\(\int \frac {(c+d x) (A+B x+C x^2+D x^3)}{(a+b x^2)^{3/2}} \, dx\) [110]

Optimal result

Integrand size = 32, antiderivative size = 167 \[ \int \frac {(c+d x) \left (A+B x+C x^2+D x^3\right )}{\left (a+b x^2\right )^{3/2}} \, dx=-\frac {b B c+A b d-a C d-a c D}{b^2 \sqrt {a+b x^2}}+\frac {\left (A b^2 c-a (b (c C+B d)-a d D)\right ) x}{a b^2 \sqrt {a+b x^2}}+\frac {(C d+c D) \sqrt {a+b x^2}}{b^2}+\frac {d D x \sqrt {a+b x^2}}{2 b^2}+\frac {(2 b (c C+B d)-3 a d D) \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{2 b^{5/2}} \] Output:

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

Mathematica [A] (verified)

Time = 1.03 (sec) , antiderivative size = 134, normalized size of antiderivative = 0.80 \[ \int \frac {(c+d x) \left (A+B x+C x^2+D x^3\right )}{\left (a+b x^2\right )^{3/2}} \, dx=\frac {\frac {\sqrt {b} \left (2 A b^2 c x+a^2 (4 C d+4 c D+3 d D x)+a b \left (-2 A d-2 B (c+d x)+x \left (-2 c C+2 C d x+2 c D x+d D x^2\right )\right )\right )}{a \sqrt {a+b x^2}}+(-2 b (c C+B d)+3 a d D) \log \left (-\sqrt {b} x+\sqrt {a+b x^2}\right )}{2 b^{5/2}} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.44 (sec) , antiderivative size = 156, normalized size of antiderivative = 0.93, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {2176, 25, 2346, 455, 224, 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.

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

\(\Big \downarrow \) 2176

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 2346

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

\(\Big \downarrow \) 455

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

\(\Big \downarrow \) 224

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

\(\Big \downarrow \) 219

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

Input:

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

Output:

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

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], 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 455

Int[((c_) + (d_.)*(x_))*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[d*(( 
a + b*x^2)^(p + 1)/(2*b*(p + 1))), x] + Simp[c   Int[(a + b*x^2)^p, x], x] 
/; FreeQ[{a, b, c, d, p}, x] &&  !LeQ[p, -1]
 

rule 2176

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

rule 2346

Int[(Pq_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{q = Expon[Pq, x], 
e = Coeff[Pq, x, Expon[Pq, x]]}, Simp[e*x^(q - 1)*((a + b*x^2)^(p + 1)/(b*( 
q + 2*p + 1))), x] + Simp[1/(b*(q + 2*p + 1))   Int[(a + b*x^2)^p*ExpandToS 
um[b*(q + 2*p + 1)*Pq - a*e*(q - 1)*x^(q - 2) - b*e*(q + 2*p + 1)*x^q, x], 
x], x]] /; FreeQ[{a, b, p}, x] && PolyQ[Pq, x] &&  !LeQ[p, -1]
 

Maple [A] (verified)

Time = 1.03 (sec) , antiderivative size = 186, normalized size of antiderivative = 1.11

method	result	size
default	\(\frac {A c x}{\sqrt {b \,x^{2}+a}\, a}-\frac {A d +B c}{b \sqrt {b \,x^{2}+a}}+\left (B d +C c \right ) \left (-\frac {x}{b \sqrt {b \,x^{2}+a}}+\frac {\ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{b^{\frac {3}{2}}}\right )+\left (C d +D c \right ) \left (\frac {x^{2}}{b \sqrt {b \,x^{2}+a}}+\frac {2 a}{b^{2} \sqrt {b \,x^{2}+a}}\right )+d D \left (\frac {x^{3}}{2 b \sqrt {b \,x^{2}+a}}-\frac {3 a \left (-\frac {x}{b \sqrt {b \,x^{2}+a}}+\frac {\ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{b^{\frac {3}{2}}}\right )}{2 b}\right )\)	\(186\)

