\(\int \frac {x^4 \sqrt {c+d x}}{(a-b x^2)^3} \, dx\) [689]

Optimal result

Integrand size = 23, antiderivative size = 278 \[ \int \frac {x^4 \sqrt {c+d x}}{\left (a-b x^2\right )^3} \, dx=\frac {a x \sqrt {c+d x}}{4 b^2 \left (a-b x^2\right )^2}-\frac {\sqrt {c+d x} \left (10 b c^2 x+a d (c-11 d x)\right )}{16 b^2 \left (b c^2-a d^2\right ) \left (a-b x^2\right )}-\frac {\left (12 b c^2-34 \sqrt {a} \sqrt {b} c d+21 a d^2\right ) \text {arctanh}\left (\frac {\sqrt [4]{b} \sqrt {c+d x}}{\sqrt {\sqrt {b} c-\sqrt {a} d}}\right )}{32 \sqrt {a} b^{11/4} \left (\sqrt {b} c-\sqrt {a} d\right )^{3/2}}+\frac {\left (12 b c^2+34 \sqrt {a} \sqrt {b} c d+21 a d^2\right ) \text {arctanh}\left (\frac {\sqrt [4]{b} \sqrt {c+d x}}{\sqrt {\sqrt {b} c+\sqrt {a} d}}\right )}{32 \sqrt {a} b^{11/4} \left (\sqrt {b} c+\sqrt {a} d\right )^{3/2}} \] Output:

1/4*a*x*(d*x+c)^(1/2)/b^2/(-b*x^2+a)^2-1/16*(d*x+c)^(1/2)*(10*b*c^2*x+a*d* 
(-11*d*x+c))/b^2/(-a*d^2+b*c^2)/(-b*x^2+a)-1/32*(12*b*c^2-34*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) 
)/a^(1/2)/b^(11/4)/(b^(1/2)*c-a^(1/2)*d)^(3/2)+1/32*(12*b*c^2+34*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))/a^(1/2)/b^(11/4)/(b^(1/2)*c+a^(1/2)*d)^(3/2)
 

Mathematica [A] (verified)

Time = 2.22 (sec) , antiderivative size = 336, normalized size of antiderivative = 1.21 \[ \int \frac {x^4 \sqrt {c+d x}}{\left (a-b x^2\right )^3} \, dx=\frac {\frac {2 b \sqrt {c+d x} \left (10 b^2 c^2 x^3+a^2 d (-c+7 d x)+a b x \left (-6 c^2+c d x-11 d^2 x^2\right )\right )}{\left (b c^2-a d^2\right ) \left (a-b x^2\right )^2}-\frac {\sqrt {-b c-\sqrt {a} \sqrt {b} d} \left (12 b c^2+34 \sqrt {a} \sqrt {b} c d+21 a d^2\right ) \arctan \left (\frac {\sqrt {-b c-\sqrt {a} \sqrt {b} d} \sqrt {c+d x}}{\sqrt {b} c+\sqrt {a} d}\right )}{\sqrt {a} \left (\sqrt {b} c+\sqrt {a} d\right )^2}+\frac {\sqrt {-b c+\sqrt {a} \sqrt {b} d} \left (12 b c^2-34 \sqrt {a} \sqrt {b} c d+21 a d^2\right ) \arctan \left (\frac {\sqrt {-b c+\sqrt {a} \sqrt {b} d} \sqrt {c+d x}}{\sqrt {b} c-\sqrt {a} d}\right )}{\sqrt {a} \left (\sqrt {b} c-\sqrt {a} d\right )^2}}{32 b^3} \] Input:

Integrate[(x^4*Sqrt[c + d*x])/(a - b*x^2)^3,x]
 

Output:

((2*b*Sqrt[c + d*x]*(10*b^2*c^2*x^3 + a^2*d*(-c + 7*d*x) + a*b*x*(-6*c^2 + 
 c*d*x - 11*d^2*x^2)))/((b*c^2 - a*d^2)*(a - b*x^2)^2) - (Sqrt[-(b*c) - Sq 
rt[a]*Sqrt[b]*d]*(12*b*c^2 + 34*Sqrt[a]*Sqrt[b]*c*d + 21*a*d^2)*ArcTan[(Sq 
rt[-(b*c) - Sqrt[a]*Sqrt[b]*d]*Sqrt[c + d*x])/(Sqrt[b]*c + Sqrt[a]*d)])/(S 
qrt[a]*(Sqrt[b]*c + Sqrt[a]*d)^2) + (Sqrt[-(b*c) + Sqrt[a]*Sqrt[b]*d]*(12* 
b*c^2 - 34*Sqrt[a]*Sqrt[b]*c*d + 21*a*d^2)*ArcTan[(Sqrt[-(b*c) + Sqrt[a]*S 
qrt[b]*d]*Sqrt[c + d*x])/(Sqrt[b]*c - Sqrt[a]*d)])/(Sqrt[a]*(Sqrt[b]*c - S 
qrt[a]*d)^2))/(32*b^3)
 

Rubi [A] (verified)

Time = 1.66 (sec) , antiderivative size = 466, normalized size of antiderivative = 1.68, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.348, Rules used = {561, 27, 1672, 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.

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

\(\Big \downarrow \) 561

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1672

\(\displaystyle \frac {2 \left (\frac {d^4 \int \frac {2 \left (8 a \left (a-\frac {b c^2}{d^2}\right ) (c+d x)^3-16 a c \left (a-\frac {b c^2}{d^2}\right ) (c+d x)^2+a \left (-\frac {8 b c^4}{d^2}+5 a c^2+\frac {3 a^2 d^2}{b}\right ) (c+d x)+a^2 c \left (c^2-\frac {a d^2}{b}\right )\right )}{\left (-\frac {b c^2}{d^2}+\frac {2 b (c+d x) c}{d^2}-\frac {b (c+d x)^2}{d^2}+a\right )^2}d\sqrt {c+d x}}{16 a b \left (b c^2-a d^2\right )}-\frac {a d^4 \sqrt {c+d x} \left (c \left (b c^2-a d^2\right )-(c+d x) \left (b c^2-a d^2\right )\right )}{8 b^2 \left (b c^2-a d^2\right ) \left (a-\frac {b c^2}{d^2}+\frac {2 b c (c+d x)}{d^2}-\frac {b (c+d x)^2}{d^2}\right )^2}\right )}{d^5}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {2 \left (\frac {d^4 \int \frac {8 a \left (a-\frac {b c^2}{d^2}\right ) (c+d x)^3-16 a c \left (a-\frac {b c^2}{d^2}\right ) (c+d x)^2+a \left (-\frac {8 b c^4}{d^2}+5 a c^2+\frac {3 a^2 d^2}{b}\right ) (c+d x)+a^2 c \left (c^2-\frac {a d^2}{b}\right )}{\left (-\frac {b c^2}{d^2}+\frac {2 b (c+d x) c}{d^2}-\frac {b (c+d x)^2}{d^2}+a\right )^2}d\sqrt {c+d x}}{8 a b \left (b c^2-a d^2\right )}-\frac {a d^4 \sqrt {c+d x} \left (c \left (b c^2-a d^2\right )-(c+d x) \left (b c^2-a d^2\right )\right )}{8 b^2 \left (b c^2-a d^2\right ) \left (a-\frac {b c^2}{d^2}+\frac {2 b c (c+d x)}{d^2}-\frac {b (c+d x)^2}{d^2}\right )^2}\right )}{d^5}\)

