\(\int \frac {(a+b x^2)^{3/2}}{(c+d x)^2} \, dx\) [1089]

Optimal result

Integrand size = 19, antiderivative size = 175 \[ \int \frac {\left (a+b x^2\right )^{3/2}}{(c+d x)^2} \, dx=-\frac {2 b c \sqrt {a+b x^2}}{d^3}+\frac {b x \sqrt {a+b x^2}}{2 d^2}-\frac {\left (b c^2+a d^2\right ) \sqrt {a+b x^2}}{d^3 (c+d x)}+\frac {3 \sqrt {b} \left (2 b c^2+a d^2\right ) \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{2 d^4}+\frac {3 b c \sqrt {b c^2+a d^2} \text {arctanh}\left (\frac {a d-b c x}{\sqrt {b c^2+a d^2} \sqrt {a+b x^2}}\right )}{d^4} \] Output:

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

Mathematica [A] (verified)

Time = 0.10 (sec) , antiderivative size = 161, normalized size of antiderivative = 0.92 \[ \int \frac {\left (a+b x^2\right )^{3/2}}{(c+d x)^2} \, dx=\frac {\frac {d \sqrt {a+b x^2} \left (-6 b c^2-2 a d^2-3 b c d x+b d^2 x^2\right )}{c+d x}-12 b c \sqrt {-b c^2-a d^2} \arctan \left (\frac {\sqrt {b} (c+d x)-d \sqrt {a+b x^2}}{\sqrt {-b c^2-a d^2}}\right )-3 \sqrt {b} \left (2 b c^2+a d^2\right ) \log \left (-\sqrt {b} x+\sqrt {a+b x^2}\right )}{2 d^4} \] Input:

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

Output:

((d*Sqrt[a + b*x^2]*(-6*b*c^2 - 2*a*d^2 - 3*b*c*d*x + b*d^2*x^2))/(c + d*x 
) - 12*b*c*Sqrt[-(b*c^2) - a*d^2]*ArcTan[(Sqrt[b]*(c + d*x) - d*Sqrt[a + b 
*x^2])/Sqrt[-(b*c^2) - a*d^2]] - 3*Sqrt[b]*(2*b*c^2 + a*d^2)*Log[-(Sqrt[b] 
*x) + Sqrt[a + b*x^2]])/(2*d^4)
 

Rubi [A] (verified)

Time = 0.56 (sec) , antiderivative size = 164, normalized size of antiderivative = 0.94, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.421, Rules used = {492, 591, 25, 719, 224, 219, 488, 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.

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

\(\Big \downarrow \) 492

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

\(\Big \downarrow \) 591

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 719

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

\(\Big \downarrow \) 224

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

\(\Big \downarrow \) 219

\(\displaystyle \frac {3 b \left (-\frac {\frac {2 c \left (a d^2+b c^2\right ) \int \frac {1}{(c+d x) \sqrt {b x^2+a}}dx}{d}-\frac {\text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right ) \left (a d^2+2 b c^2\right )}{\sqrt {b} d}}{2 d^2}-\frac {\sqrt {a+b x^2} (2 c-d x)}{2 d^2}\right )}{d}-\frac {\left (a+b x^2\right )^{3/2}}{d (c+d x)}\)

\(\Big \downarrow \) 488

\(\displaystyle \frac {3 b \left (-\frac {-\frac {2 c \left (a d^2+b c^2\right ) \int \frac {1}{b c^2+a d^2-\frac {(a d-b c x)^2}{b x^2+a}}d\frac {a d-b c x}{\sqrt {b x^2+a}}}{d}-\frac {\text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right ) \left (a d^2+2 b c^2\right )}{\sqrt {b} d}}{2 d^2}-\frac {\sqrt {a+b x^2} (2 c-d x)}{2 d^2}\right )}{d}-\frac {\left (a+b x^2\right )^{3/2}}{d (c+d x)}\)

\(\Big \downarrow \) 219

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

Input:

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

Output:

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

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], 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 488

