\(\int \frac {(a+b x^2)^{3/2}}{(c+d x)^4} \, dx\) [1108]

Optimal result

Integrand size = 19, antiderivative size = 220 \[ \int \frac {\left (a+b x^2\right )^{3/2}}{(c+d x)^4} \, dx=-\frac {\left (b c^2+a d^2\right ) \sqrt {a+b x^2}}{3 d^3 (c+d x)^3}+\frac {7 b c \sqrt {a+b x^2}}{6 d^3 (c+d x)^2}-\frac {b \left (11 b c^2+8 a d^2\right ) \sqrt {a+b x^2}}{6 d^3 \left (b c^2+a d^2\right ) (c+d x)}+\frac {b^{3/2} \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{d^4}+\frac {b^2 c \left (2 b c^2+3 a d^2\right ) \text {arctanh}\left (\frac {a d-b c x}{\sqrt {b c^2+a d^2} \sqrt {a+b x^2}}\right )}{2 d^4 \left (b c^2+a d^2\right )^{3/2}} \] Output:

-1/3*(a*d^2+b*c^2)*(b*x^2+a)^(1/2)/d^3/(d*x+c)^3+7/6*b*c*(b*x^2+a)^(1/2)/d 
^3/(d*x+c)^2-1/6*b*(8*a*d^2+11*b*c^2)*(b*x^2+a)^(1/2)/d^3/(a*d^2+b*c^2)/(d 
*x+c)+b^(3/2)*arctanh(b^(1/2)*x/(b*x^2+a)^(1/2))/d^4+1/2*b^2*c*(3*a*d^2+2* 
b*c^2)*arctanh((-b*c*x+a*d)/(a*d^2+b*c^2)^(1/2)/(b*x^2+a)^(1/2))/d^4/(a*d^ 
2+b*c^2)^(3/2)
 

Mathematica [A] (verified)

Time = 0.22 (sec) , antiderivative size = 210, normalized size of antiderivative = 0.95 \[ \int \frac {\left (a+b x^2\right )^{3/2}}{(c+d x)^4} \, dx=-\frac {\frac {d \sqrt {a+b x^2} \left (2 a^2 d^4+a b d^2 \left (5 c^2+9 c d x+8 d^2 x^2\right )+b^2 c^2 \left (6 c^2+15 c d x+11 d^2 x^2\right )\right )}{\left (b c^2+a d^2\right ) (c+d x)^3}+\frac {6 b^2 c \left (2 b c^2+3 a d^2\right ) \arctan \left (\frac {\sqrt {b} (c+d x)-d \sqrt {a+b x^2}}{\sqrt {-b c^2-a d^2}}\right )}{\left (-b c^2-a d^2\right )^{3/2}}+6 b^{3/2} \log \left (-\sqrt {b} x+\sqrt {a+b x^2}\right )}{6 d^4} \] Input:

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

Output:

-1/6*((d*Sqrt[a + b*x^2]*(2*a^2*d^4 + a*b*d^2*(5*c^2 + 9*c*d*x + 8*d^2*x^2 
) + b^2*c^2*(6*c^2 + 15*c*d*x + 11*d^2*x^2)))/((b*c^2 + a*d^2)*(c + d*x)^3 
) + (6*b^2*c*(2*b*c^2 + 3*a*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) + 6*b^(3/2)*Log[-( 
Sqrt[b]*x) + Sqrt[a + b*x^2]])/d^4
 

Rubi [A] (verified)

Time = 0.64 (sec) , antiderivative size = 234, normalized size of antiderivative = 1.06, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.421, Rules used = {492, 589, 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.

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

\(\Big \downarrow \) 492

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

\(\Big \downarrow \) 589

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 719

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

\(\Big \downarrow \) 224

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

\(\Big \downarrow \) 219

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

\(\Big \downarrow \) 488

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

\(\Big \downarrow \) 219

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

Input:

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

Output:

-1/3*(a + b*x^2)^(3/2)/(d*(c + d*x)^3) + (b*(-1/2*((c*(2*b*c^2 + a*d^2) + 
d*(3*b*c^2 + 2*a*d^2)*x)*Sqrt[a + b*x^2])/(d^2*(b*c^2 + a*d^2)*(c + d*x)^2 
) - (b*((-2*(b*c^2 + a*d^2)*ArcTanh[(Sqrt[b]*x)/Sqrt[a + b*x^2]])/(Sqrt[b] 
*d) - (c*(2*b*c^2 + 3*a*d^2)*ArcTanh[(a*d - b*c*x)/(Sqrt[b*c^2 + a*d^2]*Sq 
rt[a + b*x^2])])/(d*Sqrt[b*c^2 + a*d^2])))/(2*d^2*(b*c^2 + a*d^2))))/d
 

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 492

Int[((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[ 
(c + d*x)^(n + 1)*((a + b*x^2)^p/(d*(n + 1))), x] - Simp[2*b*(p/(d*(n + 1)) 
)   Int[x*(c + d*x)^(n + 1)*(a + b*x^2)^(p - 1), x], x] /; FreeQ[{a, b, c, 
d, n}, x] && GtQ[p, 0] && (IntegerQ[p] || LtQ[n, -1]) && NeQ[n, -1] &&  !IL 
tQ[n + 2*p + 1, 0] && IntQuadraticQ[a, 0, b, c, d, n, p, 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 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. \(2445\) vs. \(2(194)=388\).

Time = 0.42 (sec) , antiderivative size = 2446, normalized size of antiderivative = 11.12

method	result	size
default	\(\text {Expression too large to display}\)	\(2446\)

Input:

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

Output:

1/d^4*(-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)^(5/2)+1/3*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)^(5/2)-1/2*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)^(5/2)-3*b*c*d/(a*d^2+b*c^2)*(1/3*(b*(x+c/d)^2-2*b*c/d*(x+c/d)+ 
(a*d^2+b*c^2)/d^2)^(3/2)-b*c/d*(1/4*(2*b*(x+c/d)-2*b*c/d)/b*(b*(x+c/d)^2-2 
*b*c/d*(x+c/d)+(a*d^2+b*c^2)/d^2)^(1/2)+1/8*(4*b*(a*d^2+b*c^2)/d^2-4*b^2*c 
^2/d^2)/b^(3/2)*ln((-b*c/d+b*(x+c/d))/b^(1/2)+(b*(x+c/d)^2-2*b*c/d*(x+c/d) 
+(a*d^2+b*c^2)/d^2)^(1/2)))+(a*d^2+b*c^2)/d^2*((b*(x+c/d)^2-2*b*c/d*(x+c/d 
)+(a*d^2+b*c^2)/d^2)^(1/2)-b^(1/2)*c/d*ln((-b*c/d+b*(x+c/d))/b^(1/2)+(b*(x 
+c/d)^2-2*b*c/d*(x+c/d)+(a*d^2+b*c^2)/d^2)^(1/2))-(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))))+4*b/(a*d^2+b*c^2)*d^2*(1/8*(2*b*(x+c/d)-2*b*c/d)/b*(b*(x+c/d)^2-2*b* 
c/d*(x+c/d)+(a*d^2+b*c^2)/d^2)^(3/2)+3/16*(4*b*(a*d^2+b*c^2)/d^2-4*b^2*c^2 
/d^2)/b*(1/4*(2*b*(x+c/d)-2*b*c/d)/b*(b*(x+c/d)^2-2*b*c/d*(x+c/d)+(a*d^2+b 
*c^2)/d^2)^(1/2)+1/8*(4*b*(a*d^2+b*c^2)/d^2-4*b^2*c^2/d^2)/b^(3/2)*ln((-b* 
c/d+b*(x+c/d))/b^(1/2)+(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/(a*d^2+b*c^2)*d^2*(1/3*(b*(x+c/d)^2-2*b*c/d*(x+c/d)+(a*d^2+b* 
c^2)/d^2)^(3/2)-b*c/d*(1/4*(2*b*(x+c/d)-2*b*c/d)/b*(b*(x+c/d)^2-2*b*c/d...
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 612 vs. \(2 (195) = 390\).

