\(\int \frac {x^5}{\sqrt {c+d x} (a-b x^2)^3} \, dx\) [717]

Optimal result

Integrand size = 23, antiderivative size = 301 \[ \int \frac {x^5}{\sqrt {c+d x} \left (a-b x^2\right )^3} \, dx=\frac {a^2 (c-d x) \sqrt {c+d x}}{4 b^2 \left (b c^2-a d^2\right ) \left (a-b x^2\right )^2}-\frac {a \sqrt {c+d x} \left (b c^2 (16 c-17 d x)-a d^2 (10 c-11 d x)\right )}{16 b^2 \left (b c^2-a d^2\right )^2 \left (a-b x^2\right )}+\frac {\left (32 b c^2-50 \sqrt {a} \sqrt {b} c d+21 a d^2\right ) \text {arctanh}\left (\frac {\sqrt [4]{b} \sqrt {c+d x}}{\sqrt {\sqrt {b} c-\sqrt {a} d}}\right )}{32 b^{11/4} \left (\sqrt {b} c-\sqrt {a} d\right )^{5/2}}+\frac {\left (32 b c^2+50 \sqrt {a} \sqrt {b} c d+21 a d^2\right ) \text {arctanh}\left (\frac {\sqrt [4]{b} \sqrt {c+d x}}{\sqrt {\sqrt {b} c+\sqrt {a} d}}\right )}{32 b^{11/4} \left (\sqrt {b} c+\sqrt {a} d\right )^{5/2}} \] Output:

1/4*a^2*(-d*x+c)*(d*x+c)^(1/2)/b^2/(-a*d^2+b*c^2)/(-b*x^2+a)^2-1/16*a*(d*x 
+c)^(1/2)*(b*c^2*(-17*d*x+16*c)-a*d^2*(-11*d*x+10*c))/b^2/(-a*d^2+b*c^2)^2 
/(-b*x^2+a)+1/32*(32*b*c^2-50*a^(1/2)*b^(1/2)*c*d+21*a*d^2)*arctanh(b^(1/4 
)*(d*x+c)^(1/2)/(b^(1/2)*c-a^(1/2)*d)^(1/2))/b^(11/4)/(b^(1/2)*c-a^(1/2)*d 
)^(5/2)+1/32*(32*b*c^2+50*a^(1/2)*b^(1/2)*c*d+21*a*d^2)*arctanh(b^(1/4)*(d 
*x+c)^(1/2)/(b^(1/2)*c+a^(1/2)*d)^(1/2))/b^(11/4)/(b^(1/2)*c+a^(1/2)*d)^(5 
/2)
 

Mathematica [A] (verified)

Time = 2.08 (sec) , antiderivative size = 352, normalized size of antiderivative = 1.17 \[ \int \frac {x^5}{\sqrt {c+d x} \left (a-b x^2\right )^3} \, dx=\frac {\frac {2 a b \sqrt {c+d x} \left (b^2 c^2 x^2 (16 c-17 d x)+a^2 d^2 (6 c-7 d x)+a b \left (-12 c^3+13 c^2 d x-10 c d^2 x^2+11 d^3 x^3\right )\right )}{\left (b c^2-a d^2\right )^2 \left (a-b x^2\right )^2}-\frac {\sqrt {-b c-\sqrt {a} \sqrt {b} d} \left (32 b c^2+50 \sqrt {a} \sqrt {b} c d+21 a d^2\right ) \arctan \left (\frac {\sqrt {-b c-\sqrt {a} \sqrt {b} d} \sqrt {c+d x}}{\sqrt {b} c+\sqrt {a} d}\right )}{\left (\sqrt {b} c+\sqrt {a} d\right )^3}+\frac {\sqrt {b} \left (32 b c^2-50 \sqrt {a} \sqrt {b} c d+21 a d^2\right ) \arctan \left (\frac {\sqrt {-b c+\sqrt {a} \sqrt {b} d} \sqrt {c+d x}}{\sqrt {b} c-\sqrt {a} d}\right )}{\left (\sqrt {b} c-\sqrt {a} d\right )^2 \sqrt {-b c+\sqrt {a} \sqrt {b} d}}}{32 b^3} \] Input:

Integrate[x^5/(Sqrt[c + d*x]*(a - b*x^2)^3),x]
 

Output:

((2*a*b*Sqrt[c + d*x]*(b^2*c^2*x^2*(16*c - 17*d*x) + a^2*d^2*(6*c - 7*d*x) 
 + a*b*(-12*c^3 + 13*c^2*d*x - 10*c*d^2*x^2 + 11*d^3*x^3)))/((b*c^2 - a*d^ 
2)^2*(a - b*x^2)^2) - (Sqrt[-(b*c) - Sqrt[a]*Sqrt[b]*d]*(32*b*c^2 + 50*Sqr 
t[a]*Sqrt[b]*c*d + 21*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)^3 + (Sqrt[b]* 
(32*b*c^2 - 50*Sqrt[a]*Sqrt[b]*c*d + 21*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*b^3)
 

Rubi [A] (verified)

Time = 1.70 (sec) , antiderivative size = 476, normalized size of antiderivative = 1.58, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.391, Rules used = {561, 25, 27, 1517, 27, 2206, 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.