Input:

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

Output:

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

Fricas [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 442, normalized size of antiderivative = 2.65 \[ \int \frac {(c+d x) \left (A+B x+C x^2+D x^3\right )}{\left (a+b x^2\right )^{3/2}} \, dx=\left [\frac {{\left (2 \, C a^{2} b c + {\left (2 \, C a b^{2} c - {\left (3 \, D a^{2} b - 2 \, B a b^{2}\right )} d\right )} x^{2} - {\left (3 \, D a^{3} - 2 \, B a^{2} b\right )} d\right )} \sqrt {b} \log \left (-2 \, b x^{2} - 2 \, \sqrt {b x^{2} + a} \sqrt {b} x - a\right ) + 2 \, {\left (D a b^{2} d x^{3} + 2 \, {\left (D a b^{2} c + C a b^{2} d\right )} x^{2} + 2 \, {\left (2 \, D a^{2} b - B a b^{2}\right )} c + 2 \, {\left (2 \, C a^{2} b - A a b^{2}\right )} d - {\left (2 \, {\left (C a b^{2} - A b^{3}\right )} c - {\left (3 \, D a^{2} b - 2 \, B a b^{2}\right )} d\right )} x\right )} \sqrt {b x^{2} + a}}{4 \, {\left (a b^{4} x^{2} + a^{2} b^{3}\right )}}, -\frac {{\left (2 \, C a^{2} b c + {\left (2 \, C a b^{2} c - {\left (3 \, D a^{2} b - 2 \, B a b^{2}\right )} d\right )} x^{2} - {\left (3 \, D a^{3} - 2 \, B a^{2} b\right )} d\right )} \sqrt {-b} \arctan \left (\frac {\sqrt {-b} x}{\sqrt {b x^{2} + a}}\right ) - {\left (D a b^{2} d x^{3} + 2 \, {\left (D a b^{2} c + C a b^{2} d\right )} x^{2} + 2 \, {\left (2 \, D a^{2} b - B a b^{2}\right )} c + 2 \, {\left (2 \, C a^{2} b - A a b^{2}\right )} d - {\left (2 \, {\left (C a b^{2} - A b^{3}\right )} c - {\left (3 \, D a^{2} b - 2 \, B a b^{2}\right )} d\right )} x\right )} \sqrt {b x^{2} + a}}{2 \, {\left (a b^{4} x^{2} + a^{2} b^{3}\right )}}\right ] \] Input:

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

Output:

[1/4*((2*C*a^2*b*c + (2*C*a*b^2*c - (3*D*a^2*b - 2*B*a*b^2)*d)*x^2 - (3*D* 
a^3 - 2*B*a^2*b)*d)*sqrt(b)*log(-2*b*x^2 - 2*sqrt(b*x^2 + a)*sqrt(b)*x - a 
) + 2*(D*a*b^2*d*x^3 + 2*(D*a*b^2*c + C*a*b^2*d)*x^2 + 2*(2*D*a^2*b - B*a* 
b^2)*c + 2*(2*C*a^2*b - A*a*b^2)*d - (2*(C*a*b^2 - A*b^3)*c - (3*D*a^2*b - 
 2*B*a*b^2)*d)*x)*sqrt(b*x^2 + a))/(a*b^4*x^2 + a^2*b^3), -1/2*((2*C*a^2*b 
*c + (2*C*a*b^2*c - (3*D*a^2*b - 2*B*a*b^2)*d)*x^2 - (3*D*a^3 - 2*B*a^2*b) 
*d)*sqrt(-b)*arctan(sqrt(-b)*x/sqrt(b*x^2 + a)) - (D*a*b^2*d*x^3 + 2*(D*a* 
b^2*c + C*a*b^2*d)*x^2 + 2*(2*D*a^2*b - B*a*b^2)*c + 2*(2*C*a^2*b - A*a*b^ 
2)*d - (2*(C*a*b^2 - A*b^3)*c - (3*D*a^2*b - 2*B*a*b^2)*d)*x)*sqrt(b*x^2 + 
 a))/(a*b^4*x^2 + a^2*b^3)]
 