Time = 3.89 (sec) , antiderivative size = 2513, normalized size of antiderivative = 11.42 \[ \int \frac {\left (a+b x^2\right )^{3/2}}{(c+d x)^4} \, dx=\text {Too large to display} \] Input:

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

Output:

[1/12*(6*(b^3*c^7 + 2*a*b^2*c^5*d^2 + a^2*b*c^3*d^4 + (b^3*c^4*d^3 + 2*a*b 
^2*c^2*d^5 + a^2*b*d^7)*x^3 + 3*(b^3*c^5*d^2 + 2*a*b^2*c^3*d^4 + a^2*b*c*d 
^6)*x^2 + 3*(b^3*c^6*d + 2*a*b^2*c^4*d^3 + a^2*b*c^2*d^5)*x)*sqrt(b)*log(- 
2*b*x^2 - 2*sqrt(b*x^2 + a)*sqrt(b)*x - a) + 3*(2*b^3*c^6 + 3*a*b^2*c^4*d^ 
2 + (2*b^3*c^3*d^3 + 3*a*b^2*c*d^5)*x^3 + 3*(2*b^3*c^4*d^2 + 3*a*b^2*c^2*d 
^4)*x^2 + 3*(2*b^3*c^5*d + 3*a*b^2*c^3*d^3)*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*(6 
*b^3*c^6*d + 11*a*b^2*c^4*d^3 + 7*a^2*b*c^2*d^5 + 2*a^3*d^7 + (11*b^3*c^4* 
d^3 + 19*a*b^2*c^2*d^5 + 8*a^2*b*d^7)*x^2 + 3*(5*b^3*c^5*d^2 + 8*a*b^2*c^3 
*d^4 + 3*a^2*b*c*d^6)*x)*sqrt(b*x^2 + a))/(b^2*c^7*d^4 + 2*a*b*c^5*d^6 + a 
^2*c^3*d^8 + (b^2*c^4*d^7 + 2*a*b*c^2*d^9 + a^2*d^11)*x^3 + 3*(b^2*c^5*d^6 
 + 2*a*b*c^3*d^8 + a^2*c*d^10)*x^2 + 3*(b^2*c^6*d^5 + 2*a*b*c^4*d^7 + a^2* 
c^2*d^9)*x), 1/6*(3*(2*b^3*c^6 + 3*a*b^2*c^4*d^2 + (2*b^3*c^3*d^3 + 3*a*b^ 
2*c*d^5)*x^3 + 3*(2*b^3*c^4*d^2 + 3*a*b^2*c^2*d^4)*x^2 + 3*(2*b^3*c^5*d + 
3*a*b^2*c^3*d^3)*x)*sqrt(-b*c^2 - a*d^2)*arctan(sqrt(-b*c^2 - a*d^2)*(b*c* 
x - a*d)*sqrt(b*x^2 + a)/(a*b*c^2 + a^2*d^2 + (b^2*c^2 + a*b*d^2)*x^2)) + 
3*(b^3*c^7 + 2*a*b^2*c^5*d^2 + a^2*b*c^3*d^4 + (b^3*c^4*d^3 + 2*a*b^2*c^2* 
d^5 + a^2*b*d^7)*x^3 + 3*(b^3*c^5*d^2 + 2*a*b^2*c^3*d^4 + a^2*b*c*d^6)*x^2 
 + 3*(b^3*c^6*d + 2*a*b^2*c^4*d^3 + a^2*b*c^2*d^5)*x)*sqrt(b)*log(-2*b*...
 

Sympy [F]

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

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

Output:

Integral((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. 642 vs. \(2 (195) = 390\).