\(\displaystyle \int \frac {x^5}{\left (a-b x^2\right )^3 \sqrt {c+d x}} \, dx\)

\(\Big \downarrow \) 561

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1517

\(\displaystyle -\frac {2 \left (\frac {d^4 \int -\frac {2 \left (8 a \left (a-\frac {b c^2}{d^2}\right ) (c+d x)^3-24 a c \left (a-\frac {b c^2}{d^2}\right ) (c+d x)^2+a \left (-\frac {24 b c^4}{d^2}+16 a c^2+\frac {3 a^2 d^2}{b}\right ) (c+d x)+2 a c \left (\frac {4 b c^4}{d^2}-\frac {a^2 d^2}{b}\right )\right )}{\left (-\frac {b c^2}{d^2}+\frac {2 b (c+d x) c}{d^2}-\frac {b (c+d x)^2}{d^2}+a\right )^2}d\sqrt {c+d x}}{16 a b \left (b c^2-a d^2\right )}-\frac {a^2 d^6 (c-d x) \sqrt {c+d x}}{8 b^2 \left (b c^2-a d^2\right ) \left (a-\frac {b c^2}{d^2}+\frac {2 b c (c+d x)}{d^2}-\frac {b (c+d x)^2}{d^2}\right )^2}\right )}{d^6}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {2 \left (-\frac {d^4 \int \frac {8 a \left (a-\frac {b c^2}{d^2}\right ) (c+d x)^3-24 a c \left (a-\frac {b c^2}{d^2}\right ) (c+d x)^2+a \left (-\frac {24 b c^4}{d^2}+16 a c^2+\frac {3 a^2 d^2}{b}\right ) (c+d x)+2 a c \left (\frac {4 b c^4}{d^2}-\frac {a^2 d^2}{b}\right )}{\left (-\frac {b c^2}{d^2}+\frac {2 b (c+d x) c}{d^2}-\frac {b (c+d x)^2}{d^2}+a\right )^2}d\sqrt {c+d x}}{8 a b \left (b c^2-a d^2\right )}-\frac {a^2 d^6 (c-d x) \sqrt {c+d x}}{8 b^2 \left (b c^2-a d^2\right ) \left (a-\frac {b c^2}{d^2}+\frac {2 b c (c+d x)}{d^2}-\frac {b (c+d x)^2}{d^2}\right )^2}\right )}{d^6}\)

\(\Big \downarrow \) 2206

\(\displaystyle -\frac {2 \left (-\frac {d^4 \left (\frac {d^4 \int -\frac {2 a^2 \left (c \left (\frac {32 b^2 c^4}{d^4}-\frac {33 a b c^2}{d^2}+13 a^2\right )-\left (\frac {32 b^2 c^4}{d^4}-\frac {47 a b c^2}{d^2}+21 a^2\right ) (c+d x)\right )}{-\frac {b c^2}{d^2}+\frac {2 b (c+d x) c}{d^2}-\frac {b (c+d x)^2}{d^2}+a}d\sqrt {c+d x}}{8 a b \left (b c^2-a d^2\right )}+\frac {a^2 d^2 \sqrt {c+d x} \left ((c+d x) \left (17 b c^2-11 a d^2\right )+3 c d^2 \left (7 a-\frac {11 b c^2}{d^2}\right )\right )}{4 b \left (b c^2-a d^2\right ) \left (a-\frac {b c^2}{d^2}+\frac {2 b c (c+d x)}{d^2}-\frac {b (c+d x)^2}{d^2}\right )}\right )}{8 a b \left (b c^2-a d^2\right )}-\frac {a^2 d^6 (c-d x) \sqrt {c+d x}}{8 b^2 \left (b c^2-a d^2\right ) \left (a-\frac {b c^2}{d^2}+\frac {2 b c (c+d x)}{d^2}-\frac {b (c+d x)^2}{d^2}\right )^2}\right )}{d^6}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {2 \left (-\frac {d^4 \left (\frac {a^2 d^2 \sqrt {c+d x} \left ((c+d x) \left (17 b c^2-11 a d^2\right )+3 c d^2 \left (7 a-\frac {11 b c^2}{d^2}\right )\right )}{4 b \left (b c^2-a d^2\right ) \left (a-\frac {b c^2}{d^2}+\frac {2 b c (c+d x)}{d^2}-\frac {b (c+d x)^2}{d^2}\right )}-\frac {a d^4 \int \frac {c \left (\frac {32 b^2 c^4}{d^4}-\frac {33 a b c^2}{d^2}+13 a^2\right )-\left (\frac {32 b^2 c^4}{d^4}-\frac {47 a b c^2}{d^2}+21 a^2\right ) (c+d x)}{-\frac {b c^2}{d^2}+\frac {2 b (c+d x) c}{d^2}-\frac {b (c+d x)^2}{d^2}+a}d\sqrt {c+d x}}{4 b \left (b c^2-a d^2\right )}\right )}{8 a b \left (b c^2-a d^2\right )}-\frac {a^2 d^6 (c-d x) \sqrt {c+d x}}{8 b^2 \left (b c^2-a d^2\right ) \left (a-\frac {b c^2}{d^2}+\frac {2 b c (c+d x)}{d^2}-\frac {b (c+d x)^2}{d^2}\right )^2}\right )}{d^6}\)