Sympy [A] (verification not implemented)

Time = 9.72 (sec) , antiderivative size = 332, normalized size of antiderivative = 1.99 \[ \int \frac {(c+d x) \left (A+B x+C x^2+D x^3\right )}{\left (a+b x^2\right )^{3/2}} \, dx=A d \left (\begin {cases} - \frac {1}{b \sqrt {a + b x^{2}}} & \text {for}\: b \neq 0 \\\frac {x^{2}}{2 a^{\frac {3}{2}}} & \text {otherwise} \end {cases}\right ) + \frac {A c x}{a^{\frac {3}{2}} \sqrt {1 + \frac {b x^{2}}{a}}} + B c \left (\begin {cases} - \frac {1}{b \sqrt {a + b x^{2}}} & \text {for}\: b \neq 0 \\\frac {x^{2}}{2 a^{\frac {3}{2}}} & \text {otherwise} \end {cases}\right ) + B d \left (\frac {\operatorname {asinh}{\left (\frac {\sqrt {b} x}{\sqrt {a}} \right )}}{b^{\frac {3}{2}}} - \frac {x}{\sqrt {a} b \sqrt {1 + \frac {b x^{2}}{a}}}\right ) + C c \left (\frac {\operatorname {asinh}{\left (\frac {\sqrt {b} x}{\sqrt {a}} \right )}}{b^{\frac {3}{2}}} - \frac {x}{\sqrt {a} b \sqrt {1 + \frac {b x^{2}}{a}}}\right ) + C d \left (\begin {cases} \frac {2 a}{b^{2} \sqrt {a + b x^{2}}} + \frac {x^{2}}{b \sqrt {a + b x^{2}}} & \text {for}\: b \neq 0 \\\frac {x^{4}}{4 a^{\frac {3}{2}}} & \text {otherwise} \end {cases}\right ) + D c \left (\begin {cases} \frac {2 a}{b^{2} \sqrt {a + b x^{2}}} + \frac {x^{2}}{b \sqrt {a + b x^{2}}} & \text {for}\: b \neq 0 \\\frac {x^{4}}{4 a^{\frac {3}{2}}} & \text {otherwise} \end {cases}\right ) + D d \left (\frac {3 \sqrt {a} x}{2 b^{2} \sqrt {1 + \frac {b x^{2}}{a}}} - \frac {3 a \operatorname {asinh}{\left (\frac {\sqrt {b} x}{\sqrt {a}} \right )}}{2 b^{\frac {5}{2}}} + \frac {x^{3}}{2 \sqrt {a} b \sqrt {1 + \frac {b x^{2}}{a}}}\right ) \] Input:

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

Output:

A*d*Piecewise((-1/(b*sqrt(a + b*x**2)), Ne(b, 0)), (x**2/(2*a**(3/2)), Tru 
e)) + A*c*x/(a**(3/2)*sqrt(1 + b*x**2/a)) + B*c*Piecewise((-1/(b*sqrt(a + 
b*x**2)), Ne(b, 0)), (x**2/(2*a**(3/2)), True)) + B*d*(asinh(sqrt(b)*x/sqr 
t(a))/b**(3/2) - x/(sqrt(a)*b*sqrt(1 + b*x**2/a))) + C*c*(asinh(sqrt(b)*x/ 
sqrt(a))/b**(3/2) - x/(sqrt(a)*b*sqrt(1 + b*x**2/a))) + C*d*Piecewise((2*a 
/(b**2*sqrt(a + b*x**2)) + x**2/(b*sqrt(a + b*x**2)), Ne(b, 0)), (x**4/(4* 
a**(3/2)), True)) + D*c*Piecewise((2*a/(b**2*sqrt(a + b*x**2)) + x**2/(b*s 
qrt(a + b*x**2)), Ne(b, 0)), (x**4/(4*a**(3/2)), True)) + D*d*(3*sqrt(a)*x 
/(2*b**2*sqrt(1 + b*x**2/a)) - 3*a*asinh(sqrt(b)*x/sqrt(a))/(2*b**(5/2)) + 
 x**3/(2*sqrt(a)*b*sqrt(1 + b*x**2/a)))
 

