\(\int \frac {(c+d x)^{3/2}}{a-b x^2} \, dx\) [573]

Optimal result

Integrand size = 20, antiderivative size = 149 \[ \int \frac {(c+d x)^{3/2}}{a-b x^2} \, dx=-\frac {2 d \sqrt {c+d x}}{b}-\frac {\left (\sqrt {b} c-\sqrt {a} d\right )^{3/2} \text {arctanh}\left (\frac {\sqrt [4]{b} \sqrt {c+d x}}{\sqrt {\sqrt {b} c-\sqrt {a} d}}\right )}{\sqrt {a} b^{5/4}}+\frac {\left (\sqrt {b} c+\sqrt {a} d\right )^{3/2} \text {arctanh}\left (\frac {\sqrt [4]{b} \sqrt {c+d x}}{\sqrt {\sqrt {b} c+\sqrt {a} d}}\right )}{\sqrt {a} b^{5/4}} \] Output:

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

Mathematica [A] (verified)

Time = 0.36 (sec) , antiderivative size = 208, normalized size of antiderivative = 1.40 \[ \int \frac {(c+d x)^{3/2}}{a-b x^2} \, dx=\frac {-2 d \sqrt {c+d x}+\frac {\left (\sqrt {b} c+\sqrt {a} d\right )^2 \arctan \left (\frac {\sqrt {-b c-\sqrt {a} \sqrt {b} d} \sqrt {c+d x}}{\sqrt {b} c+\sqrt {a} d}\right )}{\sqrt {a} \sqrt {-b c-\sqrt {a} \sqrt {b} d}}-\frac {\left (\sqrt {b} c-\sqrt {a} d\right )^2 \arctan \left (\frac {\sqrt {-b c+\sqrt {a} \sqrt {b} d} \sqrt {c+d x}}{\sqrt {b} c-\sqrt {a} d}\right )}{\sqrt {a} \sqrt {-b c+\sqrt {a} \sqrt {b} d}}}{b} \] Input:

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

Output:

(-2*d*Sqrt[c + d*x] + ((Sqrt[b]*c + Sqrt[a]*d)^2*ArcTan[(Sqrt[-(b*c) - Sqr 
t[a]*Sqrt[b]*d]*Sqrt[c + d*x])/(Sqrt[b]*c + Sqrt[a]*d)])/(Sqrt[a]*Sqrt[-(b 
*c) - Sqrt[a]*Sqrt[b]*d]) - ((Sqrt[b]*c - Sqrt[a]*d)^2*ArcTan[(Sqrt[-(b*c) 
 + Sqrt[a]*Sqrt[b]*d]*Sqrt[c + d*x])/(Sqrt[b]*c - Sqrt[a]*d)])/(Sqrt[a]*Sq 
rt[-(b*c) + Sqrt[a]*Sqrt[b]*d]))/b
 

Rubi [A] (verified)

Time = 0.65 (sec) , antiderivative size = 167, normalized size of antiderivative = 1.12, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {481, 25, 654, 27, 1480, 221}

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.

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

\(\Big \downarrow \) 481

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 654

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1480

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

\(\Big \downarrow \) 221

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

Input:

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

Output:

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

Defintions of rubi rules used

rule 25

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

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 221

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

rule 481

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

rule 654

Int[((f_.) + (g_.)*(x_))/(Sqrt[(d_.) + (e_.)*(x_)]*((a_) + (c_.)*(x_)^2)), 
x_Symbol] :> Simp[2   Subst[Int[(e*f - d*g + g*x^2)/(c*d^2 + a*e^2 - 2*c*d* 
x^2 + c*x^4), x], x, Sqrt[d + e*x]], x] /; FreeQ[{a, c, d, e, f, g}, x]
 

rule 1480

Int[((d_) + (e_.)*(x_)^2)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] : 
> With[{q = Rt[b^2 - 4*a*c, 2]}, Simp[(e/2 + (2*c*d - b*e)/(2*q))   Int[1/( 
b/2 - q/2 + c*x^2), x], x] + Simp[(e/2 - (2*c*d - b*e)/(2*q))   Int[1/(b/2 
+ q/2 + c*x^2), x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] 
 && NeQ[c*d^2 - a*e^2, 0] && PosQ[b^2 - 4*a*c]
 

