\(\int \frac {1}{\sqrt {c+d x} (a-b x^2)^3} \, dx\) [722]

Optimal result

Integrand size = 20, antiderivative size = 315 \[ \int \frac {1}{\sqrt {c+d x} \left (a-b x^2\right )^3} \, dx=-\frac {(a d-b c x) \sqrt {c+d x}}{4 a \left (b c^2-a d^2\right ) \left (a-b x^2\right )^2}-\frac {\sqrt {c+d x} \left (a d \left (b c^2-7 a d^2\right )-6 b c \left (b c^2-2 a d^2\right ) x\right )}{16 a^2 \left (b c^2-a d^2\right )^2 \left (a-b x^2\right )}-\frac {3 \left (4 b c^2-10 \sqrt {a} \sqrt {b} c d+7 a d^2\right ) \text {arctanh}\left (\frac {\sqrt [4]{b} \sqrt {c+d x}}{\sqrt {\sqrt {b} c-\sqrt {a} d}}\right )}{32 a^{5/2} \sqrt [4]{b} \left (\sqrt {b} c-\sqrt {a} d\right )^{5/2}}+\frac {3 \left (4 b c^2+10 \sqrt {a} \sqrt {b} c d+7 a d^2\right ) \text {arctanh}\left (\frac {\sqrt [4]{b} \sqrt {c+d x}}{\sqrt {\sqrt {b} c+\sqrt {a} d}}\right )}{32 a^{5/2} \sqrt [4]{b} \left (\sqrt {b} c+\sqrt {a} d\right )^{5/2}} \] Output:

-1/4*(-b*c*x+a*d)*(d*x+c)^(1/2)/a/(-a*d^2+b*c^2)/(-b*x^2+a)^2-1/16*(d*x+c) 
^(1/2)*(a*d*(-7*a*d^2+b*c^2)-6*b*c*(-2*a*d^2+b*c^2)*x)/a^2/(-a*d^2+b*c^2)^ 
2/(-b*x^2+a)-3/32*(4*b*c^2-10*a^(1/2)*b^(1/2)*c*d+7*a*d^2)*arctanh(b^(1/4) 
*(d*x+c)^(1/2)/(b^(1/2)*c-a^(1/2)*d)^(1/2))/a^(5/2)/b^(1/4)/(b^(1/2)*c-a^( 
1/2)*d)^(5/2)+3/32*(4*b*c^2+10*a^(1/2)*b^(1/2)*c*d+7*a*d^2)*arctanh(b^(1/4 
)*(d*x+c)^(1/2)/(b^(1/2)*c+a^(1/2)*d)^(1/2))/a^(5/2)/b^(1/4)/(b^(1/2)*c+a^ 
(1/2)*d)^(5/2)
 

Mathematica [A] (verified)

Time = 1.15 (sec) , antiderivative size = 357, normalized size of antiderivative = 1.13 \[ \int \frac {1}{\sqrt {c+d x} \left (a-b x^2\right )^3} \, dx=\frac {-\frac {2 \sqrt {a} \sqrt {c+d x} \left (-11 a^3 d^3+6 b^3 c^3 x^3+a^2 b d \left (5 c^2+16 c d x+7 d^2 x^2\right )-a b^2 c x \left (10 c^2+c d x+12 d^2 x^2\right )\right )}{\left (b c^2-a d^2\right )^2 \left (a-b x^2\right )^2}+\frac {3 \left (4 b c^2+10 \sqrt {a} \sqrt {b} c d+7 a d^2\right ) \arctan \left (\frac {\sqrt {-b c-\sqrt {a} \sqrt {b} d} \sqrt {c+d x}}{\sqrt {b} c+\sqrt {a} d}\right )}{\left (\sqrt {b} c+\sqrt {a} d\right )^2 \sqrt {-b c-\sqrt {a} \sqrt {b} d}}-\frac {3 \left (4 b c^2-10 \sqrt {a} \sqrt {b} c d+7 a d^2\right ) \arctan \left (\frac {\sqrt {-b c+\sqrt {a} \sqrt {b} d} \sqrt {c+d x}}{\sqrt {b} c-\sqrt {a} d}\right )}{\left (\sqrt {b} c-\sqrt {a} d\right )^2 \sqrt {-b c+\sqrt {a} \sqrt {b} d}}}{32 a^{5/2}} \] Input:

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

Output:

((-2*Sqrt[a]*Sqrt[c + d*x]*(-11*a^3*d^3 + 6*b^3*c^3*x^3 + a^2*b*d*(5*c^2 + 
 16*c*d*x + 7*d^2*x^2) - a*b^2*c*x*(10*c^2 + c*d*x + 12*d^2*x^2)))/((b*c^2 
 - a*d^2)^2*(a - b*x^2)^2) + (3*(4*b*c^2 + 10*Sqrt[a]*Sqrt[b]*c*d + 7*a*d^ 
2)*ArcTan[(Sqrt[-(b*c) - Sqrt[a]*Sqrt[b]*d]*Sqrt[c + d*x])/(Sqrt[b]*c + Sq 
rt[a]*d)])/((Sqrt[b]*c + Sqrt[a]*d)^2*Sqrt[-(b*c) - Sqrt[a]*Sqrt[b]*d]) - 
(3*(4*b*c^2 - 10*Sqrt[a]*Sqrt[b]*c*d + 7*a*d^2)*ArcTan[(Sqrt[-(b*c) + Sqrt 
[a]*Sqrt[b]*d]*Sqrt[c + d*x])/(Sqrt[b]*c - Sqrt[a]*d)])/((Sqrt[b]*c - Sqrt 
[a]*d)^2*Sqrt[-(b*c) + Sqrt[a]*Sqrt[b]*d]))/(32*a^(5/2))
 

Rubi [A] (verified)

Time = 1.12 (sec) , antiderivative size = 401, normalized size of antiderivative = 1.27, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.450, Rules used = {496, 27, 686, 27, 654, 25, 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.

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

\(\Big \downarrow \) 496

\(\displaystyle \frac {\int \frac {6 b c^2+5 b d x c-7 a d^2}{2 \sqrt {c+d x} \left (a-b x^2\right )^2}dx}{4 a \left (b c^2-a d^2\right )}-\frac {\sqrt {c+d x} (a d-b c x)}{4 a \left (a-b x^2\right )^2 \left (b c^2-a d^2\right )}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\int \frac {6 b c^2+5 b d x c-7 a d^2}{\sqrt {c+d x} \left (a-b x^2\right )^2}dx}{8 a \left (b c^2-a d^2\right )}-\frac {\sqrt {c+d x} (a d-b c x)}{4 a \left (a-b x^2\right )^2 \left (b c^2-a d^2\right )}\)

\(\Big \downarrow \) 686

\(\displaystyle \frac {-\frac {\int -\frac {3 b \left (4 b^2 c^4-9 a b d^2 c^2+2 b d \left (b c^2-2 a d^2\right ) x c+7 a^2 d^4\right )}{2 \sqrt {c+d x} \left (a-b x^2\right )}dx}{2 a b \left (b c^2-a d^2\right )}-\frac {\sqrt {c+d x} \left (a d \left (b c^2-7 a d^2\right )-6 b c x \left (b c^2-2 a d^2\right )\right )}{2 a \left (a-b x^2\right ) \left (b c^2-a d^2\right )}}{8 a \left (b c^2-a d^2\right )}-\frac {\sqrt {c+d x} (a d-b c x)}{4 a \left (a-b x^2\right )^2 \left (b c^2-a d^2\right )}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {3 \int \frac {4 b^2 c^4-9 a b d^2 c^2+2 b d \left (b c^2-2 a d^2\right ) x c+7 a^2 d^4}{\sqrt {c+d x} \left (a-b x^2\right )}dx}{4 a \left (b c^2-a d^2\right )}-\frac {\sqrt {c+d x} \left (a d \left (b c^2-7 a d^2\right )-6 b c x \left (b c^2-2 a d^2\right )\right )}{2 a \left (a-b x^2\right ) \left (b c^2-a d^2\right )}}{8 a \left (b c^2-a d^2\right )}-\frac {\sqrt {c+d x} (a d-b c x)}{4 a \left (a-b x^2\right )^2 \left (b c^2-a d^2\right )}\)