Int[1/(((c_) + (d_.)*(x_))*Sqrt[(a_) + (b_.)*(x_)^2]), x_Symbol] :> -Subst[ 
Int[1/(b*c^2 + a*d^2 - x^2), x], x, (a*d - b*c*x)/Sqrt[a + b*x^2]] /; FreeQ 
[{a, b, c, d}, x]
 

rule 492

Int[((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[ 
(c + d*x)^(n + 1)*((a + b*x^2)^p/(d*(n + 1))), x] - Simp[2*b*(p/(d*(n + 1)) 
)   Int[x*(c + d*x)^(n + 1)*(a + b*x^2)^(p - 1), x], x] /; FreeQ[{a, b, c, 
d, n}, x] && GtQ[p, 0] && (IntegerQ[p] || LtQ[n, -1]) && NeQ[n, -1] &&  !IL 
tQ[n + 2*p + 1, 0] && IntQuadraticQ[a, 0, b, c, d, n, p, x]
 

rule 591

Int[(x_)*((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> 
Simp[(-(c + d*x)^(n + 1))*(a + b*x^2)^p*((c*(2*p + 1) - d*(n + 2*p + 1)*x)/ 
(d^2*(n + 2*p + 1)*(n + 2*p + 2))), x] + Simp[2*(p/(d^2*(n + 2*p + 1)*(n + 
2*p + 2)))   Int[(c + d*x)^n*(a + b*x^2)^(p - 1)*Simp[a*c*d*n + (b*c^2*(2*p 
 + 1) + a*d^2*(n + 2*p + 1))*x, x], x], x] /; FreeQ[{a, b, c, d, n}, x] && 
GtQ[p, 0] && LeQ[-1, n, 0] &&  !ILtQ[n + 2*p, 0]
 

rule 719

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

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(445\) vs. \(2(153)=306\).

Time = 0.46 (sec) , antiderivative size = 446, normalized size of antiderivative = 2.55