Maple [A] (verified)

Time = 0.37 (sec) , antiderivative size = 168, normalized size of antiderivative = 1.13

method	result	size
derivativedivides	\(-2 d \left (\frac {\sqrt {d x +c}}{b}-\frac {\left (a \,d^{2}+b \,c^{2}+2 \sqrt {a b \,d^{2}}\, c \right ) \operatorname {arctanh}\left (\frac {b \sqrt {d x +c}}{\sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}}\right )}{2 \sqrt {a b \,d^{2}}\, \sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}}+\frac {\left (-a \,d^{2}-b \,c^{2}+2 \sqrt {a b \,d^{2}}\, c \right ) \arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (-b c +\sqrt {a b \,d^{2}}\right ) b}}\right )}{2 \sqrt {a b \,d^{2}}\, \sqrt {\left (-b c +\sqrt {a b \,d^{2}}\right ) b}}\right )\)	\(168\)
pseudoelliptic	\(-d \left (\frac {2 \sqrt {d x +c}}{b}-\frac {\left (a \,d^{2}+b \,c^{2}+2 \sqrt {a b \,d^{2}}\, c \right ) \operatorname {arctanh}\left (\frac {b \sqrt {d x +c}}{\sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}}\right )}{\sqrt {a b \,d^{2}}\, \sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}}+\frac {\left (-a \,d^{2}-b \,c^{2}+2 \sqrt {a b \,d^{2}}\, c \right ) \arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (-b c +\sqrt {a b \,d^{2}}\right ) b}}\right )}{\sqrt {a b \,d^{2}}\, \sqrt {\left (-b c +\sqrt {a b \,d^{2}}\right ) b}}\right )\)	\(168\)
default	\(2 d \left (-\frac {\sqrt {d x +c}}{b}-\frac {\left (-a \,d^{2}-b \,c^{2}-2 \sqrt {a b \,d^{2}}\, c \right ) \operatorname {arctanh}\left (\frac {b \sqrt {d x +c}}{\sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}}\right )}{2 \sqrt {a b \,d^{2}}\, \sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}}+\frac {\left (a \,d^{2}+b \,c^{2}-2 \sqrt {a b \,d^{2}}\, c \right ) \arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (-b c +\sqrt {a b \,d^{2}}\right ) b}}\right )}{2 \sqrt {a b \,d^{2}}\, \sqrt {\left (-b c +\sqrt {a b \,d^{2}}\right ) b}}\right )\)	\(169\)
risch	\(-\frac {2 d \sqrt {d x +c}}{b}-2 d \left (\frac {\left (-a \,d^{2}-b \,c^{2}+2 \sqrt {a b \,d^{2}}\, c \right ) \arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (-b c +\sqrt {a b \,d^{2}}\right ) b}}\right )}{2 \sqrt {a b \,d^{2}}\, \sqrt {\left (-b c +\sqrt {a b \,d^{2}}\right ) b}}-\frac {\left (a \,d^{2}+b \,c^{2}+2 \sqrt {a b \,d^{2}}\, c \right ) \operatorname {arctanh}\left (\frac {b \sqrt {d x +c}}{\sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}}\right )}{2 \sqrt {a b \,d^{2}}\, \sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}}\right )\)	\(171\)

Input:

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

Output:

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 974 vs. \(2 (107) = 214\).

Time = 0.12 (sec) , antiderivative size = 974, normalized size of antiderivative = 6.54 \[ \int \frac {(c+d x)^{3/2}}{a-b x^2} \, dx =\text {Too large to display} \] Input:

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

Output:

1/2*(b*sqrt((b*c^3 + 3*a*c*d^2 + a*b^2*sqrt((9*b^2*c^4*d^2 + 6*a*b*c^2*d^4 
 + a^2*d^6)/(a*b^5)))/(a*b^2))*log(-(3*b^2*c^4*d - 2*a*b*c^2*d^3 - a^2*d^5 
)*sqrt(d*x + c) + (3*a*b^2*c^2*d^2 + a^2*b*d^4 - a*b^4*c*sqrt((9*b^2*c^4*d 
^2 + 6*a*b*c^2*d^4 + a^2*d^6)/(a*b^5)))*sqrt((b*c^3 + 3*a*c*d^2 + a*b^2*sq 
rt((9*b^2*c^4*d^2 + 6*a*b*c^2*d^4 + a^2*d^6)/(a*b^5)))/(a*b^2))) - b*sqrt( 
(b*c^3 + 3*a*c*d^2 + a*b^2*sqrt((9*b^2*c^4*d^2 + 6*a*b*c^2*d^4 + a^2*d^6)/ 
(a*b^5)))/(a*b^2))*log(-(3*b^2*c^4*d - 2*a*b*c^2*d^3 - a^2*d^5)*sqrt(d*x + 
 c) - (3*a*b^2*c^2*d^2 + a^2*b*d^4 - a*b^4*c*sqrt((9*b^2*c^4*d^2 + 6*a*b*c 
^2*d^4 + a^2*d^6)/(a*b^5)))*sqrt((b*c^3 + 3*a*c*d^2 + a*b^2*sqrt((9*b^2*c^ 
4*d^2 + 6*a*b*c^2*d^4 + a^2*d^6)/(a*b^5)))/(a*b^2))) + b*sqrt((b*c^3 + 3*a 
*c*d^2 - a*b^2*sqrt((9*b^2*c^4*d^2 + 6*a*b*c^2*d^4 + a^2*d^6)/(a*b^5)))/(a 
*b^2))*log(-(3*b^2*c^4*d - 2*a*b*c^2*d^3 - a^2*d^5)*sqrt(d*x + c) + (3*a*b 
^2*c^2*d^2 + a^2*b*d^4 + a*b^4*c*sqrt((9*b^2*c^4*d^2 + 6*a*b*c^2*d^4 + a^2 
*d^6)/(a*b^5)))*sqrt((b*c^3 + 3*a*c*d^2 - a*b^2*sqrt((9*b^2*c^4*d^2 + 6*a* 
b*c^2*d^4 + a^2*d^6)/(a*b^5)))/(a*b^2))) - b*sqrt((b*c^3 + 3*a*c*d^2 - a*b 
^2*sqrt((9*b^2*c^4*d^2 + 6*a*b*c^2*d^4 + a^2*d^6)/(a*b^5)))/(a*b^2))*log(- 
(3*b^2*c^4*d - 2*a*b*c^2*d^3 - a^2*d^5)*sqrt(d*x + c) - (3*a*b^2*c^2*d^2 + 
 a^2*b*d^4 + a*b^4*c*sqrt((9*b^2*c^4*d^2 + 6*a*b*c^2*d^4 + a^2*d^6)/(a*b^5 
)))*sqrt((b*c^3 + 3*a*c*d^2 - a*b^2*sqrt((9*b^2*c^4*d^2 + 6*a*b*c^2*d^4 + 
a^2*d^6)/(a*b^5)))/(a*b^2))) - 4*sqrt(d*x + c)*d)/b
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 314 vs. \(2 (107) = 214\).

Time = 0.17 (sec) , antiderivative size = 314, normalized size of antiderivative = 2.11 \[ \int \frac {(c+d x)^{3/2}}{a-b x^2} \, dx=-\frac {2 \, \sqrt {d x + c} d}{b} - \frac {{\left (\sqrt {a b} b^{3} c^{3} d - \sqrt {a b} a b^{2} c d^{3} + {\left (a b^{2} c^{2} d - a^{2} b d^{3}\right )} {\left | b \right |} {\left | d \right |}\right )} \arctan \left (\frac {\sqrt {d x + c}}{\sqrt {-\frac {b^{2} c + \sqrt {b^{4} c^{2} - {\left (b^{2} c^{2} - a b d^{2}\right )} b^{2}}}{b^{2}}}}\right )}{{\left (a b^{3} c - \sqrt {a b} a b^{2} d\right )} \sqrt {-b^{2} c - \sqrt {a b} b d} {\left | d \right |}} + \frac {{\left (\sqrt {a b} b^{3} c^{3} d - \sqrt {a b} a b^{2} c d^{3} - {\left (a b^{2} c^{2} d - a^{2} b d^{3}\right )} {\left | b \right |} {\left | d \right |}\right )} \arctan \left (\frac {\sqrt {d x + c}}{\sqrt {-\frac {b^{2} c - \sqrt {b^{4} c^{2} - {\left (b^{2} c^{2} - a b d^{2}\right )} b^{2}}}{b^{2}}}}\right )}{{\left (a b^{3} c + \sqrt {a b} a b^{2} d\right )} \sqrt {-b^{2} c + \sqrt {a b} b d} {\left | d \right |}} \] Input:

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

Output:

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

Mupad [B] (verification not implemented)

Time = 8.29 (sec) , antiderivative size = 1581, normalized size of antiderivative = 10.61 \[ \int \frac {(c+d x)^{3/2}}{a-b x^2} \, dx=\text {Too large to display} \] Input:

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

Output:

2*atanh((96*c^3*d^3*(a^3*b^5)^(1/2)*(c + d*x)^(1/2)*((3*c*d^2)/(4*b^2) + c 
^3/(4*a*b) - (d^3*(a^3*b^5)^(1/2))/(4*a*b^5) - (3*c^2*d*(a^3*b^5)^(1/2))/( 
4*a^2*b^4))^(1/2))/(16*a^3*c*d^7 - 48*a*b^2*c^5*d^3 + 32*a^2*b*c^3*d^5 - ( 
16*a^2*d^8*(a^3*b^5)^(1/2))/b^3 + (48*c^4*d^4*(a^3*b^5)^(1/2))/b - (32*a*c 
^2*d^6*(a^3*b^5)^(1/2))/b^2) - (32*c*d^5*(a^3*b^5)^(1/2)*(c + d*x)^(1/2)*( 
(3*c*d^2)/(4*b^2) + c^3/(4*a*b) - (d^3*(a^3*b^5)^(1/2))/(4*a*b^5) - (3*c^2 
*d*(a^3*b^5)^(1/2))/(4*a^2*b^4))^(1/2))/(48*b^3*c^5*d^3 - 32*a*b^2*c^3*d^5 
 + (16*a*d^8*(a^3*b^5)^(1/2))/b^2 - 16*a^2*b*c*d^7 - (48*c^4*d^4*(a^3*b^5) 
^(1/2))/a + (32*c^2*d^6*(a^3*b^5)^(1/2))/b) + (32*a^2*b*d^6*(c + d*x)^(1/2 
)*((3*c*d^2)/(4*b^2) + c^3/(4*a*b) - (d^3*(a^3*b^5)^(1/2))/(4*a*b^5) - (3* 
c^2*d*(a^3*b^5)^(1/2))/(4*a^2*b^4))^(1/2))/(16*a^2*c*d^7 - 48*b^2*c^5*d^3 
- (16*a*d^8*(a^3*b^5)^(1/2))/b^3 + 32*a*b*c^3*d^5 - (32*c^2*d^6*(a^3*b^5)^ 
(1/2))/b^2 + (48*c^4*d^4*(a^3*b^5)^(1/2))/(a*b)) + (96*a*b^2*c^2*d^4*(c + 
d*x)^(1/2)*((3*c*d^2)/(4*b^2) + c^3/(4*a*b) - (d^3*(a^3*b^5)^(1/2))/(4*a*b 
^5) - (3*c^2*d*(a^3*b^5)^(1/2))/(4*a^2*b^4))^(1/2))/(16*a^2*c*d^7 - 48*b^2 
*c^5*d^3 - (16*a*d^8*(a^3*b^5)^(1/2))/b^3 + 32*a*b*c^3*d^5 - (32*c^2*d^6*( 
a^3*b^5)^(1/2))/b^2 + (48*c^4*d^4*(a^3*b^5)^(1/2))/(a*b)))*((a*b^4*c^3 - a 
*d^3*(a^3*b^5)^(1/2) + 3*a^2*b^3*c*d^2 - 3*b*c^2*d*(a^3*b^5)^(1/2))/(4*a^2 
*b^5))^(1/2) - 2*atanh((32*c*d^5*(a^3*b^5)^(1/2)*(c + d*x)^(1/2)*((3*c*d^2 
)/(4*b^2) + c^3/(4*a*b) + (d^3*(a^3*b^5)^(1/2))/(4*a*b^5) + (3*c^2*d*(a...
 

