\(\int \frac {A+B x+C x^2+D x^3}{(c+d x)^3 \sqrt {a+b x^2}} \, dx\) [107]

Optimal result

Integrand size = 34, antiderivative size = 313 \[ \int \frac {A+B x+C x^2+D x^3}{(c+d x)^3 \sqrt {a+b x^2}} \, dx=-\frac {\left (c^2 C d-B c d^2+A d^3-c^3 D\right ) \sqrt {a+b x^2}}{2 d^2 \left (b c^2+a d^2\right ) (c+d x)^2}+\frac {\left (2 a d^2 \left (2 c C d-B d^2-3 c^2 D\right )+b c \left (c^2 C d+B c d^2-3 A d^3-3 c^3 D\right )\right ) \sqrt {a+b x^2}}{2 d^2 \left (b c^2+a d^2\right )^2 (c+d x)}+\frac {D \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{\sqrt {b} d^3}-\frac {\left (A b d^3 \left (2 b c^2-a d^2\right )-2 b^2 c^5 D+2 a^2 d^4 (C d-3 c D)-a b c d^2 \left (c C d-3 B d^2+5 c^2 D\right )\right ) \text {arctanh}\left (\frac {a d-b c x}{\sqrt {b c^2+a d^2} \sqrt {a+b x^2}}\right )}{2 d^3 \left (b c^2+a d^2\right )^{5/2}} \] Output:

-1/2*(A*d^3-B*c*d^2+C*c^2*d-D*c^3)*(b*x^2+a)^(1/2)/d^2/(a*d^2+b*c^2)/(d*x+ 
c)^2+1/2*(2*a*d^2*(-B*d^2+2*C*c*d-3*D*c^2)+b*c*(-3*A*d^3+B*c*d^2+C*c^2*d-3 
*D*c^3))*(b*x^2+a)^(1/2)/d^2/(a*d^2+b*c^2)^2/(d*x+c)+D*arctanh(b^(1/2)*x/( 
b*x^2+a)^(1/2))/b^(1/2)/d^3-1/2*(A*b*d^3*(-a*d^2+2*b*c^2)-2*b^2*c^5*D+2*a^ 
2*d^4*(C*d-3*D*c)-a*b*c*d^2*(-3*B*d^2+C*c*d+5*D*c^2))*arctanh((-b*c*x+a*d) 
/(a*d^2+b*c^2)^(1/2)/(b*x^2+a)^(1/2))/d^3/(a*d^2+b*c^2)^(5/2)
 

Mathematica [A] (verified)

Time = 3.04 (sec) , antiderivative size = 303, normalized size of antiderivative = 0.97 \[ \int \frac {A+B x+C x^2+D x^3}{(c+d x)^3 \sqrt {a+b x^2}} \, dx=-\frac {\frac {d \sqrt {a+b x^2} \left (a d^2 \left (5 c^3 D+d^3 (A+2 B x)+c d^2 (B-4 C x)-3 c^2 d (C-2 D x)\right )+b c \left (-B c d^2 (2 c+d x)+A d^3 (4 c+3 d x)+c^2 \left (2 c^2 D-C d^2 x+3 c d D x\right )\right )\right )}{\left (b c^2+a d^2\right )^2 (c+d x)^2}-\frac {2 \left (A b d^3 \left (-2 b c^2+a d^2\right )+2 b^2 c^5 D-2 a^2 d^4 (C d-3 c D)+a b c d^2 \left (c C d-3 B d^2+5 c^2 D\right )\right ) \arctan \left (\frac {\sqrt {b} (c+d x)-d \sqrt {a+b x^2}}{\sqrt {-b c^2-a d^2}}\right )}{\left (-b c^2-a d^2\right )^{5/2}}+\frac {2 D \log \left (-\sqrt {b} x+\sqrt {a+b x^2}\right )}{\sqrt {b}}}{2 d^3} \] Input:

Integrate[(A + B*x + C*x^2 + D*x^3)/((c + d*x)^3*Sqrt[a + b*x^2]),x]
 

Output:

-1/2*((d*Sqrt[a + b*x^2]*(a*d^2*(5*c^3*D + d^3*(A + 2*B*x) + c*d^2*(B - 4* 
C*x) - 3*c^2*d*(C - 2*D*x)) + b*c*(-(B*c*d^2*(2*c + d*x)) + A*d^3*(4*c + 3 
*d*x) + c^2*(2*c^2*D - C*d^2*x + 3*c*d*D*x))))/((b*c^2 + a*d^2)^2*(c + d*x 
)^2) - (2*(A*b*d^3*(-2*b*c^2 + a*d^2) + 2*b^2*c^5*D - 2*a^2*d^4*(C*d - 3*c 
*D) + a*b*c*d^2*(c*C*d - 3*B*d^2 + 5*c^2*D))*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)^(5/2) + (2*D 
*Log[-(Sqrt[b]*x) + Sqrt[a + b*x^2]])/Sqrt[b])/d^3
 

Rubi [A] (verified)

Time = 0.94 (sec) , antiderivative size = 358, normalized size of antiderivative = 1.14, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.294, Rules used = {2182, 25, 2182, 25, 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.

\(\displaystyle \int \frac {A+B x+C x^2+D x^3}{\sqrt {a+b x^2} (c+d x)^3} \, dx\)

\(\Big \downarrow \) 2182

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 2182

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 719

\(\displaystyle \frac {\frac {\frac {\left (2 a^2 d^4 (C d-3 c D)+A b d^3 \left (2 b c^2-a d^2\right )-a b c d^2 \left (-3 B d^2+5 c^2 D+c C d\right )-2 b^2 c^5 D\right ) \int \frac {1}{(c+d x) \sqrt {b x^2+a}}dx}{d}+\frac {2 D \left (a d^2+b c^2\right )^2 \int \frac {1}{\sqrt {b x^2+a}}dx}{d}}{d^2 \left (a d^2+b c^2\right )}+\frac {\sqrt {a+b x^2} \left (2 a d^2 \left (-B d^2-3 c^2 D+2 c C d\right )+b c \left (-3 A d^3+B c d^2-3 c^3 D+c^2 C d\right )\right )}{d^2 (c+d x) \left (a d^2+b c^2\right )}}{2 \left (a d^2+b c^2\right )}-\frac {\sqrt {a+b x^2} \left (A d^3-B c d^2+c^3 (-D)+c^2 C d\right )}{2 d^2 (c+d x)^2 \left (a d^2+b c^2\right )}\)

\(\Big \downarrow \) 224

\(\displaystyle \frac {\frac {\frac {\left (2 a^2 d^4 (C d-3 c D)+A b d^3 \left (2 b c^2-a d^2\right )-a b c d^2 \left (-3 B d^2+5 c^2 D+c C d\right )-2 b^2 c^5 D\right ) \int \frac {1}{(c+d x) \sqrt {b x^2+a}}dx}{d}+\frac {2 D \left (a d^2+b c^2\right )^2 \int \frac {1}{1-\frac {b x^2}{b x^2+a}}d\frac {x}{\sqrt {b x^2+a}}}{d}}{d^2 \left (a d^2+b c^2\right )}+\frac {\sqrt {a+b x^2} \left (2 a d^2 \left (-B d^2-3 c^2 D+2 c C d\right )+b c \left (-3 A d^3+B c d^2-3 c^3 D+c^2 C d\right )\right )}{d^2 (c+d x) \left (a d^2+b c^2\right )}}{2 \left (a d^2+b c^2\right )}-\frac {\sqrt {a+b x^2} \left (A d^3-B c d^2+c^3 (-D)+c^2 C d\right )}{2 d^2 (c+d x)^2 \left (a d^2+b c^2\right )}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {\frac {\frac {\left (2 a^2 d^4 (C d-3 c D)+A b d^3 \left (2 b c^2-a d^2\right )-a b c d^2 \left (-3 B d^2+5 c^2 D+c C d\right )-2 b^2 c^5 D\right ) \int \frac {1}{(c+d x) \sqrt {b x^2+a}}dx}{d}+\frac {2 D \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right ) \left (a d^2+b c^2\right )^2}{\sqrt {b} d}}{d^2 \left (a d^2+b c^2\right )}+\frac {\sqrt {a+b x^2} \left (2 a d^2 \left (-B d^2-3 c^2 D+2 c C d\right )+b c \left (-3 A d^3+B c d^2-3 c^3 D+c^2 C d\right )\right )}{d^2 (c+d x) \left (a d^2+b c^2\right )}}{2 \left (a d^2+b c^2\right )}-\frac {\sqrt {a+b x^2} \left (A d^3-B c d^2+c^3 (-D)+c^2 C d\right )}{2 d^2 (c+d x)^2 \left (a d^2+b c^2\right )}\)