\(\Big \downarrow \) 1480

\(\displaystyle -\frac {2 \left (-\frac {d^4 \left (\frac {a^2 d^2 \sqrt {c+d x} \left ((c+d x) \left (17 b c^2-11 a d^2\right )+3 c d^2 \left (7 a-\frac {11 b c^2}{d^2}\right )\right )}{4 b \left (b c^2-a d^2\right ) \left (a-\frac {b c^2}{d^2}+\frac {2 b c (c+d x)}{d^2}-\frac {b (c+d x)^2}{d^2}\right )}-\frac {a d^4 \left (-\frac {\left (50 \sqrt {a} \sqrt {b} c d+21 a d^2+32 b c^2\right ) \left (\sqrt {b} c-\sqrt {a} d\right )^2 \int \frac {1}{\frac {\sqrt {b} \left (\sqrt {b} c+\sqrt {a} d\right )}{d^2}-\frac {b (c+d x)}{d^2}}d\sqrt {c+d x}}{2 d^4}-\frac {\left (\sqrt {a} d+\sqrt {b} c\right )^2 \left (-50 \sqrt {a} \sqrt {b} c d+21 a d^2+32 b c^2\right ) \int \frac {1}{\frac {\sqrt {b} \left (\sqrt {b} c-\sqrt {a} d\right )}{d^2}-\frac {b (c+d x)}{d^2}}d\sqrt {c+d x}}{2 d^4}\right )}{4 b \left (b c^2-a d^2\right )}\right )}{8 a b \left (b c^2-a d^2\right )}-\frac {a^2 d^6 (c-d x) \sqrt {c+d x}}{8 b^2 \left (b c^2-a d^2\right ) \left (a-\frac {b c^2}{d^2}+\frac {2 b c (c+d x)}{d^2}-\frac {b (c+d x)^2}{d^2}\right )^2}\right )}{d^6}\)

\(\Big \downarrow \) 221

\(\displaystyle -\frac {2 \left (-\frac {d^4 \left (\frac {a^2 d^2 \sqrt {c+d x} \left ((c+d x) \left (17 b c^2-11 a d^2\right )+3 c d^2 \left (7 a-\frac {11 b c^2}{d^2}\right )\right )}{4 b \left (b c^2-a d^2\right ) \left (a-\frac {b c^2}{d^2}+\frac {2 b c (c+d x)}{d^2}-\frac {b (c+d x)^2}{d^2}\right )}-\frac {a d^4 \left (-\frac {\left (50 \sqrt {a} \sqrt {b} c d+21 a d^2+32 b c^2\right ) \left (\sqrt {b} c-\sqrt {a} d\right )^2 \text {arctanh}\left (\frac {\sqrt [4]{b} \sqrt {c+d x}}{\sqrt {\sqrt {a} d+\sqrt {b} c}}\right )}{2 b^{3/4} d^2 \sqrt {\sqrt {a} d+\sqrt {b} c}}-\frac {\left (\sqrt {a} d+\sqrt {b} c\right )^2 \left (-50 \sqrt {a} \sqrt {b} c d+21 a d^2+32 b c^2\right ) \text {arctanh}\left (\frac {\sqrt [4]{b} \sqrt {c+d x}}{\sqrt {\sqrt {b} c-\sqrt {a} d}}\right )}{2 b^{3/4} d^2 \sqrt {\sqrt {b} c-\sqrt {a} d}}\right )}{4 b \left (b c^2-a d^2\right )}\right )}{8 a b \left (b c^2-a d^2\right )}-\frac {a^2 d^6 (c-d x) \sqrt {c+d x}}{8 b^2 \left (b c^2-a d^2\right ) \left (a-\frac {b c^2}{d^2}+\frac {2 b c (c+d x)}{d^2}-\frac {b (c+d x)^2}{d^2}\right )^2}\right )}{d^6}\)

Input:

Int[x^5/(Sqrt[c + d*x]*(a - b*x^2)^3),x]
 

Output:

(-2*(-1/8*(a^2*d^6*(c - d*x)*Sqrt[c + d*x])/(b^2*(b*c^2 - a*d^2)*(a - (b*c 
^2)/d^2 + (2*b*c*(c + d*x))/d^2 - (b*(c + d*x)^2)/d^2)^2) - (d^4*((a^2*d^2 
*Sqrt[c + d*x]*(3*c*(7*a - (11*b*c^2)/d^2)*d^2 + (17*b*c^2 - 11*a*d^2)*(c 
+ d*x)))/(4*b*(b*c^2 - a*d^2)*(a - (b*c^2)/d^2 + (2*b*c*(c + d*x))/d^2 - ( 
b*(c + d*x)^2)/d^2)) - (a*d^4*(-1/2*((Sqrt[b]*c + Sqrt[a]*d)^2*(32*b*c^2 - 
 50*Sqrt[a]*Sqrt[b]*c*d + 21*a*d^2)*ArcTanh[(b^(1/4)*Sqrt[c + d*x])/Sqrt[S 
qrt[b]*c - Sqrt[a]*d]])/(b^(3/4)*d^2*Sqrt[Sqrt[b]*c - Sqrt[a]*d]) - ((Sqrt 
[b]*c - Sqrt[a]*d)^2*(32*b*c^2 + 50*Sqrt[a]*Sqrt[b]*c*d + 21*a*d^2)*ArcTan 
h[(b^(1/4)*Sqrt[c + d*x])/Sqrt[Sqrt[b]*c + Sqrt[a]*d]])/(2*b^(3/4)*d^2*Sqr 
t[Sqrt[b]*c + Sqrt[a]*d])))/(4*b*(b*c^2 - a*d^2))))/(8*a*b*(b*c^2 - a*d^2) 
)))/d^6
 

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 561

Int[(x_)^(m_.)*((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbo 
l] :> With[{k = Denominator[n]}, Simp[k/d   Subst[Int[x^(k*(n + 1) - 1)*(-c 
/d + x^k/d)^m*Simp[(b*c^2 + a*d^2)/d^2 - 2*b*c*(x^k/d^2) + b*(x^(2*k)/d^2), 
 x]^p, x], x, (c + d*x)^(1/k)], x]] /; FreeQ[{a, b, c, d, m, p}, x] && Frac 
tionQ[n] && IntegerQ[p] && IntegerQ[m]
 

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]
 

rule 1517

Int[((d_) + (e_.)*(x_)^2)^(q_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x 
_Symbol] :> With[{f = Coeff[PolynomialRemainder[(d + e*x^2)^q, a + b*x^2 + 
c*x^4, x], x, 0], g = Coeff[PolynomialRemainder[(d + e*x^2)^q, a + b*x^2 + 
c*x^4, x], x, 2]}, Simp[x*(a + b*x^2 + c*x^4)^(p + 1)*((a*b*g - f*(b^2 - 2* 
a*c) - c*(b*f - 2*a*g)*x^2)/(2*a*(p + 1)*(b^2 - 4*a*c))), x] + Simp[1/(2*a* 
(p + 1)*(b^2 - 4*a*c))   Int[(a + b*x^2 + c*x^4)^(p + 1)*ExpandToSum[2*a*(p 
 + 1)*(b^2 - 4*a*c)*PolynomialQuotient[(d + e*x^2)^q, a + b*x^2 + c*x^4, x] 
 + b^2*f*(2*p + 3) - 2*a*c*f*(4*p + 5) - a*b*g + c*(4*p + 7)*(b*f - 2*a*g)* 
x^2, x], x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ 
[c*d^2 - b*d*e + a*e^2, 0] && IGtQ[q, 1] && LtQ[p, -1]
 

rule 2206

Int[(Px_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> With[{d = 
 Coeff[PolynomialRemainder[Px, a + b*x^2 + c*x^4, x], x, 0], e = Coeff[Poly 
nomialRemainder[Px, a + b*x^2 + c*x^4, x], x, 2]}, Simp[x*(a + b*x^2 + c*x^ 
4)^(p + 1)*((a*b*e - d*(b^2 - 2*a*c) - c*(b*d - 2*a*e)*x^2)/(2*a*(p + 1)*(b 
^2 - 4*a*c))), x] + Simp[1/(2*a*(p + 1)*(b^2 - 4*a*c))   Int[(a + b*x^2 + c 
*x^4)^(p + 1)*ExpandToSum[2*a*(p + 1)*(b^2 - 4*a*c)*PolynomialQuotient[Px, 
a + b*x^2 + c*x^4, x] + b^2*d*(2*p + 3) - 2*a*c*d*(4*p + 5) - a*b*e + c*(4* 
p + 7)*(b*d - 2*a*e)*x^2, x], x], x]] /; FreeQ[{a, b, c}, x] && PolyQ[Px, x 
^2] && Expon[Px, x^2] > 1 && NeQ[b^2 - 4*a*c, 0] && LtQ[p, -1]
 

Maple [A] (verified)