Maxima [A] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 190, normalized size of antiderivative = 1.14 \[ \int \frac {(c+d x) \left (A+B x+C x^2+D x^3\right )}{\left (a+b x^2\right )^{3/2}} \, dx=\frac {D d x^{3}}{2 \, \sqrt {b x^{2} + a} b} + \frac {A c x}{\sqrt {b x^{2} + a} a} + \frac {3 \, D a d x}{2 \, \sqrt {b x^{2} + a} b^{2}} + \frac {{\left (D c + C d\right )} x^{2}}{\sqrt {b x^{2} + a} b} - \frac {3 \, D a d \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{2 \, b^{\frac {5}{2}}} - \frac {B c}{\sqrt {b x^{2} + a} b} - \frac {A d}{\sqrt {b x^{2} + a} b} - \frac {{\left (C c + B d\right )} x}{\sqrt {b x^{2} + a} b} + \frac {{\left (C c + B d\right )} \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{b^{\frac {3}{2}}} + \frac {2 \, {\left (D c + C d\right )} a}{\sqrt {b x^{2} + a} b^{2}} \] Input:

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

Output:

1/2*D*d*x^3/(sqrt(b*x^2 + a)*b) + A*c*x/(sqrt(b*x^2 + a)*a) + 3/2*D*a*d*x/ 
(sqrt(b*x^2 + a)*b^2) + (D*c + C*d)*x^2/(sqrt(b*x^2 + a)*b) - 3/2*D*a*d*ar 
csinh(b*x/sqrt(a*b))/b^(5/2) - B*c/(sqrt(b*x^2 + a)*b) - A*d/(sqrt(b*x^2 + 
 a)*b) - (C*c + B*d)*x/(sqrt(b*x^2 + a)*b) + (C*c + B*d)*arcsinh(b*x/sqrt( 
a*b))/b^(3/2) + 2*(D*c + C*d)*a/(sqrt(b*x^2 + a)*b^2)
 

Giac [A] (verification not implemented)

Time = 0.23 (sec) , antiderivative size = 175, normalized size of antiderivative = 1.05 \[ \int \frac {(c+d x) \left (A+B x+C x^2+D x^3\right )}{\left (a+b x^2\right )^{3/2}} \, dx=\frac {{\left ({\left (\frac {D d x}{b} + \frac {2 \, {\left (D a b^{3} c + C a b^{3} d\right )}}{a b^{4}}\right )} x - \frac {2 \, C a b^{3} c - 2 \, A b^{4} c - 3 \, D a^{2} b^{2} d + 2 \, B a b^{3} d}{a b^{4}}\right )} x + \frac {2 \, {\left (2 \, D a^{2} b^{2} c - B a b^{3} c + 2 \, C a^{2} b^{2} d - A a b^{3} d\right )}}{a b^{4}}}{2 \, \sqrt {b x^{2} + a}} - \frac {{\left (2 \, C b c - 3 \, D a d + 2 \, B b d\right )} \log \left ({\left | -\sqrt {b} x + \sqrt {b x^{2} + a} \right |}\right )}{2 \, b^{\frac {5}{2}}} \] Input:

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.20 (sec) , antiderivative size = 419, normalized size of antiderivative = 2.51 \[ \int \frac {(c+d x) \left (A+B x+C x^2+D x^3\right )}{\left (a+b x^2\right )^{3/2}} \, dx=\frac {-2 \sqrt {b \,x^{2}+a}\, a \,b^{2} d +8 \sqrt {b \,x^{2}+a}\, a b c d +3 \sqrt {b \,x^{2}+a}\, a b \,d^{2} x +2 \sqrt {b \,x^{2}+a}\, b^{3} c x -2 \sqrt {b \,x^{2}+a}\, b^{3} c -2 \sqrt {b \,x^{2}+a}\, b^{3} d x -2 \sqrt {b \,x^{2}+a}\, b^{2} c^{2} x +4 \sqrt {b \,x^{2}+a}\, b^{2} c d \,x^{2}+\sqrt {b \,x^{2}+a}\, b^{2} d^{2} x^{3}-3 \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}+\sqrt {b}\, x}{\sqrt {a}}\right ) a^{2} d^{2}+2 \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}+\sqrt {b}\, x}{\sqrt {a}}\right ) a \,b^{2} d +2 \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}+\sqrt {b}\, x}{\sqrt {a}}\right ) a b \,c^{2}-3 \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}+\sqrt {b}\, x}{\sqrt {a}}\right ) a b \,d^{2} x^{2}+2 \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}+\sqrt {b}\, x}{\sqrt {a}}\right ) b^{3} d \,x^{2}+2 \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}+\sqrt {b}\, x}{\sqrt {a}}\right ) b^{2} c^{2} x^{2}+2 \sqrt {b}\, a^{2} d^{2}+2 \sqrt {b}\, a \,b^{2} c -2 \sqrt {b}\, a \,b^{2} d -2 \sqrt {b}\, a b \,c^{2}+2 \sqrt {b}\, a b \,d^{2} x^{2}+2 \sqrt {b}\, b^{3} c \,x^{2}-2 \sqrt {b}\, b^{3} d \,x^{2}-2 \sqrt {b}\, b^{2} c^{2} x^{2}}{2 b^{3} \left (b \,x^{2}+a \right )} \] Input:

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

Output:

( - 2*sqrt(a + b*x**2)*a*b**2*d + 8*sqrt(a + b*x**2)*a*b*c*d + 3*sqrt(a + 
b*x**2)*a*b*d**2*x + 2*sqrt(a + b*x**2)*b**3*c*x - 2*sqrt(a + b*x**2)*b**3 
*c - 2*sqrt(a + b*x**2)*b**3*d*x - 2*sqrt(a + b*x**2)*b**2*c**2*x + 4*sqrt 
(a + b*x**2)*b**2*c*d*x**2 + sqrt(a + b*x**2)*b**2*d**2*x**3 - 3*sqrt(b)*l 
og((sqrt(a + b*x**2) + sqrt(b)*x)/sqrt(a))*a**2*d**2 + 2*sqrt(b)*log((sqrt 
(a + b*x**2) + sqrt(b)*x)/sqrt(a))*a*b**2*d + 2*sqrt(b)*log((sqrt(a + b*x* 
*2) + sqrt(b)*x)/sqrt(a))*a*b*c**2 - 3*sqrt(b)*log((sqrt(a + b*x**2) + sqr 
t(b)*x)/sqrt(a))*a*b*d**2*x**2 + 2*sqrt(b)*log((sqrt(a + b*x**2) + sqrt(b) 
*x)/sqrt(a))*b**3*d*x**2 + 2*sqrt(b)*log((sqrt(a + b*x**2) + sqrt(b)*x)/sq 
rt(a))*b**2*c**2*x**2 + 2*sqrt(b)*a**2*d**2 + 2*sqrt(b)*a*b**2*c - 2*sqrt( 
b)*a*b**2*d - 2*sqrt(b)*a*b*c**2 + 2*sqrt(b)*a*b*d**2*x**2 + 2*sqrt(b)*b** 
3*c*x**2 - 2*sqrt(b)*b**3*d*x**2 - 2*sqrt(b)*b**2*c**2*x**2)/(2*b**3*(a + 
b*x**2))