\(\Big \downarrow \) 654

\(\displaystyle \frac {\frac {3 \int -\frac {d \left (2 b^2 c^4-5 a b d^2 c^2+2 b \left (b c^2-2 a d^2\right ) (c+d x) c+7 a^2 d^4\right )}{b c^2-2 b (c+d x) c-a d^2+b (c+d x)^2}d\sqrt {c+d x}}{2 a \left (b c^2-a d^2\right )}-\frac {\sqrt {c+d x} \left (a d \left (b c^2-7 a d^2\right )-6 b c x \left (b c^2-2 a d^2\right )\right )}{2 a \left (a-b x^2\right ) \left (b c^2-a d^2\right )}}{8 a \left (b c^2-a d^2\right )}-\frac {\sqrt {c+d x} (a d-b c x)}{4 a \left (a-b x^2\right )^2 \left (b c^2-a d^2\right )}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {-\frac {3 \int \frac {d \left (2 b^2 c^4-5 a b d^2 c^2+2 b \left (b c^2-2 a d^2\right ) (c+d x) c+7 a^2 d^4\right )}{b c^2-2 b (c+d x) c-a d^2+b (c+d x)^2}d\sqrt {c+d x}}{2 a \left (b c^2-a d^2\right )}-\frac {\sqrt {c+d x} \left (a d \left (b c^2-7 a d^2\right )-6 b c x \left (b c^2-2 a d^2\right )\right )}{2 a \left (a-b x^2\right ) \left (b c^2-a d^2\right )}}{8 a \left (b c^2-a d^2\right )}-\frac {\sqrt {c+d x} (a d-b c x)}{4 a \left (a-b x^2\right )^2 \left (b c^2-a d^2\right )}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {-\frac {3 d \int \frac {2 b^2 c^4-5 a b d^2 c^2+2 b \left (b c^2-2 a d^2\right ) (c+d x) c+7 a^2 d^4}{b c^2-2 b (c+d x) c-a d^2+b (c+d x)^2}d\sqrt {c+d x}}{2 a \left (b c^2-a d^2\right )}-\frac {\sqrt {c+d x} \left (a d \left (b c^2-7 a d^2\right )-6 b c x \left (b c^2-2 a d^2\right )\right )}{2 a \left (a-b x^2\right ) \left (b c^2-a d^2\right )}}{8 a \left (b c^2-a d^2\right )}-\frac {\sqrt {c+d x} (a d-b c x)}{4 a \left (a-b x^2\right )^2 \left (b c^2-a d^2\right )}\)

\(\Big \downarrow \) 1480

\(\displaystyle \frac {-\frac {3 d \left (\frac {\sqrt {b} \left (\sqrt {b} c-\sqrt {a} d\right )^2 \left (10 \sqrt {a} \sqrt {b} c d+7 a d^2+4 b c^2\right ) \int \frac {1}{b (c+d x)-\sqrt {b} \left (\sqrt {b} c+\sqrt {a} d\right )}d\sqrt {c+d x}}{2 \sqrt {a} d}-\frac {\sqrt {b} \left (\sqrt {a} d+\sqrt {b} c\right )^2 \left (-10 \sqrt {a} \sqrt {b} c d+7 a d^2+4 b c^2\right ) \int \frac {1}{b (c+d x)-\sqrt {b} \left (\sqrt {b} c-\sqrt {a} d\right )}d\sqrt {c+d x}}{2 \sqrt {a} d}\right )}{2 a \left (b c^2-a d^2\right )}-\frac {\sqrt {c+d x} \left (a d \left (b c^2-7 a d^2\right )-6 b c x \left (b c^2-2 a d^2\right )\right )}{2 a \left (a-b x^2\right ) \left (b c^2-a d^2\right )}}{8 a \left (b c^2-a d^2\right )}-\frac {\sqrt {c+d x} (a d-b c x)}{4 a \left (a-b x^2\right )^2 \left (b c^2-a d^2\right )}\)

\(\Big \downarrow \) 221

\(\displaystyle \frac {-\frac {3 d \left (\frac {\left (\sqrt {a} d+\sqrt {b} c\right )^2 \left (-10 \sqrt {a} \sqrt {b} c d+7 a d^2+4 b c^2\right ) \text {arctanh}\left (\frac {\sqrt [4]{b} \sqrt {c+d x}}{\sqrt {\sqrt {b} c-\sqrt {a} d}}\right )}{2 \sqrt {a} \sqrt [4]{b} d \sqrt {\sqrt {b} c-\sqrt {a} d}}-\frac {\left (\sqrt {b} c-\sqrt {a} d\right )^2 \left (10 \sqrt {a} \sqrt {b} c d+7 a d^2+4 b c^2\right ) \text {arctanh}\left (\frac {\sqrt [4]{b} \sqrt {c+d x}}{\sqrt {\sqrt {a} d+\sqrt {b} c}}\right )}{2 \sqrt {a} \sqrt [4]{b} d \sqrt {\sqrt {a} d+\sqrt {b} c}}\right )}{2 a \left (b c^2-a d^2\right )}-\frac {\sqrt {c+d x} \left (a d \left (b c^2-7 a d^2\right )-6 b c x \left (b c^2-2 a d^2\right )\right )}{2 a \left (a-b x^2\right ) \left (b c^2-a d^2\right )}}{8 a \left (b c^2-a d^2\right )}-\frac {\sqrt {c+d x} (a d-b c x)}{4 a \left (a-b x^2\right )^2 \left (b c^2-a d^2\right )}\)

Input:

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

Output:

-1/4*((a*d - b*c*x)*Sqrt[c + d*x])/(a*(b*c^2 - a*d^2)*(a - b*x^2)^2) + (-1 
/2*(Sqrt[c + d*x]*(a*d*(b*c^2 - 7*a*d^2) - 6*b*c*(b*c^2 - 2*a*d^2)*x))/(a* 
(b*c^2 - a*d^2)*(a - b*x^2)) - (3*d*(((Sqrt[b]*c + Sqrt[a]*d)^2*(4*b*c^2 - 
 10*Sqrt[a]*Sqrt[b]*c*d + 7*a*d^2)*ArcTanh[(b^(1/4)*Sqrt[c + d*x])/Sqrt[Sq 
rt[b]*c - Sqrt[a]*d]])/(2*Sqrt[a]*b^(1/4)*d*Sqrt[Sqrt[b]*c - Sqrt[a]*d]) - 
 ((Sqrt[b]*c - Sqrt[a]*d)^2*(4*b*c^2 + 10*Sqrt[a]*Sqrt[b]*c*d + 7*a*d^2)*A 
rcTanh[(b^(1/4)*Sqrt[c + d*x])/Sqrt[Sqrt[b]*c + Sqrt[a]*d]])/(2*Sqrt[a]*b^ 
(1/4)*d*Sqrt[Sqrt[b]*c + Sqrt[a]*d])))/(2*a*(b*c^2 - a*d^2)))/(8*a*(b*c^2 
- a*d^2))
 

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 496

Int[((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[ 
(-(a*d + b*c*x))*(c + d*x)^(n + 1)*((a + b*x^2)^(p + 1)/(2*a*(p + 1)*(b*c^2 
 + a*d^2))), x] + Simp[1/(2*a*(p + 1)*(b*c^2 + a*d^2))   Int[(c + d*x)^n*(a 
 + b*x^2)^(p + 1)*Simp[b*c^2*(2*p + 3) + a*d^2*(n + 2*p + 3) + b*c*d*(n + 2 
*p + 4)*x, x], x], x] /; FreeQ[{a, b, c, d, n}, x] && LtQ[p, -1] && IntQuad 
raticQ[a, 0, b, c, d, n, p, x]
 

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 686

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p 
_), x_Symbol] :> Simp[(-(d + e*x)^(m + 1))*(f*a*c*e - a*g*c*d + c*(c*d*f + 
a*e*g)*x)*((a + c*x^2)^(p + 1)/(2*a*c*(p + 1)*(c*d^2 + a*e^2))), x] + Simp[ 
1/(2*a*c*(p + 1)*(c*d^2 + a*e^2))   Int[(d + e*x)^m*(a + c*x^2)^(p + 1)*Sim 
p[f*(c^2*d^2*(2*p + 3) + a*c*e^2*(m + 2*p + 3)) - a*c*d*e*g*m + c*e*(c*d*f 
+ a*e*g)*(m + 2*p + 4)*x, x], x], x] /; FreeQ[{a, c, d, e, f, g}, x] && LtQ 
[p, -1] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])
 

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 = 1.12 (sec) , antiderivative size = 505, normalized size of antiderivative = 1.60

