\(\int \frac {(a+b x^2)^{3/2}}{(c+d x)^3} \, dx\) [1098]

Optimal result

Integrand size = 19, antiderivative size = 182 \[ \int \frac {\left (a+b x^2\right )^{3/2}}{(c+d x)^3} \, dx=\frac {b \sqrt {a+b x^2}}{d^3}-\frac {\left (b c^2+a d^2\right ) \sqrt {a+b x^2}}{2 d^3 (c+d x)^2}+\frac {5 b c \sqrt {a+b x^2}}{2 d^3 (c+d x)}-\frac {3 b^{3/2} c \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{d^4}-\frac {3 b \left (2 b c^2+a d^2\right ) \text {arctanh}\left (\frac {a d-b c x}{\sqrt {b c^2+a d^2} \sqrt {a+b x^2}}\right )}{2 d^4 \sqrt {b c^2+a d^2}} \] Output:

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

Mathematica [A] (verified)

Time = 0.13 (sec) , antiderivative size = 162, normalized size of antiderivative = 0.89 \[ \int \frac {\left (a+b x^2\right )^{3/2}}{(c+d x)^3} \, dx=\frac {\frac {d \sqrt {a+b x^2} \left (6 b c^2-a d^2+9 b c d x+2 b d^2 x^2\right )}{(c+d x)^2}-\frac {6 b \left (2 b c^2+a d^2\right ) \arctan \left (\frac {\sqrt {b} (c+d x)-d \sqrt {a+b x^2}}{\sqrt {-b c^2-a d^2}}\right )}{\sqrt {-b c^2-a d^2}}+6 b^{3/2} c \log \left (-\sqrt {b} x+\sqrt {a+b x^2}\right )}{2 d^4} \] Input:

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

Output:

((d*Sqrt[a + b*x^2]*(6*b*c^2 - a*d^2 + 9*b*c*d*x + 2*b*d^2*x^2))/(c + d*x) 
^2 - (6*b*(2*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]])/Sqrt[-(b*c^2) - a*d^2] + 6*b^(3/2)*c*Log[-(Sqrt[ 
b]*x) + Sqrt[a + b*x^2]])/(2*d^4)
 

Rubi [A] (verified)

Time = 0.53 (sec) , antiderivative size = 168, normalized size of antiderivative = 0.92, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.421, Rules used = {492, 590, 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.

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

\(\Big \downarrow \) 492

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

\(\Big \downarrow \) 590

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 719

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

\(\Big \downarrow \) 224

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

\(\Big \downarrow \) 219

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

\(\Big \downarrow \) 488

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

\(\Big \downarrow \) 219

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

Input:

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

Output:

-1/2*(a + b*x^2)^(3/2)/(d*(c + d*x)^2) + (3*b*(((2*c + d*x)*Sqrt[a + b*x^2 
])/(d^2*(c + d*x)) + ((-2*Sqrt[b]*c*ArcTanh[(Sqrt[b]*x)/Sqrt[a + b*x^2]])/ 
d - ((2*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*Sqrt[b*c^2 + a*d^2]))/d^2))/(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 590

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 + 1)*x)/(d^2*( 
n + 1)*(n + 2*p + 2))), x] + Simp[2*(p/(d^2*(n + 1)*(n + 2*p + 2)))   Int[( 
c + d*x)^(n + 1)*(a + b*x^2)^(p - 1)*(a*d*(n + 1) + b*c*(2*p + 1)*x), x], x 
] /; FreeQ[{a, b, c, d}, x] && GtQ[p, 0] && LtQ[n, -1] &&  !ILtQ[n + 2*p + 
1, 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. \(882\) vs. \(2(158)=316\).