Time = 0.70 (sec) , antiderivative size = 397, normalized size of antiderivative = 1.32

method	result	size
pseudoelliptic	\(\frac {\frac {\sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}\, \left (\left (-\frac {21}{8} a^{2} d^{4}+\frac {47}{8} b \,c^{2} d^{2} a -4 b^{2} c^{4}\right ) \sqrt {a b \,d^{2}}+a b c \,d^{2} \left (a \,d^{2}-\frac {7 b \,c^{2}}{4}\right )\right ) \left (-b \,x^{2}+a \right )^{2} \arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (-b c +\sqrt {a b \,d^{2}}\right ) b}}\right )}{4}+\frac {3 \sqrt {\left (-b c +\sqrt {a b \,d^{2}}\right ) b}\, \left (\frac {2 \left (\left (\frac {21}{8} a^{2} d^{4}-\frac {47}{8} b \,c^{2} d^{2} a +4 b^{2} c^{4}\right ) \sqrt {a b \,d^{2}}+a b c \,d^{2} \left (a \,d^{2}-\frac {7 b \,c^{2}}{4}\right )\right ) \left (-b \,x^{2}+a \right )^{2} \operatorname {arctanh}\left (\frac {b \sqrt {d x +c}}{\sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}}\right )}{3}+\sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}\, \sqrt {d x +c}\, \left (\left (\frac {8}{3} c^{3} x^{2}-\frac {17}{6} c^{2} d \,x^{3}\right ) b^{2}-2 \left (-\frac {11}{12} d^{3} x^{3}+\frac {5}{6} c \,d^{2} x^{2}-\frac {13}{12} c^{2} d x +c^{3}\right ) a b +a^{2} d^{2} \left (-\frac {7 d x}{6}+c \right )\right ) \sqrt {a b \,d^{2}}\, a \right )}{8}}{\sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}\, \sqrt {\left (-b c +\sqrt {a b \,d^{2}}\right ) b}\, \sqrt {a b \,d^{2}}\, \left (-b \,x^{2}+a \right )^{2} b^{2} \left (a \,d^{2}-b \,c^{2}\right )^{2}}\)	\(397\)
derivativedivides	\(-\frac {2 \left (-\frac {a \,d^{2} \left (11 a \,d^{2}-17 b \,c^{2}\right ) \left (d x +c \right )^{\frac {7}{2}}}{32 b \left (a^{2} d^{4}-2 b \,c^{2} d^{2} a +b^{2} c^{4}\right )}+\frac {\left (43 a \,d^{2}-67 b \,c^{2}\right ) a c \,d^{2} \left (d x +c \right )^{\frac {5}{2}}}{32 b \left (a^{2} d^{4}-2 b \,c^{2} d^{2} a +b^{2} c^{4}\right )}+\frac {a \,d^{2} \left (7 a^{2} d^{4}-66 b \,c^{2} d^{2} a +83 b^{2} c^{4}\right ) \left (d x +c \right )^{\frac {3}{2}}}{32 b^{2} \left (a^{2} d^{4}-2 b \,c^{2} d^{2} a +b^{2} c^{4}\right )}-\frac {\left (13 a \,d^{2}-33 b \,c^{2}\right ) a c \,d^{2} \sqrt {d x +c}}{32 b^{2} \left (a \,d^{2}-b \,c^{2}\right )}\right )}{\left (-b \left (d x +c \right )^{2}+2 b c \left (d x +c \right )+a \,d^{2}-b \,c^{2}\right )^{2}}-\frac {\frac {\left (-8 a^{2} c \,d^{4} b +14 a \,b^{2} c^{3} d^{2}+21 \sqrt {a b \,d^{2}}\, a^{2} d^{4}-47 \sqrt {a b \,d^{2}}\, a b \,c^{2} d^{2}+32 \sqrt {a b \,d^{2}}\, b^{2} c^{4}\right ) \arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (-b c +\sqrt {a b \,d^{2}}\right ) b}}\right )}{2 b \sqrt {a b \,d^{2}}\, \sqrt {\left (-b c +\sqrt {a b \,d^{2}}\right ) b}}-\frac {\left (8 a^{2} c \,d^{4} b -14 a \,b^{2} c^{3} d^{2}+21 \sqrt {a b \,d^{2}}\, a^{2} d^{4}-47 \sqrt {a b \,d^{2}}\, a b \,c^{2} d^{2}+32 \sqrt {a b \,d^{2}}\, b^{2} c^{4}\right ) \operatorname {arctanh}\left (\frac {b \sqrt {d x +c}}{\sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}}\right )}{2 b \sqrt {a b \,d^{2}}\, \sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}}}{16 b \left (a^{2} d^{4}-2 b \,c^{2} d^{2} a +b^{2} c^{4}\right )}\)	\(556\)
default	\(-\frac {2 \left (-\frac {a \,d^{2} \left (11 a \,d^{2}-17 b \,c^{2}\right ) \left (d x +c \right )^{\frac {7}{2}}}{32 b \left (a^{2} d^{4}-2 b \,c^{2} d^{2} a +b^{2} c^{4}\right )}+\frac {\left (43 a \,d^{2}-67 b \,c^{2}\right ) a c \,d^{2} \left (d x +c \right )^{\frac {5}{2}}}{32 b \left (a^{2} d^{4}-2 b \,c^{2} d^{2} a +b^{2} c^{4}\right )}+\frac {a \,d^{2} \left (7 a^{2} d^{4}-66 b \,c^{2} d^{2} a +83 b^{2} c^{4}\right ) \left (d x +c \right )^{\frac {3}{2}}}{32 b^{2} \left (a^{2} d^{4}-2 b \,c^{2} d^{2} a +b^{2} c^{4}\right )}-\frac {\left (13 a \,d^{2}-33 b \,c^{2}\right ) a c \,d^{2} \sqrt {d x +c}}{32 b^{2} \left (a \,d^{2}-b \,c^{2}\right )}\right )}{\left (-b \left (d x +c \right )^{2}+2 b c \left (d x +c \right )+a \,d^{2}-b \,c^{2}\right )^{2}}-\frac {\frac {\left (-8 a^{2} c \,d^{4} b +14 a \,b^{2} c^{3} d^{2}+21 \sqrt {a b \,d^{2}}\, a^{2} d^{4}-47 \sqrt {a b \,d^{2}}\, a b \,c^{2} d^{2}+32 \sqrt {a b \,d^{2}}\, b^{2} c^{4}\right ) \arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (-b c +\sqrt {a b \,d^{2}}\right ) b}}\right )}{2 b \sqrt {a b \,d^{2}}\, \sqrt {\left (-b c +\sqrt {a b \,d^{2}}\right ) b}}-\frac {\left (8 a^{2} c \,d^{4} b -14 a \,b^{2} c^{3} d^{2}+21 \sqrt {a b \,d^{2}}\, a^{2} d^{4}-47 \sqrt {a b \,d^{2}}\, a b \,c^{2} d^{2}+32 \sqrt {a b \,d^{2}}\, b^{2} c^{4}\right ) \operatorname {arctanh}\left (\frac {b \sqrt {d x +c}}{\sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}}\right )}{2 b \sqrt {a b \,d^{2}}\, \sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}}}{16 b \left (a^{2} d^{4}-2 b \,c^{2} d^{2} a +b^{2} c^{4}\right )}\)	\(556\)