\(\Big \downarrow \) 488

\(\displaystyle \frac {\frac {\frac {2 D \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right ) \left (a d^2+b c^2\right )^2}{\sqrt {b} d}-\frac {\left (2 a^2 d^4 (C d-3 c D)+A b d^3 \left (2 b c^2-a d^2\right )-a b c d^2 \left (-3 B d^2+5 c^2 D+c C d\right )-2 b^2 c^5 D\right ) \int \frac {1}{b c^2+a d^2-\frac {(a d-b c x)^2}{b x^2+a}}d\frac {a d-b c x}{\sqrt {b x^2+a}}}{d}}{d^2 \left (a d^2+b c^2\right )}+\frac {\sqrt {a+b x^2} \left (2 a d^2 \left (-B d^2-3 c^2 D+2 c C d\right )+b c \left (-3 A d^3+B c d^2-3 c^3 D+c^2 C d\right )\right )}{d^2 (c+d x) \left (a d^2+b c^2\right )}}{2 \left (a d^2+b c^2\right )}-\frac {\sqrt {a+b x^2} \left (A d^3-B c d^2+c^3 (-D)+c^2 C d\right )}{2 d^2 (c+d x)^2 \left (a d^2+b c^2\right )}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {\frac {\frac {2 D \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right ) \left (a d^2+b c^2\right )^2}{\sqrt {b} d}-\frac {\text {arctanh}\left (\frac {a d-b c x}{\sqrt {a+b x^2} \sqrt {a d^2+b c^2}}\right ) \left (2 a^2 d^4 (C d-3 c D)+A b d^3 \left (2 b c^2-a d^2\right )-a b c d^2 \left (-3 B d^2+5 c^2 D+c C d\right )-2 b^2 c^5 D\right )}{d \sqrt {a d^2+b c^2}}}{d^2 \left (a d^2+b c^2\right )}+\frac {\sqrt {a+b x^2} \left (2 a d^2 \left (-B d^2-3 c^2 D+2 c C d\right )+b c \left (-3 A d^3+B c d^2-3 c^3 D+c^2 C d\right )\right )}{d^2 (c+d x) \left (a d^2+b c^2\right )}}{2 \left (a d^2+b c^2\right )}-\frac {\sqrt {a+b x^2} \left (A d^3-B c d^2+c^3 (-D)+c^2 C d\right )}{2 d^2 (c+d x)^2 \left (a d^2+b c^2\right )}\)

Input:

Int[(A + B*x + C*x^2 + D*x^3)/((c + d*x)^3*Sqrt[a + b*x^2]),x]
 

Output:

-1/2*((c^2*C*d - B*c*d^2 + A*d^3 - c^3*D)*Sqrt[a + b*x^2])/(d^2*(b*c^2 + a 
*d^2)*(c + d*x)^2) + (((2*a*d^2*(2*c*C*d - B*d^2 - 3*c^2*D) + b*c*(c^2*C*d 
 + B*c*d^2 - 3*A*d^3 - 3*c^3*D))*Sqrt[a + b*x^2])/(d^2*(b*c^2 + a*d^2)*(c 
+ d*x)) + ((2*(b*c^2 + a*d^2)^2*D*ArcTanh[(Sqrt[b]*x)/Sqrt[a + b*x^2]])/(S 
qrt[b]*d) - ((A*b*d^3*(2*b*c^2 - a*d^2) - 2*b^2*c^5*D + 2*a^2*d^4*(C*d - 3 
*c*D) - a*b*c*d^2*(c*C*d - 3*B*d^2 + 5*c^2*D))*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*(b*c^2 + 
a*d^2)))/(2*(b*c^2 + a*d^2))
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 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 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]
 

rule 2182

Int[(Pq_)*((d_) + (e_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> 
 With[{Qx = PolynomialQuotient[Pq, d + e*x, x], R = PolynomialRemainder[Pq, 
 d + e*x, x]}, Simp[e*R*(d + e*x)^(m + 1)*((a + b*x^2)^(p + 1)/((m + 1)*(b* 
d^2 + a*e^2))), x] + Simp[1/((m + 1)*(b*d^2 + a*e^2))   Int[(d + e*x)^(m + 
1)*(a + b*x^2)^p*ExpandToSum[(m + 1)*(b*d^2 + a*e^2)*Qx + b*d*R*(m + 1) - b 
*e*R*(m + 2*p + 3)*x, x], x], x]] /; FreeQ[{a, b, d, e, p}, x] && PolyQ[Pq, 
 x] && NeQ[b*d^2 + a*e^2, 0] && LtQ[m, -1]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(858\) vs. \(2(291)=582\).