Time = 0.41 (sec) , antiderivative size = 883, normalized size of antiderivative = 4.85

method	result	size
risch	\(\frac {b \sqrt {b \,x^{2}+a}}{d^{3}}-\frac {-\frac {\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}}}}{2 \left (a \,d^{2}+b \,c^{2}\right ) \left (x +\frac {c}{d}\right )^{2}}+\frac {3 b c d \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 )}{2 \left (a \,d^{2}+b \,c^{2}\right )}+\frac {b \,d^{2} \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 )}{2 \left (a \,d^{2}+b \,c^{2}\right ) \sqrt {\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}}\right )}{d^{4}}+\frac {3 b^{\frac {3}{2}} c \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{d}+\frac {2 b \left (a \,d^{2}+3 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}}}}+\frac {4 b c \left (a \,d^{2}+b \,c^{2}\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}}}{d^{3}}\)	\(883\)
default	\(\text {Expression too large to display}\)	\(1471\)

Input:

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

Output:

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 370 vs. \(2 (159) = 318\).

Time = 0.45 (sec) , antiderivative size = 1545, normalized size of antiderivative = 8.49 \[ \int \frac {\left (a+b x^2\right )^{3/2}}{(c+d x)^3} \, dx=\text {Too large to display} \] Input:

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

Output:

[1/4*(6*(b^2*c^5 + a*b*c^3*d^2 + (b^2*c^3*d^2 + a*b*c*d^4)*x^2 + 2*(b^2*c^ 
4*d + a*b*c^2*d^3)*x)*sqrt(b)*log(-2*b*x^2 + 2*sqrt(b*x^2 + a)*sqrt(b)*x - 
 a) + 3*(2*b^2*c^4 + a*b*c^2*d^2 + (2*b^2*c^2*d^2 + a*b*d^4)*x^2 + 2*(2*b^ 
2*c^3*d + a*b*c*d^3)*x)*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)) + 2*(6*b^2*c^4*d + 5*a*b*c^2* 
d^3 - a^2*d^5 + 2*(b^2*c^2*d^3 + a*b*d^5)*x^2 + 9*(b^2*c^3*d^2 + a*b*c*d^4 
)*x)*sqrt(b*x^2 + a))/(b*c^4*d^4 + a*c^2*d^6 + (b*c^2*d^6 + a*d^8)*x^2 + 2 
*(b*c^3*d^5 + a*c*d^7)*x), 1/4*(12*(b^2*c^5 + a*b*c^3*d^2 + (b^2*c^3*d^2 + 
 a*b*c*d^4)*x^2 + 2*(b^2*c^4*d + a*b*c^2*d^3)*x)*sqrt(-b)*arctan(sqrt(-b)* 
x/sqrt(b*x^2 + a)) + 3*(2*b^2*c^4 + a*b*c^2*d^2 + (2*b^2*c^2*d^2 + a*b*d^4 
)*x^2 + 2*(2*b^2*c^3*d + a*b*c*d^3)*x)*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)) + 2*(6*b^2*c^4 
*d + 5*a*b*c^2*d^3 - a^2*d^5 + 2*(b^2*c^2*d^3 + a*b*d^5)*x^2 + 9*(b^2*c^3* 
d^2 + a*b*c*d^4)*x)*sqrt(b*x^2 + a))/(b*c^4*d^4 + a*c^2*d^6 + (b*c^2*d^6 + 
 a*d^8)*x^2 + 2*(b*c^3*d^5 + a*c*d^7)*x), -1/2*(3*(2*b^2*c^4 + a*b*c^2*d^2 
 + (2*b^2*c^2*d^2 + a*b*d^4)*x^2 + 2*(2*b^2*c^3*d + a*b*c*d^3)*x)*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*(b^2*c^5 + a*b*c^3*d^2 ...
 

Sympy [F]

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

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

Output:

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

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 341 vs. \(2 (159) = 318\).