\(\Big \downarrow \) 2206

\(\displaystyle \frac {2 \left (\frac {d^4 \left (\frac {d^4 \int -\frac {2 a^2 \left (b c^2-a d^2\right ) \left (2 c \left (5 b c^2-4 a d^2\right )-\left (22 b c^2-21 a d^2\right ) (c+d x)\right )}{d^4 \left (-\frac {b c^2}{d^2}+\frac {2 b (c+d x) c}{d^2}-\frac {b (c+d x)^2}{d^2}+a\right )}d\sqrt {c+d x}}{8 a b \left (b c^2-a d^2\right )}+\frac {a \sqrt {c+d x} \left (2 c \left (5 b c^2-6 a d^2\right )-(c+d x) \left (10 b c^2-11 a d^2\right )\right )}{4 b \left (a-\frac {b c^2}{d^2}+\frac {2 b c (c+d x)}{d^2}-\frac {b (c+d x)^2}{d^2}\right )}\right )}{8 a b \left (b c^2-a d^2\right )}-\frac {a d^4 \sqrt {c+d x} \left (c \left (b c^2-a d^2\right )-(c+d x) \left (b c^2-a d^2\right )\right )}{8 b^2 \left (b c^2-a d^2\right ) \left (a-\frac {b c^2}{d^2}+\frac {2 b c (c+d x)}{d^2}-\frac {b (c+d x)^2}{d^2}\right )^2}\right )}{d^5}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {2 \left (\frac {d^4 \left (\frac {a \sqrt {c+d x} \left (2 c \left (5 b c^2-6 a d^2\right )-(c+d x) \left (10 b c^2-11 a d^2\right )\right )}{4 b \left (a-\frac {b c^2}{d^2}+\frac {2 b c (c+d x)}{d^2}-\frac {b (c+d x)^2}{d^2}\right )}-\frac {a \int \frac {2 c \left (5 b c^2-4 a d^2\right )-\left (22 b c^2-21 a d^2\right ) (c+d x)}{-\frac {b c^2}{d^2}+\frac {2 b (c+d x) c}{d^2}-\frac {b (c+d x)^2}{d^2}+a}d\sqrt {c+d x}}{4 b}\right )}{8 a b \left (b c^2-a d^2\right )}-\frac {a d^4 \sqrt {c+d x} \left (c \left (b c^2-a d^2\right )-(c+d x) \left (b c^2-a d^2\right )\right )}{8 b^2 \left (b c^2-a d^2\right ) \left (a-\frac {b c^2}{d^2}+\frac {2 b c (c+d x)}{d^2}-\frac {b (c+d x)^2}{d^2}\right )^2}\right )}{d^5}\)

\(\Big \downarrow \) 1480

\(\displaystyle \frac {2 \left (\frac {d^4 \left (\frac {a \sqrt {c+d x} \left (2 c \left (5 b c^2-6 a d^2\right )-(c+d x) \left (10 b c^2-11 a d^2\right )\right )}{4 b \left (a-\frac {b c^2}{d^2}+\frac {2 b c (c+d x)}{d^2}-\frac {b (c+d x)^2}{d^2}\right )}-\frac {a \left (\frac {\left (\sqrt {a} d+\sqrt {b} c\right ) \left (-34 \sqrt {a} \sqrt {b} c d+21 a d^2+12 b c^2\right ) \int \frac {1}{\frac {\sqrt {b} \left (\sqrt {b} c-\sqrt {a} d\right )}{d^2}-\frac {b (c+d x)}{d^2}}d\sqrt {c+d x}}{2 \sqrt {a} d}-\frac {\left (\sqrt {b} c-\sqrt {a} d\right ) \left (34 \sqrt {a} \sqrt {b} c d+21 a d^2+12 b c^2\right ) \int \frac {1}{\frac {\sqrt {b} \left (\sqrt {b} c+\sqrt {a} d\right )}{d^2}-\frac {b (c+d x)}{d^2}}d\sqrt {c+d x}}{2 \sqrt {a} d}\right )}{4 b}\right )}{8 a b \left (b c^2-a d^2\right )}-\frac {a d^4 \sqrt {c+d x} \left (c \left (b c^2-a d^2\right )-(c+d x) \left (b c^2-a d^2\right )\right )}{8 b^2 \left (b c^2-a d^2\right ) \left (a-\frac {b c^2}{d^2}+\frac {2 b c (c+d x)}{d^2}-\frac {b (c+d x)^2}{d^2}\right )^2}\right )}{d^5}\)