method	result	size
derivativedivides	\(-2 d^{5} b^{3} \left (\frac {\frac {\frac {3 \sqrt {a b \,d^{2}}\, \left (2 b c -3 \sqrt {a b \,d^{2}}\right ) \left (d x +c \right )^{\frac {3}{2}}}{4 b^{3} \left (a \,d^{2}+b \,c^{2}-2 \sqrt {a b \,d^{2}}\, c \right )}-\frac {\sqrt {a b \,d^{2}}\, \left (6 b c -11 \sqrt {a b \,d^{2}}\right ) \sqrt {d x +c}}{4 b^{3} \left (b c -\sqrt {a b \,d^{2}}\right )}}{{\left (-d x -\frac {\sqrt {a b \,d^{2}}}{b}\right )}^{2}}-\frac {3 \left (-7 a \,d^{2}-4 b \,c^{2}+10 \sqrt {a b \,d^{2}}\, c \right ) \arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (-b c +\sqrt {a b \,d^{2}}\right ) b}}\right )}{4 b \left (-a \,d^{2}-b \,c^{2}+2 \sqrt {a b \,d^{2}}\, c \right ) \sqrt {\left (-b c +\sqrt {a b \,d^{2}}\right ) b}}}{16 b \,a^{2} d^{4} \sqrt {a b \,d^{2}}}-\frac {\frac {-\frac {3 \sqrt {a b \,d^{2}}\, \left (2 b c +3 \sqrt {a b \,d^{2}}\right ) \left (d x +c \right )^{\frac {3}{2}}}{4 b^{3} \left (a \,d^{2}+b \,c^{2}+2 \sqrt {a b \,d^{2}}\, c \right )}+\frac {\sqrt {a b \,d^{2}}\, \left (6 b c +11 \sqrt {a b \,d^{2}}\right ) \sqrt {d x +c}}{4 b^{3} \left (b c +\sqrt {a b \,d^{2}}\right )}}{{\left (-d x +\frac {\sqrt {a b \,d^{2}}}{b}\right )}^{2}}+\frac {3 \left (7 a \,d^{2}+4 b \,c^{2}+10 \sqrt {a b \,d^{2}}\, c \right ) \operatorname {arctanh}\left (\frac {b \sqrt {d x +c}}{\sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}}\right )}{4 b \left (a \,d^{2}+b \,c^{2}+2 \sqrt {a b \,d^{2}}\, c \right ) \sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}}}{16 b \,a^{2} d^{4} \sqrt {a b \,d^{2}}}\right )\)	\(505\)
default	\(-2 d^{5} b^{3} \left (\frac {\frac {\frac {3 \sqrt {a b \,d^{2}}\, \left (2 b c -3 \sqrt {a b \,d^{2}}\right ) \left (d x +c \right )^{\frac {3}{2}}}{4 b^{3} \left (a \,d^{2}+b \,c^{2}-2 \sqrt {a b \,d^{2}}\, c \right )}-\frac {\sqrt {a b \,d^{2}}\, \left (6 b c -11 \sqrt {a b \,d^{2}}\right ) \sqrt {d x +c}}{4 b^{3} \left (b c -\sqrt {a b \,d^{2}}\right )}}{{\left (-d x -\frac {\sqrt {a b \,d^{2}}}{b}\right )}^{2}}-\frac {3 \left (-7 a \,d^{2}-4 b \,c^{2}+10 \sqrt {a b \,d^{2}}\, c \right ) \arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (-b c +\sqrt {a b \,d^{2}}\right ) b}}\right )}{4 b \left (-a \,d^{2}-b \,c^{2}+2 \sqrt {a b \,d^{2}}\, c \right ) \sqrt {\left (-b c +\sqrt {a b \,d^{2}}\right ) b}}}{16 b \,a^{2} d^{4} \sqrt {a b \,d^{2}}}-\frac {\frac {-\frac {3 \sqrt {a b \,d^{2}}\, \left (2 b c +3 \sqrt {a b \,d^{2}}\right ) \left (d x +c \right )^{\frac {3}{2}}}{4 b^{3} \left (a \,d^{2}+b \,c^{2}+2 \sqrt {a b \,d^{2}}\, c \right )}+\frac {\sqrt {a b \,d^{2}}\, \left (6 b c +11 \sqrt {a b \,d^{2}}\right ) \sqrt {d x +c}}{4 b^{3} \left (b c +\sqrt {a b \,d^{2}}\right )}}{{\left (-d x +\frac {\sqrt {a b \,d^{2}}}{b}\right )}^{2}}+\frac {3 \left (7 a \,d^{2}+4 b \,c^{2}+10 \sqrt {a b \,d^{2}}\, c \right ) \operatorname {arctanh}\left (\frac {b \sqrt {d x +c}}{\sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}}\right )}{4 b \left (a \,d^{2}+b \,c^{2}+2 \sqrt {a b \,d^{2}}\, c \right ) \sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}}}{16 b \,a^{2} d^{4} \sqrt {a b \,d^{2}}}\right )\)	\(505\)
pseudoelliptic	\(-\frac {d^{5} b^{3} \left (-\frac {3 \left (d x +c \right )^{\frac {3}{2}} c}{2 \left (b d x +\sqrt {a b \,d^{2}}\right )^{2} b \left (-a \,d^{2}-b \,c^{2}+2 \sqrt {a b \,d^{2}}\, c \right ) a^{2} d^{4}}+\frac {9 \left (d x +c \right )^{\frac {3}{2}}}{4 \left (b d x +\sqrt {a b \,d^{2}}\right )^{2} b a \,d^{2} \left (-a \,d^{2}-b \,c^{2}+2 \sqrt {a b \,d^{2}}\, c \right ) \sqrt {a b \,d^{2}}}+\frac {3 \sqrt {d x +c}\, c}{2 \left (b d x +\sqrt {a b \,d^{2}}\right )^{2} b \left (-b c +\sqrt {a b \,d^{2}}\right ) a^{2} d^{4}}-\frac {11 \sqrt {d x +c}}{4 \left (b d x +\sqrt {a b \,d^{2}}\right )^{2} b a \,d^{2} \left (-b c +\sqrt {a b \,d^{2}}\right ) \sqrt {a b \,d^{2}}}+\frac {21 \arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (-b c +\sqrt {a b \,d^{2}}\right ) b}}\right )}{4 b^{2} \left (-a \,d^{2}-b \,c^{2}+2 \sqrt {a b \,d^{2}}\, c \right ) \sqrt {\left (-b c +\sqrt {a b \,d^{2}}\right ) b}\, a \,d^{2} \sqrt {a b \,d^{2}}}+\frac {3 \arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (-b c +\sqrt {a b \,d^{2}}\right ) b}}\right ) c^{2}}{\left (-a \,d^{2}-b \,c^{2}+2 \sqrt {a b \,d^{2}}\, c \right ) \sqrt {\left (-b c +\sqrt {a b \,d^{2}}\right ) b}\, b \,a^{2} d^{4} \sqrt {a b \,d^{2}}}-\frac {15 \arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (-b c +\sqrt {a b \,d^{2}}\right ) b}}\right ) c}{2 b^{2} \left (-a \,d^{2}-b \,c^{2}+2 \sqrt {a b \,d^{2}}\, c \right ) \sqrt {\left (-b c +\sqrt {a b \,d^{2}}\right ) b}\, a^{2} d^{4}}+\frac {\frac {\sqrt {a b \,d^{2}}\, \left (\frac {3 \left (2 b c +3 \sqrt {a b \,d^{2}}\right ) \left (d x +c \right )^{\frac {3}{2}}}{a \,d^{2}+b \,c^{2}+2 \sqrt {a b \,d^{2}}\, c}-\frac {\left (6 b c +11 \sqrt {a b \,d^{2}}\right ) \sqrt {d x +c}}{b c +\sqrt {a b \,d^{2}}}\right )}{\left (-b d x +\sqrt {a b \,d^{2}}\right )^{2}}-\frac {3 \left (7 a \,d^{2}+4 b \,c^{2}+10 \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 )}{\left (a \,d^{2}+b \,c^{2}+2 \sqrt {a b \,d^{2}}\, c \right ) \sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}}}{4 b^{2} a^{2} d^{4} \sqrt {a b \,d^{2}}}\right )}{8}\)	\(729\)