Time = 0.07 (sec) , antiderivative size = 341, normalized size of antiderivative = 1.87 \[ \int \frac {\left (a+b x^2\right )^{3/2}}{(c+d x)^3} \, dx=\frac {3 \, \sqrt {b x^{2} + a} b^{2} c^{2}}{2 \, {\left (b c^{2} d^{3} + a d^{5}\right )}} - \frac {3 \, \sqrt {b x^{2} + a} b^{2} c x}{2 \, {\left (b c^{2} d^{2} + a d^{4}\right )}} + \frac {{\left (b x^{2} + a\right )}^{\frac {3}{2}} b c}{2 \, {\left (b c^{2} d^{2} x + a d^{4} x + b c^{3} d + a c d^{3}\right )}} - \frac {{\left (b x^{2} + a\right )}^{\frac {5}{2}}}{2 \, {\left (b c^{2} d x^{2} + a d^{3} x^{2} + 2 \, b c^{3} x + 2 \, a c d^{2} x + \frac {b c^{4}}{d} + a c^{2} d\right )}} + \frac {{\left (b x^{2} + a\right )}^{\frac {3}{2}} b}{2 \, {\left (b c^{2} d + a d^{3}\right )}} - \frac {3 \, b^{\frac {3}{2}} c \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{d^{4}} + \frac {3 \, b^{2} c^{2} \operatorname {arsinh}\left (\frac {b c x}{\sqrt {a b} {\left | d x + c \right |}} - \frac {a d}{\sqrt {a b} {\left | d x + c \right |}}\right )}{2 \, \sqrt {a + \frac {b c^{2}}{d^{2}}} d^{5}} + \frac {3 \, \sqrt {a + \frac {b c^{2}}{d^{2}}} b \operatorname {arsinh}\left (\frac {b c x}{\sqrt {a b} {\left | d x + c \right |}} - \frac {a d}{\sqrt {a b} {\left | d x + c \right |}}\right )}{2 \, d^{3}} + \frac {3 \, \sqrt {b x^{2} + a} b}{2 \, d^{3}} \] Input:

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

Output:

3/2*sqrt(b*x^2 + a)*b^2*c^2/(b*c^2*d^3 + a*d^5) - 3/2*sqrt(b*x^2 + a)*b^2* 
c*x/(b*c^2*d^2 + a*d^4) + 1/2*(b*x^2 + a)^(3/2)*b*c/(b*c^2*d^2*x + a*d^4*x 
 + b*c^3*d + a*c*d^3) - 1/2*(b*x^2 + a)^(5/2)/(b*c^2*d*x^2 + a*d^3*x^2 + 2 
*b*c^3*x + 2*a*c*d^2*x + b*c^4/d + a*c^2*d) + 1/2*(b*x^2 + a)^(3/2)*b/(b*c 
^2*d + a*d^3) - 3*b^(3/2)*c*arcsinh(b*x/sqrt(a*b))/d^4 + 3/2*b^2*c^2*arcsi 
nh(b*c*x/(sqrt(a*b)*abs(d*x + c)) - a*d/(sqrt(a*b)*abs(d*x + c)))/(sqrt(a 
+ b*c^2/d^2)*d^5) + 3/2*sqrt(a + b*c^2/d^2)*b*arcsinh(b*c*x/(sqrt(a*b)*abs 
(d*x + c)) - a*d/(sqrt(a*b)*abs(d*x + c)))/d^3 + 3/2*sqrt(b*x^2 + a)*b/d^3
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 349 vs. \(2 (159) = 318\).

Time = 0.15 (sec) , antiderivative size = 349, normalized size of antiderivative = 1.92 \[ \int \frac {\left (a+b x^2\right )^{3/2}}{(c+d x)^3} \, dx=\frac {3 \, b^{\frac {3}{2}} c \log \left ({\left | -\sqrt {b} x + \sqrt {b x^{2} + a} \right |}\right )}{d^{4}} + \frac {\sqrt {b x^{2} + a} b}{d^{3}} + \frac {3 \, {\left (2 \, b^{2} c^{2} + a b d^{2}\right )} \arctan \left (-\frac {{\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )} d + \sqrt {b} c}{\sqrt {-b c^{2} - a d^{2}}}\right )}{\sqrt {-b c^{2} - a d^{2}} d^{4}} + \frac {6 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{3} b^{2} c^{2} d + {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{3} a b d^{3} + 10 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} b^{\frac {5}{2}} c^{3} - 5 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} a b^{\frac {3}{2}} c d^{2} - 14 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )} a b^{2} c^{2} d + {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )} a^{2} b d^{3} + 5 \, a^{2} b^{\frac {3}{2}} c d^{2}}{{\left ({\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} d + 2 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )} \sqrt {b} c - a d\right )}^{2} d^{4}} \] Input:

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

Output:

3*b^(3/2)*c*log(abs(-sqrt(b)*x + sqrt(b*x^2 + a)))/d^4 + sqrt(b*x^2 + a)*b 
/d^3 + 3*(2*b^2*c^2 + a*b*d^2)*arctan(-((sqrt(b)*x - sqrt(b*x^2 + a))*d + 
sqrt(b)*c)/sqrt(-b*c^2 - a*d^2))/(sqrt(-b*c^2 - a*d^2)*d^4) + (6*(sqrt(b)* 
x - sqrt(b*x^2 + a))^3*b^2*c^2*d + (sqrt(b)*x - sqrt(b*x^2 + a))^3*a*b*d^3 
 + 10*(sqrt(b)*x - sqrt(b*x^2 + a))^2*b^(5/2)*c^3 - 5*(sqrt(b)*x - sqrt(b* 
x^2 + a))^2*a*b^(3/2)*c*d^2 - 14*(sqrt(b)*x - sqrt(b*x^2 + a))*a*b^2*c^2*d 
 + (sqrt(b)*x - sqrt(b*x^2 + a))*a^2*b*d^3 + 5*a^2*b^(3/2)*c*d^2)/(((sqrt( 
b)*x - sqrt(b*x^2 + a))^2*d + 2*(sqrt(b)*x - sqrt(b*x^2 + a))*sqrt(b)*c - 
a*d)^2*d^4)
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.18 (sec) , antiderivative size = 999, normalized size of antiderivative = 5.49 \[ \int \frac {\left (a+b x^2\right )^{3/2}}{(c+d x)^3} \, dx =\text {Too large to display} \] Input:

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

Output:

(3*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)*a*b*c**2*d**2 + 6*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)*a*b*c*d**3*x + 3*sqrt(a*d**2 + b*c**2)*lo 
g(sqrt(a + b*x**2)*sqrt(a*d**2 + b*c**2) - a*d + b*c*x)*a*b*d**4*x**2 + 6* 
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**2*c**4 + 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**2*c**3*d*x + 6*sqrt(a*d**2 + b*c**2)*log(sqr 
t(a + b*x**2)*sqrt(a*d**2 + b*c**2) - a*d + b*c*x)*b**2*c**2*d**2*x**2 - 3 
*sqrt(a*d**2 + b*c**2)*log(c + d*x)*a*b*c**2*d**2 - 6*sqrt(a*d**2 + b*c**2 
)*log(c + d*x)*a*b*c*d**3*x - 3*sqrt(a*d**2 + b*c**2)*log(c + d*x)*a*b*d** 
4*x**2 - 6*sqrt(a*d**2 + b*c**2)*log(c + d*x)*b**2*c**4 - 12*sqrt(a*d**2 + 
 b*c**2)*log(c + d*x)*b**2*c**3*d*x - 6*sqrt(a*d**2 + b*c**2)*log(c + d*x) 
*b**2*c**2*d**2*x**2 - sqrt(a + b*x**2)*a**2*d**5 + 5*sqrt(a + b*x**2)*a*b 
*c**2*d**3 + 9*sqrt(a + b*x**2)*a*b*c*d**4*x + 2*sqrt(a + b*x**2)*a*b*d**5 
*x**2 + 6*sqrt(a + b*x**2)*b**2*c**4*d + 9*sqrt(a + b*x**2)*b**2*c**3*d**2 
*x + 2*sqrt(a + b*x**2)*b**2*c**2*d**3*x**2 + 3*sqrt(b)*log(sqrt(a + b*x** 
2) - sqrt(b)*x)*a*b*c**3*d**2 + 6*sqrt(b)*log(sqrt(a + b*x**2) - sqrt(b)*x 
)*a*b*c**2*d**3*x + 3*sqrt(b)*log(sqrt(a + b*x**2) - sqrt(b)*x)*a*b*c*d**4 
*x**2 + 3*sqrt(b)*log(sqrt(a + b*x**2) - sqrt(b)*x)*b**2*c**5 + 6*sqrt(b)* 
log(sqrt(a + b*x**2) - sqrt(b)*x)*b**2*c**4*d*x + 3*sqrt(b)*log(sqrt(a ...