\(\Big \downarrow \) 221

\(\displaystyle \frac {2 \left (\frac {d^4 \left (\frac {a \sqrt {c+d x} \left (2 c \left (5 b c^2-6 a d^2\right )-(c+d x) \left (10 b c^2-11 a d^2\right )\right )}{4 b \left (a-\frac {b c^2}{d^2}+\frac {2 b c (c+d x)}{d^2}-\frac {b (c+d x)^2}{d^2}\right )}-\frac {a \left (\frac {d \left (\sqrt {a} d+\sqrt {b} c\right ) \left (-34 \sqrt {a} \sqrt {b} c d+21 a d^2+12 b c^2\right ) \text {arctanh}\left (\frac {\sqrt [4]{b} \sqrt {c+d x}}{\sqrt {\sqrt {b} c-\sqrt {a} d}}\right )}{2 \sqrt {a} b^{3/4} \sqrt {\sqrt {b} c-\sqrt {a} d}}-\frac {d \left (\sqrt {b} c-\sqrt {a} d\right ) \left (34 \sqrt {a} \sqrt {b} c d+21 a d^2+12 b c^2\right ) \text {arctanh}\left (\frac {\sqrt [4]{b} \sqrt {c+d x}}{\sqrt {\sqrt {a} d+\sqrt {b} c}}\right )}{2 \sqrt {a} b^{3/4} \sqrt {\sqrt {a} d+\sqrt {b} c}}\right )}{4 b}\right )}{8 a b \left (b c^2-a d^2\right )}-\frac {a d^4 \sqrt {c+d x} \left (c \left (b c^2-a d^2\right )-(c+d x) \left (b c^2-a d^2\right )\right )}{8 b^2 \left (b c^2-a d^2\right ) \left (a-\frac {b c^2}{d^2}+\frac {2 b c (c+d x)}{d^2}-\frac {b (c+d x)^2}{d^2}\right )^2}\right )}{d^5}\)

Input:

Int[(x^4*Sqrt[c + d*x])/(a - b*x^2)^3,x]
 

Output:

(2*(-1/8*(a*d^4*Sqrt[c + d*x]*(c*(b*c^2 - a*d^2) - (b*c^2 - a*d^2)*(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*Sqrt[c + d*x]*(2*c*(5*b*c^2 - 6*a*d^2) - (10 
*b*c^2 - 11*a*d^2)*(c + d*x)))/(4*b*(a - (b*c^2)/d^2 + (2*b*c*(c + d*x))/d 
^2 - (b*(c + d*x)^2)/d^2)) - (a*((d*(Sqrt[b]*c + Sqrt[a]*d)*(12*b*c^2 - 34 
*Sqrt[a]*Sqrt[b]*c*d + 21*a*d^2)*ArcTanh[(b^(1/4)*Sqrt[c + d*x])/Sqrt[Sqrt 
[b]*c - Sqrt[a]*d]])/(2*Sqrt[a]*b^(3/4)*Sqrt[Sqrt[b]*c - Sqrt[a]*d]) - (d* 
(Sqrt[b]*c - Sqrt[a]*d)*(12*b*c^2 + 34*Sqrt[a]*Sqrt[b]*c*d + 21*a*d^2)*Arc 
Tanh[(b^(1/4)*Sqrt[c + d*x])/Sqrt[Sqrt[b]*c + Sqrt[a]*d]])/(2*Sqrt[a]*b^(3 
/4)*Sqrt[Sqrt[b]*c + Sqrt[a]*d])))/(4*b)))/(8*a*b*(b*c^2 - a*d^2))))/d^5
 

Defintions of rubi rules used

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 1672

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

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 = 0.60 (sec) , antiderivative size = 341, normalized size of antiderivative = 1.23

