\(\int \frac {A+B x+C x^2+D x^3}{\sqrt {a+b x} (c+d x)^{9/2}} \, dx\) [137]

Optimal result

Integrand size = 34, antiderivative size = 374 \[ \int \frac {A+B x+C x^2+D x^3}{\sqrt {a+b x} (c+d x)^{9/2}} \, dx=\frac {2 \left (c^2 C d-B c d^2+A d^3-c^3 D\right ) \sqrt {a+b x}}{7 d^3 (b c-a d) (c+d x)^{7/2}}+\frac {2 \left (7 a d \left (2 c C d-B d^2-3 c^2 D\right )-b \left (8 c^2 C d-B c d^2-6 A d^3-15 c^3 D\right )\right ) \sqrt {a+b x}}{35 d^3 (b c-a d)^2 (c+d x)^{5/2}}+\frac {2 \left (35 a^2 d^2 (C d-3 c D)-14 a b d \left (c C d+2 B d^2-9 c^2 D\right )+b^2 \left (3 c^2 C d+4 B c d^2+24 A d^3-45 c^3 D\right )\right ) \sqrt {a+b x}}{105 d^3 (b c-a d)^3 (c+d x)^{3/2}}-\frac {2 \left (105 a^3 d^3 D-35 a^2 b d^2 (2 C d+3 c D)+7 a b^2 d \left (4 c C d+8 B d^2+9 c^2 D\right )-b^3 \left (6 c^2 C d+8 B c d^2+48 A d^3+15 c^3 D\right )\right ) \sqrt {a+b x}}{105 d^3 (b c-a d)^4 \sqrt {c+d x}} \] Output:

2/7*(A*d^3-B*c*d^2+C*c^2*d-D*c^3)*(b*x+a)^(1/2)/d^3/(-a*d+b*c)/(d*x+c)^(7/ 
2)+2/35*(7*a*d*(-B*d^2+2*C*c*d-3*D*c^2)-b*(-6*A*d^3-B*c*d^2+8*C*c^2*d-15*D 
*c^3))*(b*x+a)^(1/2)/d^3/(-a*d+b*c)^2/(d*x+c)^(5/2)+2/105*(35*a^2*d^2*(C*d 
-3*D*c)-14*a*b*d*(2*B*d^2+C*c*d-9*D*c^2)+b^2*(24*A*d^3+4*B*c*d^2+3*C*c^2*d 
-45*D*c^3))*(b*x+a)^(1/2)/d^3/(-a*d+b*c)^3/(d*x+c)^(3/2)-2/105*(105*a^3*d^ 
3*D-35*a^2*b*d^2*(2*C*d+3*D*c)+7*a*b^2*d*(8*B*d^2+4*C*c*d+9*D*c^2)-b^3*(48 
*A*d^3+8*B*c*d^2+6*C*c^2*d+15*D*c^3))*(b*x+a)^(1/2)/d^3/(-a*d+b*c)^4/(d*x+ 
c)^(1/2)
 

Mathematica [A] (verified)

Time = 0.33 (sec) , antiderivative size = 377, normalized size of antiderivative = 1.01 \[ \int \frac {A+B x+C x^2+D x^3}{\sqrt {a+b x} (c+d x)^{9/2}} \, dx=\frac {2 \sqrt {a+b x} \left (105 A b^3-105 a b^2 B+105 a^2 b C-105 a^3 D-\frac {15 c^2 C d (a+b x)^3}{(c+d x)^3}+\frac {15 B c d^2 (a+b x)^3}{(c+d x)^3}-\frac {15 A d^3 (a+b x)^3}{(c+d x)^3}+\frac {15 c^3 D (a+b x)^3}{(c+d x)^3}+\frac {21 b c^2 C (a+b x)^2}{(c+d x)^2}-\frac {42 b B c d (a+b x)^2}{(c+d x)^2}+\frac {42 a c C d (a+b x)^2}{(c+d x)^2}+\frac {63 A b d^2 (a+b x)^2}{(c+d x)^2}-\frac {21 a B d^2 (a+b x)^2}{(c+d x)^2}-\frac {63 a c^2 D (a+b x)^2}{(c+d x)^2}+\frac {35 b^2 B c (a+b x)}{c+d x}-\frac {70 a b c C (a+b x)}{c+d x}-\frac {105 A b^2 d (a+b x)}{c+d x}+\frac {70 a b B d (a+b x)}{c+d x}-\frac {35 a^2 C d (a+b x)}{c+d x}+\frac {105 a^2 c D (a+b x)}{c+d x}\right )}{105 (b c-a d)^4 \sqrt {c+d x}} \] Input:

Integrate[(A + B*x + C*x^2 + D*x^3)/(Sqrt[a + b*x]*(c + d*x)^(9/2)),x]
 

Output:

(2*Sqrt[a + b*x]*(105*A*b^3 - 105*a*b^2*B + 105*a^2*b*C - 105*a^3*D - (15* 
c^2*C*d*(a + b*x)^3)/(c + d*x)^3 + (15*B*c*d^2*(a + b*x)^3)/(c + d*x)^3 - 
(15*A*d^3*(a + b*x)^3)/(c + d*x)^3 + (15*c^3*D*(a + b*x)^3)/(c + d*x)^3 + 
(21*b*c^2*C*(a + b*x)^2)/(c + d*x)^2 - (42*b*B*c*d*(a + b*x)^2)/(c + d*x)^ 
2 + (42*a*c*C*d*(a + b*x)^2)/(c + d*x)^2 + (63*A*b*d^2*(a + b*x)^2)/(c + d 
*x)^2 - (21*a*B*d^2*(a + b*x)^2)/(c + d*x)^2 - (63*a*c^2*D*(a + b*x)^2)/(c 
 + d*x)^2 + (35*b^2*B*c*(a + b*x))/(c + d*x) - (70*a*b*c*C*(a + b*x))/(c + 
 d*x) - (105*A*b^2*d*(a + b*x))/(c + d*x) + (70*a*b*B*d*(a + b*x))/(c + d* 
x) - (35*a^2*C*d*(a + b*x))/(c + d*x) + (105*a^2*c*D*(a + b*x))/(c + d*x)) 
)/(105*(b*c - a*d)^4*Sqrt[c + d*x])
 

Rubi [A] (verified)

Time = 0.68 (sec) , antiderivative size = 397, normalized size of antiderivative = 1.06, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {2124, 27, 1193, 27, 87, 48}

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.

\(\displaystyle \int \frac {A+B x+C x^2+D x^3}{\sqrt {a+b x} (c+d x)^{9/2}} \, dx\)

\(\Big \downarrow \) 2124

\(\displaystyle \frac {2 \int \frac {-7 \left (a-\frac {b c}{d}\right ) D x^2+\frac {7 (b c-a d) (C d-c D) x}{d^2}+\frac {7 a d \left (-D c^2+C d c-B d^2\right )-b \left (-D c^3+C d c^2-B d^2 c-6 A d^3\right )}{d^3}}{2 \sqrt {a+b x} (c+d x)^{7/2}}dx}{7 (b c-a d)}+\frac {2 \sqrt {a+b x} \left (A+\frac {c \left (-B d^2+c^2 (-D)+c C d\right )}{d^3}\right )}{7 (c+d x)^{7/2} (b c-a d)}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\int \frac {-7 \left (a-\frac {b c}{d}\right ) D x^2+\frac {7 (b c-a d) (C d-c D) x}{d^2}+\frac {7 a d \left (-D c^2+C d c-B d^2\right )-b \left (-D c^3+C d c^2-B d^2 c-6 A d^3\right )}{d^3}}{\sqrt {a+b x} (c+d x)^{7/2}}dx}{7 (b c-a d)}+\frac {2 \sqrt {a+b x} \left (A+\frac {c \left (-B d^2+c^2 (-D)+c C d\right )}{d^3}\right )}{7 (c+d x)^{7/2} (b c-a d)}\)

\(\Big \downarrow \) 1193

\(\displaystyle \frac {\frac {2 \int \frac {\left (-10 D c^3+3 C d c^2+4 B d^2 c+24 A d^3\right ) b^2-14 a d \left (-4 D c^2+C d c+2 B d^2\right ) b+35 a^2 d^2 (C d-2 c D)+35 d (b c-a d)^2 D x}{2 d^3 \sqrt {a+b x} (c+d x)^{5/2}}dx}{5 (b c-a d)}+\frac {2 \sqrt {a+b x} \left (7 a d \left (-B d^2-3 c^2 D+2 c C d\right )-b \left (-6 A d^3-B c d^2-15 c^3 D+8 c^2 C d\right )\right )}{5 d^3 (c+d x)^{5/2} (b c-a d)}}{7 (b c-a d)}+\frac {2 \sqrt {a+b x} \left (A+\frac {c \left (-B d^2+c^2 (-D)+c C d\right )}{d^3}\right )}{7 (c+d x)^{7/2} (b c-a d)}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {\int \frac {\left (-10 D c^3+3 C d c^2+4 B d^2 c+24 A d^3\right ) b^2-14 a d \left (-4 D c^2+C d c+2 B d^2\right ) b+35 a^2 d^2 (C d-2 c D)+35 d (b c-a d)^2 D x}{\sqrt {a+b x} (c+d x)^{5/2}}dx}{5 d^3 (b c-a d)}+\frac {2 \sqrt {a+b x} \left (7 a d \left (-B d^2-3 c^2 D+2 c C d\right )-b \left (-6 A d^3-B c d^2-15 c^3 D+8 c^2 C d\right )\right )}{5 d^3 (c+d x)^{5/2} (b c-a d)}}{7 (b c-a d)}+\frac {2 \sqrt {a+b x} \left (A+\frac {c \left (-B d^2+c^2 (-D)+c C d\right )}{d^3}\right )}{7 (c+d x)^{7/2} (b c-a d)}\)