Time = 1.46 (sec) , antiderivative size = 859, normalized size of antiderivative = 2.74

method	result	size
default	\(\frac {D \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{d^{3} \sqrt {b}}-\frac {\left (C d -3 D c \right ) \ln \left (\frac {\frac {2 a \,d^{2}+2 b \,c^{2}}{d^{2}}-\frac {2 b c \left (x +\frac {c}{d}\right )}{d}+2 \sqrt {\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}\, \sqrt {b \left (x +\frac {c}{d}\right )^{2}-\frac {2 b c \left (x +\frac {c}{d}\right )}{d}+\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}}{x +\frac {c}{d}}\right )}{d^{4} \sqrt {\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}}+\frac {\left (B \,d^{2}-2 C c d +3 D c^{2}\right ) \left (-\frac {d^{2} \sqrt {b \left (x +\frac {c}{d}\right )^{2}-\frac {2 b c \left (x +\frac {c}{d}\right )}{d}+\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}}{\left (a \,d^{2}+b \,c^{2}\right ) \left (x +\frac {c}{d}\right )}-\frac {b c d \ln \left (\frac {\frac {2 a \,d^{2}+2 b \,c^{2}}{d^{2}}-\frac {2 b c \left (x +\frac {c}{d}\right )}{d}+2 \sqrt {\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}\, \sqrt {b \left (x +\frac {c}{d}\right )^{2}-\frac {2 b c \left (x +\frac {c}{d}\right )}{d}+\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}}{x +\frac {c}{d}}\right )}{\left (a \,d^{2}+b \,c^{2}\right ) \sqrt {\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}}\right )}{d^{5}}+\frac {\left (A \,d^{3}-B c \,d^{2}+C \,c^{2} d -D c^{3}\right ) \left (-\frac {d^{2} \sqrt {b \left (x +\frac {c}{d}\right )^{2}-\frac {2 b c \left (x +\frac {c}{d}\right )}{d}+\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}}{2 \left (a \,d^{2}+b \,c^{2}\right ) \left (x +\frac {c}{d}\right )^{2}}+\frac {3 b c d \left (-\frac {d^{2} \sqrt {b \left (x +\frac {c}{d}\right )^{2}-\frac {2 b c \left (x +\frac {c}{d}\right )}{d}+\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}}{\left (a \,d^{2}+b \,c^{2}\right ) \left (x +\frac {c}{d}\right )}-\frac {b c d \ln \left (\frac {\frac {2 a \,d^{2}+2 b \,c^{2}}{d^{2}}-\frac {2 b c \left (x +\frac {c}{d}\right )}{d}+2 \sqrt {\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}\, \sqrt {b \left (x +\frac {c}{d}\right )^{2}-\frac {2 b c \left (x +\frac {c}{d}\right )}{d}+\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}}{x +\frac {c}{d}}\right )}{\left (a \,d^{2}+b \,c^{2}\right ) \sqrt {\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}}\right )}{2 \left (a \,d^{2}+b \,c^{2}\right )}+\frac {b \,d^{2} \ln \left (\frac {\frac {2 a \,d^{2}+2 b \,c^{2}}{d^{2}}-\frac {2 b c \left (x +\frac {c}{d}\right )}{d}+2 \sqrt {\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}\, \sqrt {b \left (x +\frac {c}{d}\right )^{2}-\frac {2 b c \left (x +\frac {c}{d}\right )}{d}+\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}}{x +\frac {c}{d}}\right )}{2 \left (a \,d^{2}+b \,c^{2}\right ) \sqrt {\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}}\right )}{d^{6}}\)	\(859\)

Input:

int((D*x^3+C*x^2+B*x+A)/(d*x+c)^3/(b*x^2+a)^(1/2),x,method=_RETURNVERBOSE)
 

Output:

D/d^3*ln(b^(1/2)*x+(b*x^2+a)^(1/2))/b^(1/2)-1/d^4*(C*d-3*D*c)/((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/ 
d^5*(B*d^2-2*C*c*d+3*D*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/d^6 
*(A*d^3-B*c*d^2+C*c^2*d-D*c^3)*(-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)))
 

Fricas [F(-1)]

Timed out. \[ \int \frac {A+B x+C x^2+D x^3}{(c+d x)^3 \sqrt {a+b x^2}} \, dx=\text {Timed out} \] Input:

integrate((D*x^3+C*x^2+B*x+A)/(d*x+c)^3/(b*x^2+a)^(1/2),x, algorithm="fric 
as")
 

Output:

Timed out
 

Sympy [F]

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

integrate((D*x**3+C*x**2+B*x+A)/(d*x+c)**3/(b*x**2+a)**(1/2),x)
 

Output:

Integral((A + B*x + C*x**2 + D*x**3)/(sqrt(a + b*x**2)*(c + d*x)**3), x)
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1281 vs. \(2 (294) = 588\).