Input:

int(1/(d*x+c)^(1/2)/(-b*x^2+a)^3,x,method=_RETURNVERBOSE)
 

Output:

-2*d^5*b^3*(1/16/b/a^2/d^4/(a*b*d^2)^(1/2)*((3/4*(a*b*d^2)^(1/2)/b^3*(2*b* 
c-3*(a*b*d^2)^(1/2))/(a*d^2+b*c^2-2*(a*b*d^2)^(1/2)*c)*(d*x+c)^(3/2)-1/4*( 
a*b*d^2)^(1/2)/b^3*(6*b*c-11*(a*b*d^2)^(1/2))/(b*c-(a*b*d^2)^(1/2))*(d*x+c 
)^(1/2))/(-d*x-(a*b*d^2)^(1/2)/b)^2-3/4*(-7*a*d^2-4*b*c^2+10*(a*b*d^2)^(1/ 
2)*c)/b/(-a*d^2-b*c^2+2*(a*b*d^2)^(1/2)*c)/((-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)))-1/16/b/a^2/d^4 
/(a*b*d^2)^(1/2)*((-3/4*(a*b*d^2)^(1/2)/b^3*(2*b*c+3*(a*b*d^2)^(1/2))/(a*d 
^2+b*c^2+2*(a*b*d^2)^(1/2)*c)*(d*x+c)^(3/2)+1/4*(a*b*d^2)^(1/2)/b^3*(6*b*c 
+11*(a*b*d^2)^(1/2))/(b*c+(a*b*d^2)^(1/2))*(d*x+c)^(1/2))/(-d*x+(a*b*d^2)^ 
(1/2)/b)^2+3/4*(7*a*d^2+4*b*c^2+10*(a*b*d^2)^(1/2)*c)/b/(a*d^2+b*c^2+2*(a* 
b*d^2)^(1/2)*c)/((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))))
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 5776 vs. \(2 (258) = 516\).

Time = 2.25 (sec) , antiderivative size = 5776, normalized size of antiderivative = 18.34 \[ \int \frac {1}{\sqrt {c+d x} \left (a-b x^2\right )^3} \, dx=\text {Too large to display} \] Input:

integrate(1/(d*x+c)^(1/2)/(-b*x^2+a)^3,x, algorithm="fricas")
 

Output:

Too large to include
 

Sympy [F(-1)]

Timed out. \[ \int \frac {1}{\sqrt {c+d x} \left (a-b x^2\right )^3} \, dx=\text {Timed out} \] Input:

integrate(1/(d*x+c)**(1/2)/(-b*x**2+a)**3,x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \frac {1}{\sqrt {c+d x} \left (a-b x^2\right )^3} \, dx=\int { -\frac {1}{{\left (b x^{2} - a\right )}^{3} \sqrt {d x + c}} \,d x } \] Input:

integrate(1/(d*x+c)^(1/2)/(-b*x^2+a)^3,x, algorithm="maxima")
 

Output:

-integrate(1/((b*x^2 - a)^3*sqrt(d*x + c)), x)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1642 vs. \(2 (258) = 516\).

