4*x**3 + 4*sqrt(a + b*x**2)*b**4*c**3*d*x**3 + 3*sqr 
t(b)*log((sqrt(a + b*x**2) + sqrt(b)*x)/sqrt(a))*a**4*d**4 + 12*sqrt(b)*lo 
g((sqrt(a + b*x**2) + sqrt(b)*x)/sqrt(a))*a**3*b*c*d**3 + 6*sqrt(b)*log((s 
qrt(a + b*x**2) + sqrt(b)*x)/sqrt(a))*a**3*b*d**4*x**2 + 24*sqrt(b)*log((s 
qrt(a + b*x**2) + sqrt(b)*x)/sqrt(a))*a**2*b**2*c*d**3*x**2 + 3*sqrt(b)*lo 
g((sqrt(a + b*x**2) + sqrt(b)*x)/sqrt(a))*a**2*b**2*d**4*x**4 + 12*sqrt(b) 
*log((sqrt(a + b*x**2) + sqrt(b)*x)/sqrt(a))*a*b**3*c*d**3*x**4 + 6*sqrt(b 
)*a**3*b*c**2*d**2 - 2*sqrt(b)*a**2*b**2*c**4 + 4*sqrt(b)*a**2*b**2*c**3*d 
 + 12*sqrt(b)*a**2*b**2*c**2*d**2*x**2 - 4*sqrt(b)*a*b**3*c**4*x**2 + 8*sq 
rt(b)*a*b**3*c**3*d*x**2 + 6*sqrt(b)*a*b**3*c**2*d**2*x**4 - 2*sqrt(b)*b** 
4*c**4*x**4 + 4*sqrt(b)*b**4*c**3*d*x**4)/(3*a*b**3*(a**2 + 2*a*b*x**2 + b 
**2*x**4))