\(\int \frac {x (a+b x^2)^{3/2}}{(c+d x)^4} \, dx\) [1107]

Optimal result

Integrand size = 20, antiderivative size = 262 \[ \int \frac {x \left (a+b x^2\right )^{3/2}}{(c+d x)^4} \, dx=\frac {b \sqrt {a+b x^2}}{d^4}+\frac {c \left (b c^2+a d^2\right ) \sqrt {a+b x^2}}{3 d^4 (c+d x)^3}-\frac {\left (10 b c^2+3 a d^2\right ) \sqrt {a+b x^2}}{6 d^4 (c+d x)^2}+\frac {b c \left (26 b c^2+23 a d^2\right ) \sqrt {a+b x^2}}{6 d^4 \left (b c^2+a d^2\right ) (c+d x)}-\frac {4 b^{3/2} c \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{d^5}-\frac {b \left (8 b^2 c^4+12 a b c^2 d^2+3 a^2 d^4\right ) \text {arctanh}\left (\frac {a d-b c x}{\sqrt {b c^2+a d^2} \sqrt {a+b x^2}}\right )}{2 d^5 \left (b c^2+a d^2\right )^{3/2}} \] Output:

b*(b*x^2+a)^(1/2)/d^4+1/3*c*(a*d^2+b*c^2)*(b*x^2+a)^(1/2)/d^4/(d*x+c)^3-1/ 
6*(3*a*d^2+10*b*c^2)*(b*x^2+a)^(1/2)/d^4/(d*x+c)^2+1/6*b*c*(23*a*d^2+26*b* 
c^2)*(b*x^2+a)^(1/2)/d^4/(a*d^2+b*c^2)/(d*x+c)-4*b^(3/2)*c*arctanh(b^(1/2) 
*x/(b*x^2+a)^(1/2))/d^5-1/2*b*(3*a^2*d^4+12*a*b*c^2*d^2+8*b^2*c^4)*arctanh 
((-b*c*x+a*d)/(a*d^2+b*c^2)^(1/2)/(b*x^2+a)^(1/2))/d^5/(a*d^2+b*c^2)^(3/2)
 

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(697\) vs. \(2(262)=524\).

Time = 4.70 (sec) , antiderivative size = 697, normalized size of antiderivative = 2.66 \[ \int \frac {x \left (a+b x^2\right )^{3/2}}{(c+d x)^4} \, dx=\frac {-\frac {a d \left (24 b^3 c^2 x^2 \left (4 c^3+11 c^2 d x+9 c d^2 x^2+d^3 x^3\right ) \left (\sqrt {b} x-\sqrt {a+b x^2}\right )+a^3 d^4 (c+3 d x) \left (-4 \sqrt {b} x+\sqrt {a+b x^2}\right )+a^2 b d \left (d \sqrt {a+b x^2} \left (-20 c^3-51 c^2 d x-33 c d^2 x^2+18 d^3 x^3\right )+\sqrt {b} \left (3 c^4+65 c^3 d x+153 c^2 d^2 x^2+111 c d^3 x^3-18 d^4 x^4\right )\right )-6 a b^2 c \left (\sqrt {a+b x^2} \left (4 c^4+12 c^3 d x+24 c^2 d^2 x^2+35 c d^3 x^3+26 d^4 x^4\right )-\sqrt {b} x \left (12 c^4+34 c^3 d x+42 c^2 d^2 x^2+37 c d^3 x^3+26 d^4 x^4\right )\right )\right )}{\left (b c^2+a d^2\right ) (c+d x)^3 \left (a^2+8 a b x^2+8 b^2 x^4-4 a \sqrt {b} x \sqrt {a+b x^2}-8 b^{3/2} x^3 \sqrt {a+b x^2}\right )}+\frac {48 b^3 c^4 \arctan \left (\frac {\sqrt {b} (c+d x)-d \sqrt {a+b x^2}}{\sqrt {-b c^2-a d^2}}\right )}{\left (-b c^2-a d^2\right )^{3/2}}+\frac {72 a b^2 c^2 d^2 \arctan \left (\frac {\sqrt {b} (c+d x)-d \sqrt {a+b x^2}}{\sqrt {-b c^2-a d^2}}\right )}{\left (-b c^2-a d^2\right )^{3/2}}+\frac {18 a^2 b d^4 \arctan \left (\frac {\sqrt {b} (c+d x)-d \sqrt {a+b x^2}}{\sqrt {-b c^2-a d^2}}\right )}{\left (-b c^2-a d^2\right )^{3/2}}+\frac {2 b^{3/2} \left (13 c^4+27 c^3 d x+9 c^2 d^2 x^2-9 c d^3 x^3-3 d^4 x^4+12 c (c+d x)^3 \log \left (-\sqrt {b} x+\sqrt {a+b x^2}\right )\right )}{(c+d x)^3}}{6 d^5} \] Input:

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

Output:

(-((a*d*(24*b^3*c^2*x^2*(4*c^3 + 11*c^2*d*x + 9*c*d^2*x^2 + d^3*x^3)*(Sqrt 
[b]*x - Sqrt[a + b*x^2]) + a^3*d^4*(c + 3*d*x)*(-4*Sqrt[b]*x + Sqrt[a + b* 
x^2]) + a^2*b*d*(d*Sqrt[a + b*x^2]*(-20*c^3 - 51*c^2*d*x - 33*c*d^2*x^2 + 
18*d^3*x^3) + Sqrt[b]*(3*c^4 + 65*c^3*d*x + 153*c^2*d^2*x^2 + 111*c*d^3*x^ 
3 - 18*d^4*x^4)) - 6*a*b^2*c*(Sqrt[a + b*x^2]*(4*c^4 + 12*c^3*d*x + 24*c^2 
*d^2*x^2 + 35*c*d^3*x^3 + 26*d^4*x^4) - Sqrt[b]*x*(12*c^4 + 34*c^3*d*x + 4 
2*c^2*d^2*x^2 + 37*c*d^3*x^3 + 26*d^4*x^4))))/((b*c^2 + a*d^2)*(c + d*x)^3 
*(a^2 + 8*a*b*x^2 + 8*b^2*x^4 - 4*a*Sqrt[b]*x*Sqrt[a + b*x^2] - 8*b^(3/2)* 
x^3*Sqrt[a + b*x^2]))) + (48*b^3*c^4*ArcTan[(Sqrt[b]*(c + d*x) - d*Sqrt[a 
+ b*x^2])/Sqrt[-(b*c^2) - a*d^2]])/(-(b*c^2) - a*d^2)^(3/2) + (72*a*b^2*c^ 
2*d^2*ArcTan[(Sqrt[b]*(c + d*x) - d*Sqrt[a + b*x^2])/Sqrt[-(b*c^2) - a*d^2 
]])/(-(b*c^2) - a*d^2)^(3/2) + (18*a^2*b*d^4*ArcTan[(Sqrt[b]*(c + d*x) - d 
*Sqrt[a + b*x^2])/Sqrt[-(b*c^2) - a*d^2]])/(-(b*c^2) - a*d^2)^(3/2) + (2*b 
^(3/2)*(13*c^4 + 27*c^3*d*x + 9*c^2*d^2*x^2 - 9*c*d^3*x^3 - 3*d^4*x^4 + 12 
*c*(c + d*x)^3*Log[-(Sqrt[b]*x) + Sqrt[a + b*x^2]]))/(c + d*x)^3)/(6*d^5)
 

Rubi [A] (verified)

Time = 0.78 (sec) , antiderivative size = 277, normalized size of antiderivative = 1.06, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.450, Rules used = {589, 27, 681, 27, 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.

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

\(\Big \downarrow \) 589

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 681

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 719

\(\displaystyle -\frac {b \left (-\frac {\frac {\left (3 a^2 d^4+12 a b c^2 d^2+8 b^2 c^4\right ) \int \frac {1}{(c+d x) \sqrt {b x^2+a}}dx}{d}-\frac {8 b c \left (a d^2+b c^2\right ) \int \frac {1}{\sqrt {b x^2+a}}dx}{d}}{d^2}-\frac {\sqrt {a+b x^2} \left (d x \left (3 a d^2+4 b c^2\right )+8 c \left (a d^2+b c^2\right )\right )}{d^2 (c+d x)}\right )}{2 d^2 \left (a d^2+b c^2\right )}-\frac {\left (a+b x^2\right )^{3/2} \left (3 d x \left (a d^2+2 b c^2\right )+c \left (a d^2+4 b c^2\right )\right )}{6 d^2 (c+d x)^3 \left (a d^2+b c^2\right )}\)