method	result	size
pseudoelliptic	\(\frac {\frac {13 d \left (\frac {\left (-21 a \,d^{2}+22 b \,c^{2}\right ) \sqrt {a b \,d^{2}}}{13}+b c \left (a \,d^{2}-\frac {12 b \,c^{2}}{13}\right )\right ) \sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}\, \left (-b \,x^{2}+a \right )^{2} \arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (-b c +\sqrt {a b \,d^{2}}\right ) b}}\right )}{2}+\left (\frac {13 d \left (\frac {\left (21 a \,d^{2}-22 b \,c^{2}\right ) \sqrt {a b \,d^{2}}}{13}+b c \left (a \,d^{2}-\frac {12 b \,c^{2}}{13}\right )\right ) \left (-b \,x^{2}+a \right )^{2} \operatorname {arctanh}\left (\frac {b \sqrt {d x +c}}{\sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}}\right )}{2}+\sqrt {d x +c}\, \sqrt {a b \,d^{2}}\, \sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}\, \left (-10 b^{2} c^{2} x^{3}+6 x \left (\frac {11}{6} d^{2} x^{2}-\frac {1}{6} c d x +c^{2}\right ) a b +a^{2} d \left (-7 d x +c \right )\right )\right ) \sqrt {\left (-b c +\sqrt {a b \,d^{2}}\right ) b}}{16 \sqrt {a b \,d^{2}}\, \sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}\, \sqrt {\left (-b c +\sqrt {a b \,d^{2}}\right ) b}\, b^{2} \left (a \,d^{2}-b \,c^{2}\right ) \left (-b \,x^{2}+a \right )^{2}}\)	\(341\)
default	\(2 d \left (\frac {\frac {\left (11 a \,d^{2}-10 b \,c^{2}\right ) \left (d x +c \right )^{\frac {7}{2}}}{32 \left (a \,d^{2}-b \,c^{2}\right ) b}-\frac {\left (17 a \,d^{2}-15 b \,c^{2}\right ) c \left (d x +c \right )^{\frac {5}{2}}}{16 b \left (a \,d^{2}-b \,c^{2}\right )}-\frac {\left (7 a^{2} d^{4}-41 b \,c^{2} d^{2} a +30 b^{2} c^{4}\right ) \left (d x +c \right )^{\frac {3}{2}}}{32 b^{2} \left (a \,d^{2}-b \,c^{2}\right )}+\frac {c \left (4 a \,d^{2}-5 b \,c^{2}\right ) \sqrt {d x +c}}{16 b^{2}}}{\left (-b \left (d x +c \right )^{2}+2 b c \left (d x +c \right )+a \,d^{2}-b \,c^{2}\right )^{2}}+\frac {\frac {\left (13 a b c \,d^{2}-12 c^{3} b^{2}-21 \sqrt {a b \,d^{2}}\, a \,d^{2}+22 \sqrt {a b \,d^{2}}\, b \,c^{2}\right ) \arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (-b c +\sqrt {a b \,d^{2}}\right ) b}}\right )}{2 b \sqrt {a b \,d^{2}}\, \sqrt {\left (-b c +\sqrt {a b \,d^{2}}\right ) b}}-\frac {\left (-13 a b c \,d^{2}+12 c^{3} b^{2}-21 \sqrt {a b \,d^{2}}\, a \,d^{2}+22 \sqrt {a b \,d^{2}}\, b \,c^{2}\right ) \operatorname {arctanh}\left (\frac {b \sqrt {d x +c}}{\sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}}\right )}{2 b \sqrt {a b \,d^{2}}\, \sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}}}{32 b \left (a \,d^{2}-b \,c^{2}\right )}\right )\)	\(420\)
derivativedivides	\(-2 d \left (-\frac {\frac {\left (11 a \,d^{2}-10 b \,c^{2}\right ) \left (d x +c \right )^{\frac {7}{2}}}{32 \left (a \,d^{2}-b \,c^{2}\right ) b}-\frac {\left (17 a \,d^{2}-15 b \,c^{2}\right ) c \left (d x +c \right )^{\frac {5}{2}}}{16 b \left (a \,d^{2}-b \,c^{2}\right )}-\frac {\left (7 a^{2} d^{4}-41 b \,c^{2} d^{2} a +30 b^{2} c^{4}\right ) \left (d x +c \right )^{\frac {3}{2}}}{32 b^{2} \left (a \,d^{2}-b \,c^{2}\right )}+\frac {c \left (4 a \,d^{2}-5 b \,c^{2}\right ) \sqrt {d x +c}}{16 b^{2}}}{\left (-b \left (d x +c \right )^{2}+2 b c \left (d x +c \right )+a \,d^{2}-b \,c^{2}\right )^{2}}-\frac {\frac {\left (13 a b c \,d^{2}-12 c^{3} b^{2}-21 \sqrt {a b \,d^{2}}\, a \,d^{2}+22 \sqrt {a b \,d^{2}}\, b \,c^{2}\right ) \arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (-b c +\sqrt {a b \,d^{2}}\right ) b}}\right )}{2 b \sqrt {a b \,d^{2}}\, \sqrt {\left (-b c +\sqrt {a b \,d^{2}}\right ) b}}-\frac {\left (-13 a b c \,d^{2}+12 c^{3} b^{2}-21 \sqrt {a b \,d^{2}}\, a \,d^{2}+22 \sqrt {a b \,d^{2}}\, b \,c^{2}\right ) \operatorname {arctanh}\left (\frac {b \sqrt {d x +c}}{\sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}}\right )}{2 b \sqrt {a b \,d^{2}}\, \sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}}}{32 b \left (a \,d^{2}-b \,c^{2}\right )}\right )\)	\(421\)