Time = 0.10 (sec) , antiderivative size = 1281, normalized size of antiderivative = 4.09 \[ \int \frac {A+B x+C x^2+D x^3}{(c+d x)^3 \sqrt {a+b x^2}} \, dx=\text {Too large to display} \] Input:

integrate((D*x^3+C*x^2+B*x+A)/(d*x+c)^3/(b*x^2+a)^(1/2),x, algorithm="maxi 
ma")
 

Output:

3/2*sqrt(b*x^2 + a)*D*b*c^4/(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) - 3/2*sqrt(b*x^2 + a)*C*b*c^3/(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) + 3/2*sqrt(b*x^2 + a)*B*b*c^2/(b^2*c^4*d*x + 2*a*b*c^2*d^3*x + a 
^2*d^5*x + b^2*c^5 + 2*a*b*c^3*d^2 + a^2*c*d^4) + 1/2*sqrt(b*x^2 + a)*D*c^ 
3/(b*c^2*d^4*x^2 + a*d^6*x^2 + 2*b*c^3*d^3*x + 2*a*c*d^5*x + b*c^4*d^2 + a 
*c^2*d^4) - 3/2*sqrt(b*x^2 + a)*A*b*c/(b^2*c^4*x + 2*a*b*c^2*d^2*x + a^2*d 
^4*x + b^2*c^5/d + 2*a*b*c^3*d + a^2*c*d^3) - 1/2*sqrt(b*x^2 + a)*C*c^2/(b 
*c^2*d^3*x^2 + a*d^5*x^2 + 2*b*c^3*d^2*x + 2*a*c*d^4*x + b*c^4*d + a*c^2*d 
^3) - 3*sqrt(b*x^2 + a)*D*c^2/(b*c^2*d^3*x + a*d^5*x + b*c^3*d^2 + a*c*d^4 
) + 1/2*sqrt(b*x^2 + a)*B*c/(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) + 2*sqrt(b*x^2 + a)*C*c/(b*c^2*d^2*x + a*d^4 
*x + b*c^3*d + a*c*d^3) - 1/2*sqrt(b*x^2 + a)*A/(b*c^2*d*x^2 + a*d^3*x^2 + 
 2*b*c^3*x + 2*a*c*d^2*x + b*c^4/d + a*c^2*d) - sqrt(b*x^2 + a)*B/(b*c^2*d 
*x + a*d^3*x + b*c^3 + a*c*d^2) + D*arcsinh(b*x/sqrt(a*b))/(sqrt(b)*d^3) - 
 3/2*D*b^2*c^5*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)^(5/2)*d^8) + 3/2*C*b^2*c^4*arcsinh(b*c*x/(sqr 
t(a*b)*abs(d*x + c)) - a*d/(sqrt(a*b)*abs(d*x + c)))/((a + b*c^2/d^2)^(5/2 
)*d^7) + 7/2*D*b*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^6) - 3/2*B*b^2*c^3*arcsinh(b...
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1159 vs. \(2 (294) = 588\).

Time = 0.27 (sec) , antiderivative size = 1159, normalized size of antiderivative = 3.70 \[ \int \frac {A+B x+C x^2+D x^3}{(c+d x)^3 \sqrt {a+b x^2}} \, dx=\text {Too large to display} \] Input:

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

Output:

-(2*D*b^2*c^5 + 5*D*a*b*c^3*d^2 + C*a*b*c^2*d^3 - 2*A*b^2*c^2*d^3 + 6*D*a^ 
2*c*d^4 - 3*B*a*b*c*d^4 - 2*C*a^2*d^5 + A*a*b*d^5)*arctan(-((sqrt(b)*x - s 
qrt(b*x^2 + a))*d + sqrt(b)*c)/sqrt(-b*c^2 - a*d^2))/((b^2*c^4*d^3 + 2*a*b 
*c^2*d^5 + a^2*d^7)*sqrt(-b*c^2 - a*d^2)) - D*log(abs(-sqrt(b)*x + sqrt(b* 
x^2 + a)))/(sqrt(b)*d^3) - (4*(sqrt(b)*x - sqrt(b*x^2 + a))^3*D*b^2*c^5*d 
- 2*(sqrt(b)*x - sqrt(b*x^2 + a))^3*C*b^2*c^4*d^2 + 7*(sqrt(b)*x - sqrt(b* 
x^2 + a))^3*D*a*b*c^3*d^3 - 5*(sqrt(b)*x - sqrt(b*x^2 + a))^3*C*a*b*c^2*d^ 
4 + 2*(sqrt(b)*x - sqrt(b*x^2 + a))^3*A*b^2*c^2*d^4 + 3*(sqrt(b)*x - sqrt( 
b*x^2 + a))^3*B*a*b*c*d^5 - (sqrt(b)*x - sqrt(b*x^2 + a))^3*A*a*b*d^6 + 6* 
(sqrt(b)*x - sqrt(b*x^2 + a))^2*D*b^(5/2)*c^6 - 2*(sqrt(b)*x - sqrt(b*x^2 
+ a))^2*C*b^(5/2)*c^5*d + 9*(sqrt(b)*x - sqrt(b*x^2 + a))^2*D*a*b^(3/2)*c^ 
4*d^2 - 2*(sqrt(b)*x - sqrt(b*x^2 + a))^2*B*b^(5/2)*c^4*d^2 - 7*(sqrt(b)*x 
 - sqrt(b*x^2 + a))^2*C*a*b^(3/2)*c^3*d^3 + 6*(sqrt(b)*x - sqrt(b*x^2 + a) 
)^2*A*b^(5/2)*c^3*d^3 - 6*(sqrt(b)*x - sqrt(b*x^2 + a))^2*D*a^2*sqrt(b)*c^ 
2*d^4 + 5*(sqrt(b)*x - sqrt(b*x^2 + a))^2*B*a*b^(3/2)*c^2*d^4 + 4*(sqrt(b) 
*x - sqrt(b*x^2 + a))^2*C*a^2*sqrt(b)*c*d^5 - 3*(sqrt(b)*x - sqrt(b*x^2 + 
a))^2*A*a*b^(3/2)*c*d^5 - 2*(sqrt(b)*x - sqrt(b*x^2 + a))^2*B*a^2*sqrt(b)* 
d^6 - 8*(sqrt(b)*x - sqrt(b*x^2 + a))*D*a*b^2*c^5*d + 2*(sqrt(b)*x - sqrt( 
b*x^2 + a))*C*a*b^2*c^4*d^2 - 17*(sqrt(b)*x - sqrt(b*x^2 + a))*D*a^2*b*c^3 
*d^3 + 4*(sqrt(b)*x - sqrt(b*x^2 + a))*B*a*b^2*c^3*d^3 + 11*(sqrt(b)*x ...
 

Mupad [F(-1)]

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

int((A + B*x + C*x^2 + x^3*D)/((a + b*x^2)^(1/2)*(c + d*x)^3),x)
 

Output:

int((A + B*x + C*x^2 + x^3*D)/((a + b*x^2)^(1/2)*(c + d*x)^3), x)
 

Reduce [B] (verification not implemented)

Time = 0.18 (sec) , antiderivative size = 2742, normalized size of antiderivative = 8.76 \[ \int \frac {A+B x+C x^2+D x^3}{(c+d x)^3 \sqrt {a+b x^2}} \, dx =\text {Too large to display} \] Input:

int((D*x^3+C*x^2+B*x+A)/(d*x+c)^3/(b*x^2+a)^(1/2),x)
 

Output:

(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**2*c**2*d**4 + 2*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**2*c*d**5*x + 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**2*d**6*x**2 + 4*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 + 8*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 + 4*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 - 2*sqrt(a*d**2 + b*c**2)*log( - sq 
rt(a + b*x**2)*sqrt(a*d**2 + b*c**2) - a*d + b*c*x)*a*b**3*c**4*d**2 - 4*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**3*c**3*d**3*x - 3*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**3*c**3*d**3 - 2*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* 
*3*c**2*d**4*x**2 - 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)*a*b**3*c**2*d**4*x - 3*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**3*c*d* 
*5*x**2 + 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)*a*b**2*c**5*d**2 + 12*sqrt(a*d**2 + b*c**2)*log( - sq 
rt(a + b*x**2)*sqrt(a*d**2 + b*c**2) - a*d + b*c*x)*a*b**2*c**4*d**3*x ...