\(\Big \downarrow \) 224

\(\displaystyle -\frac {b \left (-\frac {\frac {\left (3 a^2 d^4+12 a b c^2 d^2+8 b^2 c^4\right ) \int \frac {1}{(c+d x) \sqrt {b x^2+a}}dx}{d}-\frac {8 b c \left (a d^2+b c^2\right ) \int \frac {1}{1-\frac {b x^2}{b x^2+a}}d\frac {x}{\sqrt {b x^2+a}}}{d}}{d^2}-\frac {\sqrt {a+b x^2} \left (d x \left (3 a d^2+4 b c^2\right )+8 c \left (a d^2+b c^2\right )\right )}{d^2 (c+d x)}\right )}{2 d^2 \left (a d^2+b c^2\right )}-\frac {\left (a+b x^2\right )^{3/2} \left (3 d x \left (a d^2+2 b c^2\right )+c \left (a d^2+4 b c^2\right )\right )}{6 d^2 (c+d x)^3 \left (a d^2+b c^2\right )}\)

\(\Big \downarrow \) 219

\(\displaystyle -\frac {b \left (-\frac {\frac {\left (3 a^2 d^4+12 a b c^2 d^2+8 b^2 c^4\right ) \int \frac {1}{(c+d x) \sqrt {b x^2+a}}dx}{d}-\frac {8 \sqrt {b} c \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right ) \left (a d^2+b c^2\right )}{d}}{d^2}-\frac {\sqrt {a+b x^2} \left (d x \left (3 a d^2+4 b c^2\right )+8 c \left (a d^2+b c^2\right )\right )}{d^2 (c+d x)}\right )}{2 d^2 \left (a d^2+b c^2\right )}-\frac {\left (a+b x^2\right )^{3/2} \left (3 d x \left (a d^2+2 b c^2\right )+c \left (a d^2+4 b c^2\right )\right )}{6 d^2 (c+d x)^3 \left (a d^2+b c^2\right )}\)

\(\Big \downarrow \) 488

\(\displaystyle -\frac {b \left (-\frac {-\frac {\left (3 a^2 d^4+12 a b c^2 d^2+8 b^2 c^4\right ) \int \frac {1}{b c^2+a d^2-\frac {(a d-b c x)^2}{b x^2+a}}d\frac {a d-b c x}{\sqrt {b x^2+a}}}{d}-\frac {8 \sqrt {b} c \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right ) \left (a d^2+b c^2\right )}{d}}{d^2}-\frac {\sqrt {a+b x^2} \left (d x \left (3 a d^2+4 b c^2\right )+8 c \left (a d^2+b c^2\right )\right )}{d^2 (c+d x)}\right )}{2 d^2 \left (a d^2+b c^2\right )}-\frac {\left (a+b x^2\right )^{3/2} \left (3 d x \left (a d^2+2 b c^2\right )+c \left (a d^2+4 b c^2\right )\right )}{6 d^2 (c+d x)^3 \left (a d^2+b c^2\right )}\)

\(\Big \downarrow \) 219

\(\displaystyle -\frac {b \left (-\frac {-\frac {\left (3 a^2 d^4+12 a b c^2 d^2+8 b^2 c^4\right ) \text {arctanh}\left (\frac {a d-b c x}{\sqrt {a+b x^2} \sqrt {a d^2+b c^2}}\right )}{d \sqrt {a d^2+b c^2}}-\frac {8 \sqrt {b} c \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right ) \left (a d^2+b c^2\right )}{d}}{d^2}-\frac {\sqrt {a+b x^2} \left (d x \left (3 a d^2+4 b c^2\right )+8 c \left (a d^2+b c^2\right )\right )}{d^2 (c+d x)}\right )}{2 d^2 \left (a d^2+b c^2\right )}-\frac {\left (a+b x^2\right )^{3/2} \left (3 d x \left (a d^2+2 b c^2\right )+c \left (a d^2+4 b c^2\right )\right )}{6 d^2 (c+d x)^3 \left (a d^2+b c^2\right )}\)

Input:

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

Output:

-1/6*((c*(4*b*c^2 + a*d^2) + 3*d*(2*b*c^2 + a*d^2)*x)*(a + b*x^2)^(3/2))/( 
d^2*(b*c^2 + a*d^2)*(c + d*x)^3) - (b*(-(((8*c*(b*c^2 + a*d^2) + d*(4*b*c^ 
2 + 3*a*d^2)*x)*Sqrt[a + b*x^2])/(d^2*(c + d*x))) - ((-8*Sqrt[b]*c*(b*c^2 
+ a*d^2)*ArcTanh[(Sqrt[b]*x)/Sqrt[a + b*x^2]])/d - ((8*b^2*c^4 + 12*a*b*c^ 
2*d^2 + 3*a^2*d^4)*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^2*(b*c^2 + a*d^2))
 

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 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 589

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*(a*d^2 + b*c^2*(2*p + 1)) - d*( 
a*d^2*(n + 1) + b*c^2*(n - 2*p + 1))*x)/(d^2*(n + 1)*(n + 2)*(b*c^2 + a*d^2 
))), x] + Simp[b*(p/(d^2*(n + 1)*(n + 2)*(b*c^2 + a*d^2)))   Int[(c + d*x)^ 
(n + 2)*(a + b*x^2)^(p - 1)*Simp[2*a*c*d*(n + 2) - (2*a*d^2*(n + 1) - 2*b*c 
^2*(2*p + 1))*x, x], x], x] /; FreeQ[{a, b, c, d}, x] && GtQ[p, 0] && LtQ[n 
, -2] && LtQ[n + 2*p, 0] &&  !ILtQ[n + 2*p + 3, 0]
 

rule 681

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p 
_.), x_Symbol] :> Simp[(d + e*x)^(m + 1)*(e*f*(m + 2*p + 2) - d*g*(2*p + 1) 
 + e*g*(m + 1)*x)*((a + c*x^2)^p/(e^2*(m + 1)*(m + 2*p + 2))), x] + Simp[p/ 
(e^2*(m + 1)*(m + 2*p + 2))   Int[(d + e*x)^(m + 1)*(a + c*x^2)^(p - 1)*Sim 
p[g*(2*a*e + 2*a*e*m) + (g*(2*c*d + 4*c*d*p) - 2*c*e*f*(m + 2*p + 2))*x, x] 
, x], x] /; FreeQ[{a, c, d, e, f, g, m}, x] && GtQ[p, 0] && (LtQ[m, -1] || 
EqQ[p, 1] || (IntegerQ[p] &&  !RationalQ[m])) && NeQ[m, -1] &&  !ILtQ[m + 2 
*p + 1, 0] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])
 

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. \(1672\) vs. \(2(234)=468\).

Time = 0.46 (sec) , antiderivative size = 1673, normalized size of antiderivative = 6.39

method	result	size
risch	\(\text {Expression too large to display}\)	\(1673\)
default	\(\text {Expression too large to display}\)	\(3919\)

Input:

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

Output:

b*(b*x^2+a)^(1/2)/d^4-1/d^4*(-1/d^4*(a^2*d^4+6*a*b*c^2*d^2+5*b^2*c^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)))+c*(a^2*d^4+2*a*b*c^2*d^2+b^2*c^4)/d^5* 
(-1/3/(a*d^2+b*c^2)*d^2/(x+c/d)^3*(b*(x+c/d)^2-2*b*c/d*(x+c/d)+(a*d^2+b*c^ 
2)/d^2)^(1/2)+5/3*b*c*d/(a*d^2+b*c^2)*(-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 
)))-2/3*b/(a*d^2+b*c^2)*d^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...
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 708 vs. \(2 (235) = 470\).

Time = 6.54 (sec) , antiderivative size = 2897, normalized size of antiderivative = 11.06 \[ \int \frac {x \left (a+b x^2\right )^{3/2}}{(c+d x)^4} \, dx=\text {Too large to display} \] Input:

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

Output:

[1/12*(24*(b^3*c^8 + 2*a*b^2*c^6*d^2 + a^2*b*c^4*d^4 + (b^3*c^5*d^3 + 2*a* 
b^2*c^3*d^5 + a^2*b*c*d^7)*x^3 + 3*(b^3*c^6*d^2 + 2*a*b^2*c^4*d^4 + a^2*b* 
c^2*d^6)*x^2 + 3*(b^3*c^7*d + 2*a*b^2*c^5*d^3 + a^2*b*c^3*d^5)*x)*sqrt(b)* 
log(-2*b*x^2 + 2*sqrt(b*x^2 + a)*sqrt(b)*x - a) + 3*(8*b^3*c^7 + 12*a*b^2* 
c^5*d^2 + 3*a^2*b*c^3*d^4 + (8*b^3*c^4*d^3 + 12*a*b^2*c^2*d^5 + 3*a^2*b*d^ 
7)*x^3 + 3*(8*b^3*c^5*d^2 + 12*a*b^2*c^3*d^4 + 3*a^2*b*c*d^6)*x^2 + 3*(8*b 
^3*c^6*d + 12*a*b^2*c^4*d^3 + 3*a^2*b*c^2*d^5)*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 
*(24*b^3*c^7*d + 44*a*b^2*c^5*d^3 + 19*a^2*b*c^3*d^5 - a^3*c*d^7 + 6*(b^3* 
c^4*d^4 + 2*a*b^2*c^2*d^6 + a^2*b*d^8)*x^3 + (44*b^3*c^5*d^3 + 85*a*b^2*c^ 
3*d^5 + 41*a^2*b*c*d^7)*x^2 + 3*(20*b^3*c^6*d^2 + 37*a*b^2*c^4*d^4 + 16*a^ 
2*b*c^2*d^6 - a^3*d^8)*x)*sqrt(b*x^2 + a))/(b^2*c^7*d^5 + 2*a*b*c^5*d^7 + 
a^2*c^3*d^9 + (b^2*c^4*d^8 + 2*a*b*c^2*d^10 + a^2*d^12)*x^3 + 3*(b^2*c^5*d 
^7 + 2*a*b*c^3*d^9 + a^2*c*d^11)*x^2 + 3*(b^2*c^6*d^6 + 2*a*b*c^4*d^8 + a^ 
2*c^2*d^10)*x), 1/12*(48*(b^3*c^8 + 2*a*b^2*c^6*d^2 + a^2*b*c^4*d^4 + (b^3 
*c^5*d^3 + 2*a*b^2*c^3*d^5 + a^2*b*c*d^7)*x^3 + 3*(b^3*c^6*d^2 + 2*a*b^2*c 
^4*d^4 + a^2*b*c^2*d^6)*x^2 + 3*(b^3*c^7*d + 2*a*b^2*c^5*d^3 + a^2*b*c^3*d 
^5)*x)*sqrt(-b)*arctan(sqrt(-b)*x/sqrt(b*x^2 + a)) + 3*(8*b^3*c^7 + 12*a*b 
^2*c^5*d^2 + 3*a^2*b*c^3*d^4 + (8*b^3*c^4*d^3 + 12*a*b^2*c^2*d^5 + 3*a^...
 

Sympy [F]

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

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

Output:

Integral(x*(a + b*x**2)**(3/2)/(c + d*x)**4, x)
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 825 vs. \(2 (235) = 470\).

Time = 0.10 (sec) , antiderivative size = 825, normalized size of antiderivative = 3.15 \[ \int \frac {x \left (a+b x^2\right )^{3/2}}{(c+d x)^4} \, dx =\text {Too large to display} \] Input:

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

Output:

-1/2*sqrt(b*x^2 + a)*b^3*c^4/(b^2*c^4*d^4 + 2*a*b*c^2*d^6 + a^2*d^8) + 1/2 
*sqrt(b*x^2 + a)*b^3*c^3*x/(b^2*c^4*d^3 + 2*a*b*c^2*d^5 + a^2*d^7) - 1/6*( 
b*x^2 + a)^(3/2)*b^2*c^3/(b^2*c^4*d^3*x + 2*a*b*c^2*d^5*x + a^2*d^7*x + b^ 
2*c^5*d^2 + 2*a*b*c^3*d^4 + a^2*c*d^6) + 1/6*(b*x^2 + a)^(5/2)*b*c^2/(b^2* 
c^4*d^2*x^2 + 2*a*b*c^2*d^4*x^2 + a^2*d^6*x^2 + 2*b^2*c^5*d*x + 4*a*b*c^3* 
d^3*x + 2*a^2*c*d^5*x + b^2*c^6 + 2*a*b*c^4*d^2 + a^2*c^2*d^4) - 1/6*(b*x^ 
2 + a)^(3/2)*b^2*c^2/(b^2*c^4*d^2 + 2*a*b*c^2*d^4 + a^2*d^6) + 3*sqrt(b*x^ 
2 + a)*b^2*c^2/(b*c^2*d^4 + a*d^6) - 5/2*sqrt(b*x^2 + a)*b^2*c*x/(b*c^2*d^ 
3 + a*d^5) + 1/3*(b*x^2 + a)^(5/2)*c/(b*c^2*d^3*x^3 + a*d^5*x^3 + 3*b*c^3* 
d^2*x^2 + 3*a*c*d^4*x^2 + 3*b*c^4*d*x + 3*a*c^2*d^3*x + b*c^5 + a*c^3*d^2) 
 + 7/6*(b*x^2 + a)^(3/2)*b*c/(b*c^2*d^3*x + a*d^5*x + b*c^3*d^2 + a*c*d^4) 
 - 1/2*(b*x^2 + a)^(5/2)/(b*c^2*d^2*x^2 + a*d^4*x^2 + 2*b*c^3*d*x + 2*a*c* 
d^3*x + b*c^4 + a*c^2*d^2) + 1/2*(b*x^2 + a)^(3/2)*b/(b*c^2*d^2 + a*d^4) - 
 4*b^(3/2)*c*arcsinh(b*x/sqrt(a*b))/d^5 - 1/2*b^3*c^4*arcsinh(b*c*x/(sqrt( 
a*b)*abs(d*x + c)) - a*d/(sqrt(a*b)*abs(d*x + c)))/((a + b*c^2/d^2)^(3/2)* 
d^8) + 3*b^2*c^2*arcsinh(b*c*x/(sqrt(a*b)*abs(d*x + c)) - a*d/(sqrt(a*b)*a 
bs(d*x + c)))/(sqrt(a + b*c^2/d^2)*d^6) + 3/2*sqrt(a + b*c^2/d^2)*b*arcsin 
h(b*c*x/(sqrt(a*b)*abs(d*x + c)) - a*d/(sqrt(a*b)*abs(d*x + c)))/d^4 + 3/2 
*sqrt(b*x^2 + a)*b/d^4
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 694 vs. \(2 (235) = 470\).

Time = 0.20 (sec) , antiderivative size = 694, normalized size of antiderivative = 2.65 \[ \int \frac {x \left (a+b x^2\right )^{3/2}}{(c+d x)^4} \, dx=\frac {{\left (8 \, b^{3} c^{4} + 12 \, a b^{2} c^{2} d^{2} + 3 \, a^{2} b d^{4}\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 )}{{\left (b c^{2} d^{5} + a d^{7}\right )} \sqrt {-b c^{2} - a d^{2}}} + \frac {4 \, b^{\frac {3}{2}} c \log \left ({\left | -\sqrt {b} x + \sqrt {b x^{2} + a} \right |}\right )}{d^{5}} + \frac {\sqrt {b x^{2} + a} b}{d^{4}} + \frac {36 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{5} b^{3} c^{4} d^{2} + 36 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{5} a b^{2} c^{2} d^{4} + 3 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{5} a^{2} b d^{6} + 120 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{4} b^{\frac {7}{2}} c^{5} d + 84 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{4} a b^{\frac {5}{2}} c^{3} d^{3} - 21 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{4} a^{2} b^{\frac {3}{2}} c d^{5} + 104 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{3} b^{4} c^{6} - 64 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{3} a b^{3} c^{4} d^{2} - 138 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{3} a^{2} b^{2} c^{2} d^{4} - 192 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} a b^{\frac {7}{2}} c^{5} d - 114 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} a^{2} b^{\frac {5}{2}} c^{3} d^{3} + 48 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} a^{3} b^{\frac {3}{2}} c d^{5} + 120 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )} a^{2} b^{3} c^{4} d^{2} + 102 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )} a^{3} b^{2} c^{2} d^{4} - 3 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )} a^{4} b d^{6} - 26 \, a^{3} b^{\frac {5}{2}} c^{3} d^{3} - 23 \, a^{4} b^{\frac {3}{2}} c d^{5}}{3 \, {\left (b c^{2} d^{5} + a d^{7}\right )} {\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 )}^{3}} \] Input:

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

Output:

(8*b^3*c^4 + 12*a*b^2*c^2*d^2 + 3*a^2*b*d^4)*arctan(-((sqrt(b)*x - sqrt(b* 
x^2 + a))*d + sqrt(b)*c)/sqrt(-b*c^2 - a*d^2))/((b*c^2*d^5 + a*d^7)*sqrt(- 
b*c^2 - a*d^2)) + 4*b^(3/2)*c*log(abs(-sqrt(b)*x + sqrt(b*x^2 + a)))/d^5 + 
 sqrt(b*x^2 + a)*b/d^4 + 1/3*(36*(sqrt(b)*x - sqrt(b*x^2 + a))^5*b^3*c^4*d 
^2 + 36*(sqrt(b)*x - sqrt(b*x^2 + a))^5*a*b^2*c^2*d^4 + 3*(sqrt(b)*x - sqr 
t(b*x^2 + a))^5*a^2*b*d^6 + 120*(sqrt(b)*x - sqrt(b*x^2 + a))^4*b^(7/2)*c^ 
5*d + 84*(sqrt(b)*x - sqrt(b*x^2 + a))^4*a*b^(5/2)*c^3*d^3 - 21*(sqrt(b)*x 
 - sqrt(b*x^2 + a))^4*a^2*b^(3/2)*c*d^5 + 104*(sqrt(b)*x - sqrt(b*x^2 + a) 
)^3*b^4*c^6 - 64*(sqrt(b)*x - sqrt(b*x^2 + a))^3*a*b^3*c^4*d^2 - 138*(sqrt 
(b)*x - sqrt(b*x^2 + a))^3*a^2*b^2*c^2*d^4 - 192*(sqrt(b)*x - sqrt(b*x^2 + 
 a))^2*a*b^(7/2)*c^5*d - 114*(sqrt(b)*x - sqrt(b*x^2 + a))^2*a^2*b^(5/2)*c 
^3*d^3 + 48*(sqrt(b)*x - sqrt(b*x^2 + a))^2*a^3*b^(3/2)*c*d^5 + 120*(sqrt( 
b)*x - sqrt(b*x^2 + a))*a^2*b^3*c^4*d^2 + 102*(sqrt(b)*x - sqrt(b*x^2 + a) 
)*a^3*b^2*c^2*d^4 - 3*(sqrt(b)*x - sqrt(b*x^2 + a))*a^4*b*d^6 - 26*a^3*b^( 
5/2)*c^3*d^3 - 23*a^4*b^(3/2)*c*d^5)/((b*c^2*d^5 + a*d^7)*((sqrt(b)*x - sq 
rt(b*x^2 + a))^2*d + 2*(sqrt(b)*x - sqrt(b*x^2 + a))*sqrt(b)*c - a*d)^3)
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.22 (sec) , antiderivative size = 2166, normalized size of antiderivative = 8.27 \[ \int \frac {x \left (a+b x^2\right )^{3/2}}{(c+d x)^4} \, dx =\text {Too large to display} \] Input:

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

Output:

(9*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**2*b*c**3*d**4 + 27*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**2*b*c**2*d**5*x + 27*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**2*b* 
c*d**6*x**2 + 9*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**2*b*d**7*x**3 + 36*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**2*c**5*d**2 + 108*s 
qrt(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**2*c**4*d**3*x + 108*sqrt(a*d**2 + b*c**2)*log(sqrt(a + b*x**2)*s 
qrt(a*d**2 + b*c**2) - a*d + b*c*x)*a*b**2*c**3*d**4*x**2 + 36*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** 
2*c**2*d**5*x**3 + 24*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**3*c**7 + 72*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**3*c**6*d*x + 72*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**3*c**5*d**2*x**2 + 24*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**3*c**4*d**3*x**3 - 9*sqrt(a*d**2 + b*c** 
2)*log(c + d*x)*a**2*b*c**3*d**4 - 27*sqrt(a*d**2 + b*c**2)*log(c + d*x)*a 
**2*b*c**2*d**5*x - 27*sqrt(a*d**2 + b*c**2)*log(c + d*x)*a**2*b*c*d**6*x* 
*2 - 9*sqrt(a*d**2 + b*c**2)*log(c + d*x)*a**2*b*d**7*x**3 - 36*sqrt(a*...