method	result	size
risch	\(-\frac {\left (-d x +4 c \right ) \sqrt {b \,x^{2}+a}\, b}{2 d^{3}}+\frac {\frac {2 \left (a^{2} d^{4}+2 b \,c^{2} d^{2} a +b^{2} c^{4}\right ) \left (-\frac {d^{2} \sqrt {b \left (x +\frac {c}{d}\right )^{2}-\frac {2 b c \left (x +\frac {c}{d}\right )}{d}+\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}}{\left (a \,d^{2}+b \,c^{2}\right ) \left (x +\frac {c}{d}\right )}-\frac {b c d \ln \left (\frac {\frac {2 a \,d^{2}+2 b \,c^{2}}{d^{2}}-\frac {2 b c \left (x +\frac {c}{d}\right )}{d}+2 \sqrt {\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}\, \sqrt {b \left (x +\frac {c}{d}\right )^{2}-\frac {2 b c \left (x +\frac {c}{d}\right )}{d}+\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}}{x +\frac {c}{d}}\right )}{\left (a \,d^{2}+b \,c^{2}\right ) \sqrt {\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}}\right )}{d^{3}}+\frac {3 \sqrt {b}\, \left (a \,d^{2}+2 b \,c^{2}\right ) \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{d}+\frac {8 b c \left (a \,d^{2}+b \,c^{2}\right ) \ln \left (\frac {\frac {2 a \,d^{2}+2 b \,c^{2}}{d^{2}}-\frac {2 b c \left (x +\frac {c}{d}\right )}{d}+2 \sqrt {\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}\, \sqrt {b \left (x +\frac {c}{d}\right )^{2}-\frac {2 b c \left (x +\frac {c}{d}\right )}{d}+\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}}{x +\frac {c}{d}}\right )}{d^{2} \sqrt {\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}}}{2 d^{3}}\)	\(446\)
default	\(\frac {-\frac {d^{2} \left (b \left (x +\frac {c}{d}\right )^{2}-\frac {2 b c \left (x +\frac {c}{d}\right )}{d}+\frac {a \,d^{2}+b \,c^{2}}{d^{2}}\right )^{\frac {5}{2}}}{\left (a \,d^{2}+b \,c^{2}\right ) \left (x +\frac {c}{d}\right )}-\frac {3 b c d \left (\frac {\left (b \left (x +\frac {c}{d}\right )^{2}-\frac {2 b c \left (x +\frac {c}{d}\right )}{d}+\frac {a \,d^{2}+b \,c^{2}}{d^{2}}\right )^{\frac {3}{2}}}{3}-\frac {b c \left (\frac {\left (2 b \left (x +\frac {c}{d}\right )-\frac {2 b c}{d}\right ) \sqrt {b \left (x +\frac {c}{d}\right )^{2}-\frac {2 b c \left (x +\frac {c}{d}\right )}{d}+\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}}{4 b}+\frac {\left (\frac {4 b \left (a \,d^{2}+b \,c^{2}\right )}{d^{2}}-\frac {4 b^{2} c^{2}}{d^{2}}\right ) \ln \left (\frac {-\frac {b c}{d}+b \left (x +\frac {c}{d}\right )}{\sqrt {b}}+\sqrt {b \left (x +\frac {c}{d}\right )^{2}-\frac {2 b c \left (x +\frac {c}{d}\right )}{d}+\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}\right )}{8 b^{\frac {3}{2}}}\right )}{d}+\frac {\left (a \,d^{2}+b \,c^{2}\right ) \left (\sqrt {b \left (x +\frac {c}{d}\right )^{2}-\frac {2 b c \left (x +\frac {c}{d}\right )}{d}+\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}-\frac {\sqrt {b}\, c \ln \left (\frac {-\frac {b c}{d}+b \left (x +\frac {c}{d}\right )}{\sqrt {b}}+\sqrt {b \left (x +\frac {c}{d}\right )^{2}-\frac {2 b c \left (x +\frac {c}{d}\right )}{d}+\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}\right )}{d}-\frac {\left (a \,d^{2}+b \,c^{2}\right ) \ln \left (\frac {\frac {2 a \,d^{2}+2 b \,c^{2}}{d^{2}}-\frac {2 b c \left (x +\frac {c}{d}\right )}{d}+2 \sqrt {\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}\, \sqrt {b \left (x +\frac {c}{d}\right )^{2}-\frac {2 b c \left (x +\frac {c}{d}\right )}{d}+\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}}{x +\frac {c}{d}}\right )}{d^{2} \sqrt {\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}}\right )}{d^{2}}\right )}{a \,d^{2}+b \,c^{2}}+\frac {4 b \,d^{2} \left (\frac {\left (2 b \left (x +\frac {c}{d}\right )-\frac {2 b c}{d}\right ) \left (b \left (x +\frac {c}{d}\right )^{2}-\frac {2 b c \left (x +\frac {c}{d}\right )}{d}+\frac {a \,d^{2}+b \,c^{2}}{d^{2}}\right )^{\frac {3}{2}}}{8 b}+\frac {3 \left (\frac {4 b \left (a \,d^{2}+b \,c^{2}\right )}{d^{2}}-\frac {4 b^{2} c^{2}}{d^{2}}\right ) \left (\frac {\left (2 b \left (x +\frac {c}{d}\right )-\frac {2 b c}{d}\right ) \sqrt {b \left (x +\frac {c}{d}\right )^{2}-\frac {2 b c \left (x +\frac {c}{d}\right )}{d}+\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}}{4 b}+\frac {\left (\frac {4 b \left (a \,d^{2}+b \,c^{2}\right )}{d^{2}}-\frac {4 b^{2} c^{2}}{d^{2}}\right ) \ln \left (\frac {-\frac {b c}{d}+b \left (x +\frac {c}{d}\right )}{\sqrt {b}}+\sqrt {b \left (x +\frac {c}{d}\right )^{2}-\frac {2 b c \left (x +\frac {c}{d}\right )}{d}+\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}\right )}{8 b^{\frac {3}{2}}}\right )}{16 b}\right )}{a \,d^{2}+b \,c^{2}}}{d^{2}}\)	\(872\)

Input:

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

Output:

-1/2*(-d*x+4*c)*(b*x^2+a)^(1/2)*b/d^3+1/2/d^3*(2*(a^2*d^4+2*a*b*c^2*d^2+b^ 
2*c^4)/d^3*(-1/(a*d^2+b*c^2)*d^2/(x+c/d)*(b*(x+c/d)^2-2*b*c/d*(x+c/d)+(a*d 
^2+b*c^2)/d^2)^(1/2)-b*c*d/(a*d^2+b*c^2)/((a*d^2+b*c^2)/d^2)^(1/2)*ln((2*( 
a*d^2+b*c^2)/d^2-2*b*c/d*(x+c/d)+2*((a*d^2+b*c^2)/d^2)^(1/2)*(b*(x+c/d)^2- 
2*b*c/d*(x+c/d)+(a*d^2+b*c^2)/d^2)^(1/2))/(x+c/d)))+3*b^(1/2)*(a*d^2+2*b*c 
^2)/d*ln(b^(1/2)*x+(b*x^2+a)^(1/2))+8/d^2*b*c*(a*d^2+b*c^2)/((a*d^2+b*c^2) 
/d^2)^(1/2)*ln((2*(a*d^2+b*c^2)/d^2-2*b*c/d*(x+c/d)+2*((a*d^2+b*c^2)/d^2)^ 
(1/2)*(b*(x+c/d)^2-2*b*c/d*(x+c/d)+(a*d^2+b*c^2)/d^2)^(1/2))/(x+c/d)))
 

Fricas [A] (verification not implemented)

Time = 0.30 (sec) , antiderivative size = 888, normalized size of antiderivative = 5.07 \[ \int \frac {\left (a+b x^2\right )^{3/2}}{(c+d x)^2} \, dx =\text {Too large to display} \] Input:

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [A] (verification not implemented)

Time = 0.05 (sec) , antiderivative size = 148, normalized size of antiderivative = 0.85 \[ \int \frac {\left (a+b x^2\right )^{3/2}}{(c+d x)^2} \, dx=-\frac {{\left (b x^{2} + a\right )}^{\frac {3}{2}}}{d^{2} x + c d} + \frac {3 \, \sqrt {b x^{2} + a} b x}{2 \, d^{2}} + \frac {3 \, b^{\frac {3}{2}} c^{2} \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{d^{4}} + \frac {3 \, a \sqrt {b} \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{2 \, d^{2}} - \frac {3 \, \sqrt {a + \frac {b c^{2}}{d^{2}}} b c \operatorname {arsinh}\left (\frac {b c x}{\sqrt {a b} {\left | d x + c \right |}} - \frac {a d}{\sqrt {a b} {\left | d x + c \right |}}\right )}{d^{3}} - \frac {3 \, \sqrt {b x^{2} + a} b c}{d^{3}} \] Input:

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

Output:

-(b*x^2 + a)^(3/2)/(d^2*x + c*d) + 3/2*sqrt(b*x^2 + a)*b*x/d^2 + 3*b^(3/2) 
*c^2*arcsinh(b*x/sqrt(a*b))/d^4 + 3/2*a*sqrt(b)*arcsinh(b*x/sqrt(a*b))/d^2 
 - 3*sqrt(a + b*c^2/d^2)*b*c*arcsinh(b*c*x/(sqrt(a*b)*abs(d*x + c)) - a*d/ 
(sqrt(a*b)*abs(d*x + c)))/d^3 - 3*sqrt(b*x^2 + a)*b*c/d^3
 

Giac [F(-1)]

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

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

Output:

Timed out
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.21 (sec) , antiderivative size = 411, normalized size of antiderivative = 2.35 \[ \int \frac {\left (a+b x^2\right )^{3/2}}{(c+d x)^2} \, dx=\frac {12 \sqrt {a \,d^{2}+b \,c^{2}}\, \mathrm {log}\left (-\sqrt {b \,x^{2}+a}\, \sqrt {a \,d^{2}+b \,c^{2}}-a d +b c x \right ) b \,c^{2}+12 \sqrt {a \,d^{2}+b \,c^{2}}\, \mathrm {log}\left (-\sqrt {b \,x^{2}+a}\, \sqrt {a \,d^{2}+b \,c^{2}}-a d +b c x \right ) b c d x -12 \sqrt {a \,d^{2}+b \,c^{2}}\, \mathrm {log}\left (d x +c \right ) b \,c^{2}-12 \sqrt {a \,d^{2}+b \,c^{2}}\, \mathrm {log}\left (d x +c \right ) b c d x -4 \sqrt {b \,x^{2}+a}\, a \,d^{3}-12 \sqrt {b \,x^{2}+a}\, b \,c^{2} d -6 \sqrt {b \,x^{2}+a}\, b c \,d^{2} x +2 \sqrt {b \,x^{2}+a}\, b \,d^{3} x^{2}-3 \sqrt {b}\, \mathrm {log}\left (\sqrt {b \,x^{2}+a}-\sqrt {b}\, x \right ) a c \,d^{2}-3 \sqrt {b}\, \mathrm {log}\left (\sqrt {b \,x^{2}+a}-\sqrt {b}\, x \right ) a \,d^{3} x -6 \sqrt {b}\, \mathrm {log}\left (\sqrt {b \,x^{2}+a}-\sqrt {b}\, x \right ) b \,c^{3}-6 \sqrt {b}\, \mathrm {log}\left (\sqrt {b \,x^{2}+a}-\sqrt {b}\, x \right ) b \,c^{2} d x +3 \sqrt {b}\, \mathrm {log}\left (\sqrt {b \,x^{2}+a}+\sqrt {b}\, x \right ) a c \,d^{2}+3 \sqrt {b}\, \mathrm {log}\left (\sqrt {b \,x^{2}+a}+\sqrt {b}\, x \right ) a \,d^{3} x +6 \sqrt {b}\, \mathrm {log}\left (\sqrt {b \,x^{2}+a}+\sqrt {b}\, x \right ) b \,c^{3}+6 \sqrt {b}\, \mathrm {log}\left (\sqrt {b \,x^{2}+a}+\sqrt {b}\, x \right ) b \,c^{2} d x}{4 d^{4} \left (d x +c \right )} \] Input:

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

Output:

(12*sqrt(a*d**2 + b*c**2)*log( - sqrt(a + b*x**2)*sqrt(a*d**2 + b*c**2) - 
a*d + b*c*x)*b*c**2 + 12*sqrt(a*d**2 + b*c**2)*log( - sqrt(a + b*x**2)*sqr 
t(a*d**2 + b*c**2) - a*d + b*c*x)*b*c*d*x - 12*sqrt(a*d**2 + b*c**2)*log(c 
 + d*x)*b*c**2 - 12*sqrt(a*d**2 + b*c**2)*log(c + d*x)*b*c*d*x - 4*sqrt(a 
+ b*x**2)*a*d**3 - 12*sqrt(a + b*x**2)*b*c**2*d - 6*sqrt(a + b*x**2)*b*c*d 
**2*x + 2*sqrt(a + b*x**2)*b*d**3*x**2 - 3*sqrt(b)*log(sqrt(a + b*x**2) - 
sqrt(b)*x)*a*c*d**2 - 3*sqrt(b)*log(sqrt(a + b*x**2) - sqrt(b)*x)*a*d**3*x 
 - 6*sqrt(b)*log(sqrt(a + b*x**2) - sqrt(b)*x)*b*c**3 - 6*sqrt(b)*log(sqrt 
(a + b*x**2) - sqrt(b)*x)*b*c**2*d*x + 3*sqrt(b)*log(sqrt(a + b*x**2) + sq 
rt(b)*x)*a*c*d**2 + 3*sqrt(b)*log(sqrt(a + b*x**2) + sqrt(b)*x)*a*d**3*x + 
 6*sqrt(b)*log(sqrt(a + b*x**2) + sqrt(b)*x)*b*c**3 + 6*sqrt(b)*log(sqrt(a 
 + b*x**2) + sqrt(b)*x)*b*c**2*d*x)/(4*d**4*(c + d*x))