Input:

int(x^5/(d*x+c)^(1/2)/(-b*x^2+a)^3,x,method=_RETURNVERBOSE)
 

Output:

3/8/(a*b*d^2)^(1/2)/(a*d^2-b*c^2)^2/((b*c+(a*b*d^2)^(1/2))*b)^(1/2)*(2/3*( 
(b*c+(a*b*d^2)^(1/2))*b)^(1/2)*((-21/8*a^2*d^4+47/8*b*c^2*d^2*a-4*b^2*c^4) 
*(a*b*d^2)^(1/2)+a*b*c*d^2*(a*d^2-7/4*b*c^2))*(-b*x^2+a)^2*arctan(b*(d*x+c 
)^(1/2)/((-b*c+(a*b*d^2)^(1/2))*b)^(1/2))+((-b*c+(a*b*d^2)^(1/2))*b)^(1/2) 
*(2/3*((21/8*a^2*d^4-47/8*b*c^2*d^2*a+4*b^2*c^4)*(a*b*d^2)^(1/2)+a*b*c*d^2 
*(a*d^2-7/4*b*c^2))*(-b*x^2+a)^2*arctanh(b*(d*x+c)^(1/2)/((b*c+(a*b*d^2)^( 
1/2))*b)^(1/2))+((b*c+(a*b*d^2)^(1/2))*b)^(1/2)*(d*x+c)^(1/2)*((8/3*c^3*x^ 
2-17/6*c^2*d*x^3)*b^2-2*(-11/12*d^3*x^3+5/6*c*d^2*x^2-13/12*c^2*d*x+c^3)*a 
*b+a^2*d^2*(-7/6*d*x+c))*(a*b*d^2)^(1/2)*a))/((-b*c+(a*b*d^2)^(1/2))*b)^(1 
/2)/b^2/(-b*x^2+a)^2
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 6629 vs. \(2 (248) = 496\).

Time = 13.13 (sec) , antiderivative size = 6629, normalized size of antiderivative = 22.02 \[ \int \frac {x^5}{\sqrt {c+d x} \left (a-b x^2\right )^3} \, dx=\text {Too large to display} \] Input:

integrate(x^5/(d*x+c)^(1/2)/(-b*x^2+a)^3,x, algorithm="fricas")
 

Output:

Too large to include
 

Sympy [F(-1)]

Timed out. \[ \int \frac {x^5}{\sqrt {c+d x} \left (a-b x^2\right )^3} \, dx=\text {Timed out} \] Input:

integrate(x**5/(d*x+c)**(1/2)/(-b*x**2+a)**3,x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \frac {x^5}{\sqrt {c+d x} \left (a-b x^2\right )^3} \, dx=\int { -\frac {x^{5}}{{\left (b x^{2} - a\right )}^{3} \sqrt {d x + c}} \,d x } \] Input:

integrate(x^5/(d*x+c)^(1/2)/(-b*x^2+a)^3,x, algorithm="maxima")
 

Output:

-integrate(x^5/((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. 1602 vs. \(2 (248) = 496\).