Time = 0.09 (sec) , antiderivative size = 642, normalized size of antiderivative = 2.92 \[ \int \frac {\left (a+b x^2\right )^{3/2}}{(c+d x)^4} \, dx=\frac {\sqrt {b x^{2} + a} b^{3} c^{3}}{2 \, {\left (b^{2} c^{4} d^{3} + 2 \, a b c^{2} d^{5} + a^{2} d^{7}\right )}} - \frac {\sqrt {b x^{2} + a} b^{3} c^{2} x}{2 \, {\left (b^{2} c^{4} d^{2} + 2 \, a b c^{2} d^{4} + a^{2} d^{6}\right )}} + \frac {{\left (b x^{2} + a\right )}^{\frac {3}{2}} b^{2} c^{2}}{6 \, {\left (b^{2} c^{4} d^{2} x + 2 \, a b c^{2} d^{4} x + a^{2} d^{6} x + b^{2} c^{5} d + 2 \, a b c^{3} d^{3} + a^{2} c d^{5}\right )}} - \frac {{\left (b x^{2} + a\right )}^{\frac {5}{2}} b c}{6 \, {\left (b^{2} c^{4} d x^{2} + 2 \, a b c^{2} d^{3} x^{2} + a^{2} d^{5} x^{2} + 2 \, b^{2} c^{5} x + 4 \, a b c^{3} d^{2} x + 2 \, a^{2} c d^{4} x + \frac {b^{2} c^{6}}{d} + 2 \, a b c^{4} d + a^{2} c^{2} d^{3}\right )}} + \frac {{\left (b x^{2} + a\right )}^{\frac {3}{2}} b^{2} c}{6 \, {\left (b^{2} c^{4} d + 2 \, a b c^{2} d^{3} + a^{2} d^{5}\right )}} - \frac {3 \, \sqrt {b x^{2} + a} b^{2} c}{2 \, {\left (b c^{2} d^{3} + a d^{5}\right )}} + \frac {\sqrt {b x^{2} + a} b^{2} x}{b c^{2} d^{2} + a d^{4}} - \frac {{\left (b x^{2} + a\right )}^{\frac {5}{2}}}{3 \, {\left (b c^{2} d^{2} x^{3} + a d^{4} x^{3} + 3 \, b c^{3} d x^{2} + 3 \, a c d^{3} x^{2} + 3 \, b c^{4} x + 3 \, a c^{2} d^{2} x + \frac {b c^{5}}{d} + a c^{3} d\right )}} - \frac {2 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} b}{3 \, {\left (b c^{2} d^{2} x + a d^{4} x + b c^{3} d + a c d^{3}\right )}} + \frac {b^{\frac {3}{2}} \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{d^{4}} + \frac {b^{3} c^{3} \operatorname {arsinh}\left (\frac {b c x}{\sqrt {a b} {\left | d x + c \right |}} - \frac {a d}{\sqrt {a b} {\left | d x + c \right |}}\right )}{2 \, {\left (a + \frac {b c^{2}}{d^{2}}\right )}^{\frac {3}{2}} d^{7}} - \frac {3 \, b^{2} c \operatorname {arsinh}\left (\frac {b c x}{\sqrt {a b} {\left | d x + c \right |}} - \frac {a d}{\sqrt {a b} {\left | d x + c \right |}}\right )}{2 \, \sqrt {a + \frac {b c^{2}}{d^{2}}} d^{5}} \] Input:

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

Output:

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 604 vs. \(2 (195) = 390\).

Time = 0.20 (sec) , antiderivative size = 604, normalized size of antiderivative = 2.75 \[ \int \frac {\left (a+b x^2\right )^{3/2}}{(c+d x)^4} \, dx=\frac {{\left (2 \, b^{3} c^{3} + 3 \, a b^{2} c d^{2}\right )} \arctan \left (\frac {{\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )} d + \sqrt {b} c}{\sqrt {-b c^{2} - a d^{2}}}\right )}{{\left (b c^{2} d^{4} + a d^{6}\right )} \sqrt {-b c^{2} - a d^{2}}} - \frac {b^{\frac {3}{2}} \log \left ({\left | -\sqrt {b} x + \sqrt {b x^{2} + a} \right |}\right )}{d^{4}} - \frac {18 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{5} b^{3} c^{3} d^{2} + 15 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{5} a b^{2} c d^{4} + 54 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{4} b^{\frac {7}{2}} c^{4} d + 27 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{4} a b^{\frac {5}{2}} c^{2} d^{3} - 12 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{4} a^{2} b^{\frac {3}{2}} d^{5} + 44 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{3} b^{4} c^{5} - 34 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{3} a b^{3} c^{3} d^{2} - 48 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{3} a^{2} b^{2} c d^{4} - 78 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} a b^{\frac {7}{2}} c^{4} d - 36 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} a^{2} b^{\frac {5}{2}} c^{2} d^{3} + 12 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} a^{3} b^{\frac {3}{2}} d^{5} + 48 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )} a^{2} b^{3} c^{3} d^{2} + 33 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )} a^{3} b^{2} c d^{4} - 11 \, a^{3} b^{\frac {5}{2}} c^{2} d^{3} - 8 \, a^{4} b^{\frac {3}{2}} d^{5}}{3 \, {\left (b c^{2} d^{4} + a d^{6}\right )} {\left ({\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} d + 2 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )} \sqrt {b} c - a d\right )}^{3}} \] Input:

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

Output:

(2*b^3*c^3 + 3*a*b^2*c*d^2)*arctan(((sqrt(b)*x - sqrt(b*x^2 + a))*d + sqrt 
(b)*c)/sqrt(-b*c^2 - a*d^2))/((b*c^2*d^4 + a*d^6)*sqrt(-b*c^2 - a*d^2)) - 
b^(3/2)*log(abs(-sqrt(b)*x + sqrt(b*x^2 + a)))/d^4 - 1/3*(18*(sqrt(b)*x - 
sqrt(b*x^2 + a))^5*b^3*c^3*d^2 + 15*(sqrt(b)*x - sqrt(b*x^2 + a))^5*a*b^2* 
c*d^4 + 54*(sqrt(b)*x - sqrt(b*x^2 + a))^4*b^(7/2)*c^4*d + 27*(sqrt(b)*x - 
 sqrt(b*x^2 + a))^4*a*b^(5/2)*c^2*d^3 - 12*(sqrt(b)*x - sqrt(b*x^2 + a))^4 
*a^2*b^(3/2)*d^5 + 44*(sqrt(b)*x - sqrt(b*x^2 + a))^3*b^4*c^5 - 34*(sqrt(b 
)*x - sqrt(b*x^2 + a))^3*a*b^3*c^3*d^2 - 48*(sqrt(b)*x - sqrt(b*x^2 + a))^ 
3*a^2*b^2*c*d^4 - 78*(sqrt(b)*x - sqrt(b*x^2 + a))^2*a*b^(7/2)*c^4*d - 36* 
(sqrt(b)*x - sqrt(b*x^2 + a))^2*a^2*b^(5/2)*c^2*d^3 + 12*(sqrt(b)*x - sqrt 
(b*x^2 + a))^2*a^3*b^(3/2)*d^5 + 48*(sqrt(b)*x - sqrt(b*x^2 + a))*a^2*b^3* 
c^3*d^2 + 33*(sqrt(b)*x - sqrt(b*x^2 + a))*a^3*b^2*c*d^4 - 11*a^3*b^(5/2)* 
c^2*d^3 - 8*a^4*b^(3/2)*d^5)/((b*c^2*d^4 + a*d^6)*((sqrt(b)*x - sqrt(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. \[ \int \frac {\left (a+b x^2\right )^{3/2}}{(c+d x)^4} \, dx=\int \frac {{\left (b\,x^2+a\right )}^{3/2}}{{\left (c+d\,x\right )}^4} \,d x \] Input:

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.25 (sec) , antiderivative size = 1734, normalized size of antiderivative = 7.88 \[ \int \frac {\left (a+b x^2\right )^{3/2}}{(c+d x)^4} \, dx =\text {Too large to display} \] Input:

int((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*b**2*c**4*d**2 + 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*b**2*c**3*d**3*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*b**2*c**2*d**4*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*b**2*c*d**5*x**3 + 6*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 + 18*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*x + 18*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**2*x**2 + 6*sqr 
t(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**3*d**3*x**3 - 9*sqrt(a*d**2 + b*c**2)*log(c + d*x)*a*b**2*c* 
*4*d**2 - 27*sqrt(a*d**2 + b*c**2)*log(c + d*x)*a*b**2*c**3*d**3*x - 27*sq 
rt(a*d**2 + b*c**2)*log(c + d*x)*a*b**2*c**2*d**4*x**2 - 9*sqrt(a*d**2 + b 
*c**2)*log(c + d*x)*a*b**2*c*d**5*x**3 - 6*sqrt(a*d**2 + b*c**2)*log(c + d 
*x)*b**3*c**6 - 18*sqrt(a*d**2 + b*c**2)*log(c + d*x)*b**3*c**5*d*x - 18*s 
qrt(a*d**2 + b*c**2)*log(c + d*x)*b**3*c**4*d**2*x**2 - 6*sqrt(a*d**2 + b* 
c**2)*log(c + d*x)*b**3*c**3*d**3*x**3 - 2*sqrt(a + b*x**2)*a**3*d**7 - 7* 
sqrt(a + b*x**2)*a**2*b*c**2*d**5 - 9*sqrt(a + b*x**2)*a**2*b*c*d**6*x - 8 
*sqrt(a + b*x**2)*a**2*b*d**7*x**2 - 11*sqrt(a + b*x**2)*a*b**2*c**4*d*...