\(\Big \downarrow \) 87

\(\displaystyle \frac {\frac {\frac {2 \sqrt {a+b x} \left (35 a^2 d^2 (C d-3 c D)-14 a b d \left (2 B d^2-9 c^2 D+c C d\right )+b^2 \left (24 A d^3+4 B c d^2-45 c^3 D+3 c^2 C d\right )\right )}{3 (c+d x)^{3/2} (b c-a d)}-\frac {\left (105 a^3 d^3 D-35 a^2 b d^2 (3 c D+2 C d)+7 a b^2 d \left (8 B d^2+9 c^2 D+4 c C d\right )-\left (b^3 \left (48 A d^3+8 B c d^2+15 c^3 D+6 c^2 C d\right )\right )\right ) \int \frac {1}{\sqrt {a+b x} (c+d x)^{3/2}}dx}{3 (b c-a d)}}{5 d^3 (b c-a d)}+\frac {2 \sqrt {a+b x} \left (7 a d \left (-B d^2-3 c^2 D+2 c C d\right )-b \left (-6 A d^3-B c d^2-15 c^3 D+8 c^2 C d\right )\right )}{5 d^3 (c+d x)^{5/2} (b c-a d)}}{7 (b c-a d)}+\frac {2 \sqrt {a+b x} \left (A+\frac {c \left (-B d^2+c^2 (-D)+c C d\right )}{d^3}\right )}{7 (c+d x)^{7/2} (b c-a d)}\)

\(\Big \downarrow \) 48

\(\displaystyle \frac {\frac {\frac {2 \sqrt {a+b x} \left (35 a^2 d^2 (C d-3 c D)-14 a b d \left (2 B d^2-9 c^2 D+c C d\right )+b^2 \left (24 A d^3+4 B c d^2-45 c^3 D+3 c^2 C d\right )\right )}{3 (c+d x)^{3/2} (b c-a d)}-\frac {2 \sqrt {a+b x} \left (105 a^3 d^3 D-35 a^2 b d^2 (3 c D+2 C d)+7 a b^2 d \left (8 B d^2+9 c^2 D+4 c C d\right )-\left (b^3 \left (48 A d^3+8 B c d^2+15 c^3 D+6 c^2 C d\right )\right )\right )}{3 \sqrt {c+d x} (b c-a d)^2}}{5 d^3 (b c-a d)}+\frac {2 \sqrt {a+b x} \left (7 a d \left (-B d^2-3 c^2 D+2 c C d\right )-b \left (-6 A d^3-B c d^2-15 c^3 D+8 c^2 C d\right )\right )}{5 d^3 (c+d x)^{5/2} (b c-a d)}}{7 (b c-a d)}+\frac {2 \sqrt {a+b x} \left (A+\frac {c \left (-B d^2+c^2 (-D)+c C d\right )}{d^3}\right )}{7 (c+d x)^{7/2} (b c-a d)}\)

Input:

Int[(A + B*x + C*x^2 + D*x^3)/(Sqrt[a + b*x]*(c + d*x)^(9/2)),x]
 

Output:

(2*(A + (c*(c*C*d - B*d^2 - c^2*D))/d^3)*Sqrt[a + b*x])/(7*(b*c - a*d)*(c 
+ d*x)^(7/2)) + ((2*(7*a*d*(2*c*C*d - B*d^2 - 3*c^2*D) - b*(8*c^2*C*d - B* 
c*d^2 - 6*A*d^3 - 15*c^3*D))*Sqrt[a + b*x])/(5*d^3*(b*c - a*d)*(c + d*x)^( 
5/2)) + ((2*(35*a^2*d^2*(C*d - 3*c*D) - 14*a*b*d*(c*C*d + 2*B*d^2 - 9*c^2* 
D) + b^2*(3*c^2*C*d + 4*B*c*d^2 + 24*A*d^3 - 45*c^3*D))*Sqrt[a + b*x])/(3* 
(b*c - a*d)*(c + d*x)^(3/2)) - (2*(105*a^3*d^3*D - 35*a^2*b*d^2*(2*C*d + 3 
*c*D) + 7*a*b^2*d*(4*c*C*d + 8*B*d^2 + 9*c^2*D) - b^3*(6*c^2*C*d + 8*B*c*d 
^2 + 48*A*d^3 + 15*c^3*D))*Sqrt[a + b*x])/(3*(b*c - a*d)^2*Sqrt[c + d*x])) 
/(5*d^3*(b*c - a*d)))/(7*(b*c - a*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 48