Time = 0.34 (sec) , antiderivative size = 1602, normalized size of antiderivative = 5.32 \[ \int \frac {x^5}{\sqrt {c+d x} \left (a-b x^2\right )^3} \, dx=\text {Too large to display} \] Input:

integrate(x^5/(d*x+c)^(1/2)/(-b*x^2+a)^3,x, algorithm="giac")
 

Output:

1/32*((b^4*c^4*d - 2*a*b^3*c^2*d^3 + a^2*b^2*d^5)^2*(32*sqrt(a*b)*b^2*c^4 
- 47*sqrt(a*b)*a*b*c^2*d^2 + 21*sqrt(a*b)*a^2*d^4)*abs(b) - (32*b^7*c^9 - 
97*a*b^6*c^7*d^2 + 111*a^2*b^5*c^5*d^4 - 59*a^3*b^4*c^3*d^6 + 13*a^4*b^3*c 
*d^8)*abs(b^4*c^4*d - 2*a*b^3*c^2*d^3 + a^2*b^2*d^5)*abs(b) + 2*(7*sqrt(a* 
b)*b^10*c^12*d^2 - 32*sqrt(a*b)*a*b^9*c^10*d^4 + 58*sqrt(a*b)*a^2*b^8*c^8* 
d^6 - 52*sqrt(a*b)*a^3*b^7*c^6*d^8 + 23*sqrt(a*b)*a^4*b^6*c^4*d^10 - 4*sqr 
t(a*b)*a^5*b^5*c^2*d^12)*abs(b))*arctan(sqrt(d*x + c)/sqrt(-(b^5*c^5 - 2*a 
*b^4*c^3*d^2 + a^2*b^3*c*d^4 + sqrt((b^5*c^5 - 2*a*b^4*c^3*d^2 + a^2*b^3*c 
*d^4)^2 - (b^5*c^6 - 3*a*b^4*c^4*d^2 + 3*a^2*b^3*c^2*d^4 - a^3*b^2*d^6)*(b 
^5*c^4 - 2*a*b^4*c^2*d^2 + a^2*b^3*d^4)))/(b^5*c^4 - 2*a*b^4*c^2*d^2 + a^2 
*b^3*d^4)))/((b^10*c^9 - 4*a*b^9*c^7*d^2 + 6*a^2*b^8*c^5*d^4 - 4*a^3*b^7*c 
^3*d^6 + a^4*b^6*c*d^8 - sqrt(a*b)*b^9*c^8*d + 4*sqrt(a*b)*a*b^8*c^6*d^3 - 
 6*sqrt(a*b)*a^2*b^7*c^4*d^5 + 4*sqrt(a*b)*a^3*b^6*c^2*d^7 - sqrt(a*b)*a^4 
*b^5*d^9)*sqrt(-b^2*c - sqrt(a*b)*b*d)*abs(b^4*c^4*d - 2*a*b^3*c^2*d^3 + a 
^2*b^2*d^5)) - 1/32*((b^4*c^4*d - 2*a*b^3*c^2*d^3 + a^2*b^2*d^5)^2*(32*sqr 
t(a*b)*b^2*c^4 - 47*sqrt(a*b)*a*b*c^2*d^2 + 21*sqrt(a*b)*a^2*d^4)*abs(b) + 
 (32*b^7*c^9 - 97*a*b^6*c^7*d^2 + 111*a^2*b^5*c^5*d^4 - 59*a^3*b^4*c^3*d^6 
 + 13*a^4*b^3*c*d^8)*abs(b^4*c^4*d - 2*a*b^3*c^2*d^3 + a^2*b^2*d^5)*abs(b) 
 + 2*(7*sqrt(a*b)*b^10*c^12*d^2 - 32*sqrt(a*b)*a*b^9*c^10*d^4 + 58*sqrt(a* 
b)*a^2*b^8*c^8*d^6 - 52*sqrt(a*b)*a^3*b^7*c^6*d^8 + 23*sqrt(a*b)*a^4*b^...
 