Input:

int(x^4*(d*x+c)^(1/2)/(-b*x^2+a)^3,x,method=_RETURNVERBOSE)
 

Output:

1/16/(a*b*d^2)^(1/2)/((b*c+(a*b*d^2)^(1/2))*b)^(1/2)/((-b*c+(a*b*d^2)^(1/2 
))*b)^(1/2)*(13/2*d*(1/13*(-21*a*d^2+22*b*c^2)*(a*b*d^2)^(1/2)+b*c*(a*d^2- 
12/13*b*c^2))*((b*c+(a*b*d^2)^(1/2))*b)^(1/2)*(-b*x^2+a)^2*arctan(b*(d*x+c 
)^(1/2)/((-b*c+(a*b*d^2)^(1/2))*b)^(1/2))+(13/2*d*(1/13*(21*a*d^2-22*b*c^2 
)*(a*b*d^2)^(1/2)+b*c*(a*d^2-12/13*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))+(d*x+c)^(1/2)*(a*b*d^2)^(1/2)*((b*c 
+(a*b*d^2)^(1/2))*b)^(1/2)*(-10*b^2*c^2*x^3+6*x*(11/6*d^2*x^2-1/6*c*d*x+c^ 
2)*a*b+a^2*d*(-7*d*x+c)))*((-b*c+(a*b*d^2)^(1/2))*b)^(1/2))/b^2/(a*d^2-b*c 
^2)/(-b*x^2+a)^2
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 4643 vs. \(2 (223) = 446\).

Time = 0.95 (sec) , antiderivative size = 4643, normalized size of antiderivative = 16.70 \[ \int \frac {x^4 \sqrt {c+d x}}{\left (a-b x^2\right )^3} \, dx=\text {Too large to display} \] Input:

integrate(x^4*(d*x+c)^(1/2)/(-b*x^2+a)^3,x, algorithm="fricas")
 

Output:

Too large to include
 

Sympy [F(-1)]

Timed out. \[ \int \frac {x^4 \sqrt {c+d x}}{\left (a-b x^2\right )^3} \, dx=\text {Timed out} \] Input:

integrate(x**4*(d*x+c)**(1/2)/(-b*x**2+a)**3,x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \frac {x^4 \sqrt {c+d x}}{\left (a-b x^2\right )^3} \, dx=\int { -\frac {\sqrt {d x + c} x^{4}}{{\left (b x^{2} - a\right )}^{3}} \,d x } \] Input:

integrate(x^4*(d*x+c)^(1/2)/(-b*x^2+a)^3,x, algorithm="maxima")
 

Output:

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1038 vs. \(2 (223) = 446\).

Time = 0.29 (sec) , antiderivative size = 1038, normalized size of antiderivative = 3.73 \[ \int \frac {x^4 \sqrt {c+d x}}{\left (a-b x^2\right )^3} \, dx =\text {Too large to display} \] Input:

integrate(x^4*(d*x+c)^(1/2)/(-b*x^2+a)^3,x, algorithm="giac")
 

Output:

-1/32*((b^3*c^2*d - a*b^2*d^3)^2*(22*a*b*c^2*d - 21*a^2*d^3)*abs(b) - 2*(5 
*sqrt(a*b)*b^4*c^5*d - 9*sqrt(a*b)*a*b^3*c^3*d^3 + 4*sqrt(a*b)*a^2*b^2*c*d 
^5)*abs(b^3*c^2*d - a*b^2*d^3)*abs(b) - (12*b^8*c^8*d - 37*a*b^7*c^6*d^3 + 
 38*a^2*b^6*c^4*d^5 - 13*a^3*b^5*c^2*d^7)*abs(b))*arctan(sqrt(d*x + c)/sqr 
t(-(b^4*c^3 - a*b^3*c*d^2 + sqrt((b^4*c^3 - a*b^3*c*d^2)^2 - (b^4*c^4 - 2* 
a*b^3*c^2*d^2 + a^2*b^2*d^4)*(b^4*c^2 - a*b^3*d^2)))/(b^4*c^2 - a*b^3*d^2) 
))/((a*b^7*c^4*d - 2*a^2*b^6*c^2*d^3 + a^3*b^5*d^5 - sqrt(a*b)*b^7*c^5 + 2 
*sqrt(a*b)*a*b^6*c^3*d^2 - sqrt(a*b)*a^2*b^5*c*d^4)*sqrt(-b^2*c - sqrt(a*b 
)*b*d)*abs(b^3*c^2*d - a*b^2*d^3)) - 1/32*((b^3*c^2*d - a*b^2*d^3)^2*(22*a 
*b*c^2*d - 21*a^2*d^3)*abs(b) + 2*(5*sqrt(a*b)*b^4*c^5*d - 9*sqrt(a*b)*a*b 
^3*c^3*d^3 + 4*sqrt(a*b)*a^2*b^2*c*d^5)*abs(b^3*c^2*d - a*b^2*d^3)*abs(b) 
- (12*b^8*c^8*d - 37*a*b^7*c^6*d^3 + 38*a^2*b^6*c^4*d^5 - 13*a^3*b^5*c^2*d 
^7)*abs(b))*arctan(sqrt(d*x + c)/sqrt(-(b^4*c^3 - a*b^3*c*d^2 - sqrt((b^4* 
c^3 - a*b^3*c*d^2)^2 - (b^4*c^4 - 2*a*b^3*c^2*d^2 + a^2*b^2*d^4)*(b^4*c^2 
- a*b^3*d^2)))/(b^4*c^2 - a*b^3*d^2)))/((a*b^7*c^4*d - 2*a^2*b^6*c^2*d^3 + 
 a^3*b^5*d^5 + sqrt(a*b)*b^7*c^5 - 2*sqrt(a*b)*a*b^6*c^3*d^2 + sqrt(a*b)*a 
^2*b^5*c*d^4)*sqrt(-b^2*c + sqrt(a*b)*b*d)*abs(b^3*c^2*d - a*b^2*d^3)) + 1 
/16*(10*(d*x + c)^(7/2)*b^2*c^2*d - 30*(d*x + c)^(5/2)*b^2*c^3*d + 30*(d*x 
 + c)^(3/2)*b^2*c^4*d - 10*sqrt(d*x + c)*b^2*c^5*d - 11*(d*x + c)^(7/2)*a* 
b*d^3 + 34*(d*x + c)^(5/2)*a*b*c*d^3 - 41*(d*x + c)^(3/2)*a*b*c^2*d^3 +...
 

Mupad [B] (verification not implemented)

Time = 9.79 (sec) , antiderivative size = 7189, normalized size of antiderivative = 25.86 \[ \int \frac {x^4 \sqrt {c+d x}}{\left (a-b x^2\right )^3} \, dx=\text {Too large to display} \] Input:

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

Output:

atan(((((40960*a*b^8*c^5*d^3 + 32768*a^3*b^6*c*d^7 - 73728*a^2*b^7*c^3*d^5 
)/(4096*(b^7*c^4 + a^2*b^5*d^4 - 2*a*b^6*c^2*d^2)) - ((c + d*x)^(1/2)*(409 
6*a*b^8*c^5*d^2 + 4096*a^3*b^6*c*d^6 - 8192*a^2*b^7*c^3*d^4)*(-(441*a^3*d^ 
7*(a^3*b^11)^(1/2) - 144*a*b^9*c^7 + 105*a^4*b^6*c*d^6 + 356*a^2*b^8*c^5*d 
^2 - 321*a^3*b^7*c^3*d^4 - 384*b^3*c^6*d*(a^3*b^11)^(1/2) + 1248*a*b^2*c^4 
*d^3*(a^3*b^11)^(1/2) - 1301*a^2*b*c^2*d^5*(a^3*b^11)^(1/2))/(4096*(a^2*b^ 
14*c^6 - a^5*b^11*d^6 - 3*a^3*b^13*c^4*d^2 + 3*a^4*b^12*c^2*d^4)))^(1/2))/ 
(64*(b^4*c^4 + a^2*b^2*d^4 - 2*a*b^3*c^2*d^2)))*(-(441*a^3*d^7*(a^3*b^11)^ 
(1/2) - 144*a*b^9*c^7 + 105*a^4*b^6*c*d^6 + 356*a^2*b^8*c^5*d^2 - 321*a^3* 
b^7*c^3*d^4 - 384*b^3*c^6*d*(a^3*b^11)^(1/2) + 1248*a*b^2*c^4*d^3*(a^3*b^1 
1)^(1/2) - 1301*a^2*b*c^2*d^5*(a^3*b^11)^(1/2))/(4096*(a^2*b^14*c^6 - a^5* 
b^11*d^6 - 3*a^3*b^13*c^4*d^2 + 3*a^4*b^12*c^2*d^4)))^(1/2) + ((c + d*x)^( 
1/2)*(441*a^3*d^8 + 144*b^3*c^6*d^2 + 172*a*b^2*c^4*d^4 - 755*a^2*b*c^2*d^ 
6))/(64*(b^4*c^4 + a^2*b^2*d^4 - 2*a*b^3*c^2*d^2)))*(-(441*a^3*d^7*(a^3*b^ 
11)^(1/2) - 144*a*b^9*c^7 + 105*a^4*b^6*c*d^6 + 356*a^2*b^8*c^5*d^2 - 321* 
a^3*b^7*c^3*d^4 - 384*b^3*c^6*d*(a^3*b^11)^(1/2) + 1248*a*b^2*c^4*d^3*(a^3 
*b^11)^(1/2) - 1301*a^2*b*c^2*d^5*(a^3*b^11)^(1/2))/(4096*(a^2*b^14*c^6 - 
a^5*b^11*d^6 - 3*a^3*b^13*c^4*d^2 + 3*a^4*b^12*c^2*d^4)))^(1/2)*1i - (((40 
960*a*b^8*c^5*d^3 + 32768*a^3*b^6*c*d^7 - 73728*a^2*b^7*c^3*d^5)/(4096*(b^ 
7*c^4 + a^2*b^5*d^4 - 2*a*b^6*c^2*d^2)) + ((c + d*x)^(1/2)*(4096*a*b^8*...
 

Reduce [B] (verification not implemented)

Time = 3.60 (sec) , antiderivative size = 2568, normalized size of antiderivative = 9.24 \[ \int \frac {x^4 \sqrt {c+d x}}{\left (a-b x^2\right )^3} \, dx =\text {Too large to display} \] Input:

int(x^4*(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**4 + 70*sqrt(a)*sqrt(sqrt(b)*sqr 
t(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**2 + 84*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*d**4*x**2 - 2 
4*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**2*b**2*c**4 - 140*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**2*b**2*c**2*d**2*x**2 - 42*sqrt(a)*sqrt(sqrt(b)*sqrt(a)*d - b*c)*a 
tan((sqrt(c + d*x)*b)/(sqrt(b)*sqrt(sqrt(b)*sqrt(a)*d - b*c)))*a**2*b**2*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**3*c**4*x**2 + 70*sqrt(a)*sq 
rt(sqrt(b)*sqrt(a)*d - b*c)*atan((sqrt(c + d*x)*b)/(sqrt(b)*sqrt(sqrt(b)*s 
qrt(a)*d - b*c)))*a*b**3*c**2*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 
**4*c**4*x**4 - 16*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*c*d**3 + 20*sqrt(b)*sq 
rt(sqrt(b)*sqrt(a)*d - b*c)*atan((sqrt(c + d*x)*b)/(sqrt(b)*sqrt(sqrt(b)*s 
qrt(a)*d - b*c)))*a**3*b*c**3*d + 32*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**3*b...