Reduce [B] (verification not implemented)

Time = 0.20 (sec) , antiderivative size = 268, normalized size of antiderivative = 1.80 \[ \int \frac {(c+d x)^{3/2}}{a-b x^2} \, dx=\frac {-2 \sqrt {a}\, \sqrt {\sqrt {b}\, \sqrt {a}\, d -b c}\, \mathit {atan} \left (\frac {\sqrt {d x +c}\, b}{\sqrt {b}\, \sqrt {\sqrt {b}\, \sqrt {a}\, d -b c}}\right ) b c +2 \sqrt {b}\, \sqrt {\sqrt {b}\, \sqrt {a}\, d -b c}\, \mathit {atan} \left (\frac {\sqrt {d x +c}\, b}{\sqrt {b}\, \sqrt {\sqrt {b}\, \sqrt {a}\, d -b c}}\right ) a d -\sqrt {a}\, \sqrt {\sqrt {b}\, \sqrt {a}\, d +b c}\, \mathrm {log}\left (-\sqrt {\sqrt {b}\, \sqrt {a}\, d +b c}+\sqrt {b}\, \sqrt {d x +c}\right ) b c +\sqrt {a}\, \sqrt {\sqrt {b}\, \sqrt {a}\, d +b c}\, \mathrm {log}\left (\sqrt {\sqrt {b}\, \sqrt {a}\, d +b c}+\sqrt {b}\, \sqrt {d x +c}\right ) b c -\sqrt {b}\, \sqrt {\sqrt {b}\, \sqrt {a}\, d +b c}\, \mathrm {log}\left (-\sqrt {\sqrt {b}\, \sqrt {a}\, d +b c}+\sqrt {b}\, \sqrt {d x +c}\right ) a d +\sqrt {b}\, \sqrt {\sqrt {b}\, \sqrt {a}\, d +b c}\, \mathrm {log}\left (\sqrt {\sqrt {b}\, \sqrt {a}\, d +b c}+\sqrt {b}\, \sqrt {d x +c}\right ) a d -4 \sqrt {d x +c}\, a b d}{2 a \,b^{2}} \] Input:

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

Output:

( - 2*sqrt(a)*sqrt(sqrt(b)*sqrt(a)*d - b*c)*atan((sqrt(c + d*x)*b)/(sqrt(b 
)*sqrt(sqrt(b)*sqrt(a)*d - b*c)))*b*c + 2*sqrt(b)*sqrt(sqrt(b)*sqrt(a)*d - 
 b*c)*atan((sqrt(c + d*x)*b)/(sqrt(b)*sqrt(sqrt(b)*sqrt(a)*d - b*c)))*a*d 
- sqrt(a)*sqrt(sqrt(b)*sqrt(a)*d + b*c)*log( - sqrt(sqrt(b)*sqrt(a)*d + b* 
c) + sqrt(b)*sqrt(c + d*x))*b*c + sqrt(a)*sqrt(sqrt(b)*sqrt(a)*d + b*c)*lo 
g(sqrt(sqrt(b)*sqrt(a)*d + b*c) + sqrt(b)*sqrt(c + d*x))*b*c - sqrt(b)*sqr 
t(sqrt(b)*sqrt(a)*d + b*c)*log( - sqrt(sqrt(b)*sqrt(a)*d + b*c) + sqrt(b)* 
sqrt(c + d*x))*a*d + sqrt(b)*sqrt(sqrt(b)*sqrt(a)*d + b*c)*log(sqrt(sqrt(b 
)*sqrt(a)*d + b*c) + sqrt(b)*sqrt(c + d*x))*a*d - 4*sqrt(c + d*x)*a*b*d)/( 
2*a*b**2)