Mupad [B] (verification not implemented)

Time = 11.36 (sec) , antiderivative size = 9778, normalized size of antiderivative = 32.49 \[ \int \frac {x^5}{\sqrt {c+d x} \left (a-b x^2\right )^3} \, dx=\text {Too large to display} \] Input:

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

Output:

(((11*a^2*d^4 - 17*a*b*c^2*d^2)*(c + d*x)^(7/2))/(16*b*(a*d^2 - b*c^2)^2) 
+ ((13*a^2*c*d^4 - 33*a*b*c^3*d^2)*(c + d*x)^(1/2))/(16*b^2*(a*d^2 - b*c^2 
)) - ((c + d*x)^(3/2)*(7*a^3*d^6 + 83*a*b^2*c^4*d^2 - 66*a^2*b*c^2*d^4))/( 
16*b^2*(a*d^2 - b*c^2)^2) - (c*(43*a^2*d^4 - 67*a*b*c^2*d^2)*(c + d*x)^(5/ 
2))/(16*b*(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(((((131072*a*b^10*c^9*d^2 + 53248 
*a^5*b^6*c*d^10 - 397312*a^2*b^9*c^7*d^4 + 454656*a^3*b^8*c^5*d^6 - 241664 
*a^4*b^7*c^3*d^8)/(4096*(b^9*c^8 + a^4*b^5*d^8 - 4*a*b^8*c^6*d^2 + 6*a^2*b 
^7*c^4*d^4 - 4*a^3*b^6*c^2*d^6)) - ((c + d*x)^(1/2)*((1024*b^10*c^9 - 441* 
a^4*d^9*(a*b^11)^(1/2) - 1916*a*b^9*c^7*d^2 + 105*a^4*b^6*c*d^8 - 1920*b^4 
*c^8*d*(a*b^11)^(1/2) + 1501*a^2*b^8*c^5*d^4 - 570*a^3*b^7*c^3*d^6 - 4669* 
a^2*b^2*c^4*d^5*(a*b^11)^(1/2) + 4640*a*b^3*c^6*d^3*(a*b^11)^(1/2) + 2246* 
a^3*b*c^2*d^7*(a*b^11)^(1/2))/(4096*(b^16*c^10 - a^5*b^11*d^10 - 5*a*b^15* 
c^8*d^2 + 10*a^2*b^14*c^6*d^4 - 10*a^3*b^13*c^4*d^6 + 5*a^4*b^12*c^2*d^8)) 
)^(1/2)*(4096*a*b^10*c^9*d^2 + 4096*a^5*b^6*c*d^10 - 16384*a^2*b^9*c^7*d^4 
 + 24576*a^3*b^8*c^5*d^6 - 16384*a^4*b^7*c^3*d^8))/(64*(b^6*c^8 + a^4*b^2* 
d^8 - 4*a*b^5*c^6*d^2 + 6*a^2*b^4*c^4*d^4 - 4*a^3*b^3*c^2*d^6)))*((1024*b^ 
10*c^9 - 441*a^4*d^9*(a*b^11)^(1/2) - 1916*a*b^9*c^7*d^2 + 105*a^4*b^6*c*d 
^8 - 1920*b^4*c^8*d*(a*b^11)^(1/2) + 1501*a^2*b^8*c^5*d^4 - 570*a^3*b^7...
 

Reduce [B] (verification not implemented)

Time = 3.91 (sec) , antiderivative size = 3124, normalized size of antiderivative = 10.38 \[ \int \frac {x^5}{\sqrt {c+d x} \left (a-b x^2\right )^3} \, dx =\text {Too large to display} \] Input:

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

Output:

( - 42*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*d**5 + 110*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*c**2*d**3 + 84*sqrt(a)*sqrt(sqrt(b)*sqrt(a)*d - b*c)*atan((sqr 
t(c + d*x)*b)/(sqrt(b)*sqrt(sqrt(b)*sqrt(a)*d - b*c)))*a**3*b*d**5*x**2 - 
92*sqrt(a)*sqrt(sqrt(b)*sqrt(a)*d - b*c)*atan((sqrt(c + d*x)*b)/(sqrt(b)*s 
qrt(sqrt(b)*sqrt(a)*d - b*c)))*a**2*b**2*c**4*d - 220*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**2*c**2*d**3*x**2 - 42*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** 
2*d**5*x**4 + 184*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**3*c**4*d*x**2 + 110*sqr 
t(a)*sqrt(sqrt(b)*sqrt(a)*d - b*c)*atan((sqrt(c + d*x)*b)/(sqrt(b)*sqrt(sq 
rt(b)*sqrt(a)*d - b*c)))*a*b**3*c**2*d**3*x**4 - 92*sqrt(a)*sqrt(sqrt(b)*s 
qrt(a)*d - b*c)*atan((sqrt(c + d*x)*b)/(sqrt(b)*sqrt(sqrt(b)*sqrt(a)*d - b 
*c)))*b**4*c**4*d*x**4 - 26*sqrt(b)*sqrt(sqrt(b)*sqrt(a)*d - b*c)*atan((sq 
rt(c + d*x)*b)/(sqrt(b)*sqrt(sqrt(b)*sqrt(a)*d - b*c)))*a**4*c*d**4 + 66*s 
qrt(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**3*b*c**3*d**2 + 52*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...