Time = 0.28 (sec) , antiderivative size = 1642, normalized size of antiderivative = 5.21 \[ \int \frac {1}{\sqrt {c+d x} \left (a-b x^2\right )^3} \, dx=\text {Too large to display} \] Input:

integrate(1/(d*x+c)^(1/2)/(-b*x^2+a)^3,x, algorithm="giac")
 

Output:

-3/32*(2*(a^2*b^2*c^4*d - 2*a^3*b*c^2*d^3 + a^4*d^5)^2*(b^2*c^3*d - 2*a*b* 
c*d^3)*abs(b) + (2*sqrt(a*b)*a*b^4*c^8*d - 9*sqrt(a*b)*a^2*b^3*c^6*d^3 + 1 
9*sqrt(a*b)*a^3*b^2*c^4*d^5 - 19*sqrt(a*b)*a^4*b*c^2*d^7 + 7*sqrt(a*b)*a^5 
*d^9)*abs(a^2*b^2*c^4*d - 2*a^3*b*c^2*d^3 + a^4*d^5)*abs(b) - (4*a^3*b^7*c 
^13*d - 25*a^4*b^6*c^11*d^3 + 67*a^5*b^5*c^9*d^5 - 98*a^6*b^4*c^7*d^7 + 82 
*a^7*b^3*c^5*d^9 - 37*a^8*b^2*c^3*d^11 + 7*a^9*b*c*d^13)*abs(b))*arctan(sq 
rt(d*x + c)/sqrt(-(a^2*b^3*c^5 - 2*a^3*b^2*c^3*d^2 + a^4*b*c*d^4 + sqrt((a 
^2*b^3*c^5 - 2*a^3*b^2*c^3*d^2 + a^4*b*c*d^4)^2 - (a^2*b^3*c^6 - 3*a^3*b^2 
*c^4*d^2 + 3*a^4*b*c^2*d^4 - a^5*d^6)*(a^2*b^3*c^4 - 2*a^3*b^2*c^2*d^2 + a 
^4*b*d^4)))/(a^2*b^3*c^4 - 2*a^3*b^2*c^2*d^2 + a^4*b*d^4)))/((a^4*b^5*c^8* 
d - 4*a^5*b^4*c^6*d^3 + 6*a^6*b^3*c^4*d^5 - 4*a^7*b^2*c^2*d^7 + a^8*b*d^9 
- sqrt(a*b)*a^3*b^5*c^9 + 4*sqrt(a*b)*a^4*b^4*c^7*d^2 - 6*sqrt(a*b)*a^5*b^ 
3*c^5*d^4 + 4*sqrt(a*b)*a^6*b^2*c^3*d^6 - sqrt(a*b)*a^7*b*c*d^8)*sqrt(-b^2 
*c - sqrt(a*b)*b*d)*abs(a^2*b^2*c^4*d - 2*a^3*b*c^2*d^3 + a^4*d^5)) - 3/32 
*(2*(a^2*b^2*c^4*d - 2*a^3*b*c^2*d^3 + a^4*d^5)^2*(sqrt(a*b)*b*c^3*d - 2*s 
qrt(a*b)*a*c*d^3)*abs(b) - (2*a^2*b^4*c^8*d - 9*a^3*b^3*c^6*d^3 + 19*a^4*b 
^2*c^4*d^5 - 19*a^5*b*c^2*d^7 + 7*a^6*d^9)*abs(a^2*b^2*c^4*d - 2*a^3*b*c^2 
*d^3 + a^4*d^5)*abs(b) - (4*sqrt(a*b)*a^3*b^6*c^13*d - 25*sqrt(a*b)*a^4*b^ 
5*c^11*d^3 + 67*sqrt(a*b)*a^5*b^4*c^9*d^5 - 98*sqrt(a*b)*a^6*b^3*c^7*d^7 + 
 82*sqrt(a*b)*a^7*b^2*c^5*d^9 - 37*sqrt(a*b)*a^8*b*c^3*d^11 + 7*sqrt(a*...
 

Mupad [B] (verification not implemented)

Time = 11.23 (sec) , antiderivative size = 8961, normalized size of antiderivative = 28.45 \[ \int \frac {1}{\sqrt {c+d x} \left (a-b x^2\right )^3} \, dx=\text {Too large to display} \] Input:

int(1/((a - b*x^2)^3*(c + d*x)^(1/2)),x)
 

Output:

- ((d*(c + d*x)^(3/2)*(9*b^3*c^5 - 22*a*b^2*c^3*d^2 + a^2*b*c*d^4))/(8*a^2 
*(a*d^2 - b*c^2)^2) - (d*(c + d*x)^(1/2)*(11*a^2*d^4 - 6*b^2*c^4 + 15*a*b* 
c^2*d^2))/(16*a^2*(a*d^2 - b*c^2)) + (3*b*d*(b^2*c^3 - 2*a*b*c*d^2)*(c + d 
*x)^(7/2))/(8*a^2*(a*d^2 - b*c^2)^2) + (b*d*(c + d*x)^(5/2)*(7*a^2*d^4 - 1 
8*b^2*c^4 + 35*a*b*c^2*d^2))/(16*a^2*(a*d^2 - b*c^2)^2))/(b^2*(c + d*x)^4 
+ a^2*d^4 + b^2*c^4 + (6*b^2*c^2 - 2*a*b*d^2)*(c + d*x)^2 - (4*b^2*c^3 - 4 
*a*b*c*d^2)*(c + d*x) - 4*b^2*c*(c + d*x)^3 - 2*a*b*c^2*d^2) - atan(((((3* 
(14336*a^9*b^3*d^11 + 4096*a^5*b^7*c^8*d^3 - 18432*a^6*b^6*c^6*d^5 + 38912 
*a^7*b^5*c^4*d^7 - 38912*a^8*b^4*c^2*d^9))/(2048*(a^10*d^8 + a^6*b^4*c^8 - 
 4*a^9*b*c^2*d^6 - 4*a^7*b^3*c^6*d^2 + 6*a^8*b^2*c^4*d^4)) - ((c + d*x)^(1 
/2)*(-(9*(16*a^5*b^5*c^9 - 49*a^2*d^9*(a^15*b)^(1/2) - 84*a^6*b^4*c^7*d^2 
+ 189*a^7*b^3*c^5*d^4 - 210*a^8*b^2*c^3*d^6 - 21*b^2*c^4*d^5*(a^15*b)^(1/2 
) + 105*a^9*b*c*d^8 + 54*a*b*c^2*d^7*(a^15*b)^(1/2)))/(4096*(a^15*b*d^10 - 
 a^10*b^6*c^10 + 5*a^11*b^5*c^8*d^2 - 10*a^12*b^4*c^6*d^4 + 10*a^13*b^3*c^ 
4*d^6 - 5*a^14*b^2*c^2*d^8)))^(1/2)*(4096*a^9*b^4*c*d^10 + 4096*a^5*b^8*c^ 
9*d^2 - 16384*a^6*b^7*c^7*d^4 + 24576*a^7*b^6*c^5*d^6 - 16384*a^8*b^5*c^3* 
d^8))/(64*(a^8*d^8 + a^4*b^4*c^8 - 4*a^7*b*c^2*d^6 - 4*a^5*b^3*c^6*d^2 + 6 
*a^6*b^2*c^4*d^4)))*(-(9*(16*a^5*b^5*c^9 - 49*a^2*d^9*(a^15*b)^(1/2) - 84* 
a^6*b^4*c^7*d^2 + 189*a^7*b^3*c^5*d^4 - 210*a^8*b^2*c^3*d^6 - 21*b^2*c^4*d 
^5*(a^15*b)^(1/2) + 105*a^9*b*c*d^8 + 54*a*b*c^2*d^7*(a^15*b)^(1/2)))/(...
 

Reduce [B] (verification not implemented)

Time = 3.73 (sec) , antiderivative size = 3161, normalized size of antiderivative = 10.03 \[ \int \frac {1}{\sqrt {c+d x} \left (a-b x^2\right )^3} \, dx =\text {Too large to display} \] Input:

int(1/(d*x+c)^(1/2)/(-b*x^2+a)^3,x)
 

Output:

(66*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)))*a**4*b*c*d**4 - 66*sqrt(a)*sqrt(sqrt(b)*sq 
rt(a)*d - b*c)*atan((sqrt(c + d*x)*b)/(sqrt(b)*sqrt(sqrt(b)*sqrt(a)*d - b* 
c)))*a**3*b**2*c**3*d**2 - 132*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)))*a**3*b**2*c*d** 
4*x**2 + 24*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)))*a**2*b**3*c**5 + 132*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)))*a**2*b**3*c**3*d**2*x**2 + 66*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)))*a 
**2*b**3*c*d**4*x**4 - 48*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)))*a*b**4*c**5*x**2 - 6 
6*sqrt(a)*sqrt(sqrt(b)*sqrt(a)*d - b*c)*atan((sqrt(c + d*x)*b)/(sqrt(b)*sq 
rt(sqrt(b)*sqrt(a)*d - b*c)))*a*b**4*c**3*d**2*x**4 + 24*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**5*c**5*x**4 + 42*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**5*d**5 - 30* 
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**4*b*c**2*d**3 - 84*sqrt(b)*sqrt(sqrt(b)*sqr 
t(a)*d - b*c)*atan((sqrt(c + d*x)*b)/(sqrt(b)*sqrt(sqrt(b)*sqrt(a)*d - ...