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp 
[(a + b*x)^(m + 1)*((c + d*x)^(n + 1)/((b*c - a*d)*(m + 1))), x] /; FreeQ[{ 
a, b, c, d, m, n}, x] && EqQ[m + n + 2, 0] && NeQ[m, -1]
 

rule 87

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p 
_.), x_] :> Simp[(-(b*e - a*f))*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(f*(p 
+ 1)*(c*f - d*e))), x] - Simp[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p 
+ 1)))/(f*(p + 1)*(c*f - d*e))   Int[(c + d*x)^n*(e + f*x)^(p + 1), x], x] 
/; FreeQ[{a, b, c, d, e, f, n}, x] && LtQ[p, -1] && ( !LtQ[n, -1] || Intege 
rQ[p] ||  !(IntegerQ[n] ||  !(EqQ[e, 0] ||  !(EqQ[c, 0] || LtQ[p, n]))))
 

rule 1193

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_)*((a_.) + (b_.)*(x_) 
 + (c_.)*(x_)^2)^(p_.), x_Symbol] :> With[{Qx = PolynomialQuotient[(a + b*x 
 + c*x^2)^p, d + e*x, x], R = PolynomialRemainder[(a + b*x + c*x^2)^p, d + 
e*x, x]}, Simp[R*(d + e*x)^(m + 1)*((f + g*x)^(n + 1)/((m + 1)*(e*f - d*g)) 
), x] + Simp[1/((m + 1)*(e*f - d*g))   Int[(d + e*x)^(m + 1)*(f + g*x)^n*Ex 
pandToSum[(m + 1)*(e*f - d*g)*Qx - g*R*(m + n + 2), x], x], x]] /; FreeQ[{a 
, b, c, d, e, f, g, n}, x] && IGtQ[p, 0] && ILtQ[2*m, -2] &&  !IntegerQ[n] 
&&  !(EqQ[m, -2] && EqQ[p, 1] && EqQ[2*c*d - b*e, 0])
 

rule 2124

Int[(Px_)*((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] : 
> With[{Qx = PolynomialQuotient[Px, a + b*x, x], R = PolynomialRemainder[Px 
, a + b*x, x]}, Simp[R*(a + b*x)^(m + 1)*((c + d*x)^(n + 1)/((m + 1)*(b*c - 
 a*d))), x] + Simp[1/((m + 1)*(b*c - a*d))   Int[(a + b*x)^(m + 1)*(c + d*x 
)^n*ExpandToSum[(m + 1)*(b*c - a*d)*Qx - d*R*(m + n + 2), x], x], x]] /; Fr 
eeQ[{a, b, c, d, n}, x] && PolyQ[Px, x] && LtQ[m, -1] && (IntegerQ[m] ||  ! 
ILtQ[n, -1])
 

Maple [A] (verified)

Time = 0.59 (sec) , antiderivative size = 550, normalized size of antiderivative = 1.47

method	result	size
default	\(-\frac {2 \sqrt {b x +a}\, \left (-48 A \,b^{3} d^{3} x^{3}+56 B a \,b^{2} d^{3} x^{3}-8 B \,b^{3} c \,d^{2} x^{3}-70 C \,a^{2} b \,d^{3} x^{3}+28 C a \,b^{2} c \,d^{2} x^{3}-6 C \,b^{3} c^{2} d \,x^{3}+105 D a^{3} d^{3} x^{3}-105 D a^{2} b c \,d^{2} x^{3}+63 D a \,b^{2} c^{2} d \,x^{3}-15 D b^{3} c^{3} x^{3}+24 A a \,b^{2} d^{3} x^{2}-168 A \,b^{3} c \,d^{2} x^{2}-28 B \,a^{2} b \,d^{3} x^{2}+200 B a \,b^{2} c \,d^{2} x^{2}-28 B \,b^{3} c^{2} d \,x^{2}+35 C \,a^{3} d^{3} x^{2}-259 C \,a^{2} b c \,d^{2} x^{2}+101 C a \,b^{2} c^{2} d \,x^{2}-21 C \,b^{3} c^{3} x^{2}+210 D a^{3} c \,d^{2} x^{2}-84 D a^{2} b \,c^{2} d \,x^{2}+18 D a \,b^{2} c^{3} x^{2}-18 A \,a^{2} b \,d^{3} x +84 A a \,b^{2} c \,d^{2} x -210 A \,b^{3} c^{2} d x +21 B \,a^{3} d^{3} x -101 B \,a^{2} b c \,d^{2} x +259 B a \,b^{2} c^{2} d x -35 B \,b^{3} c^{3} x +28 C \,a^{3} c \,d^{2} x -200 C \,a^{2} b \,c^{2} d x +28 C a \,b^{2} c^{3} x +168 D a^{3} c^{2} d x -24 D a^{2} b \,c^{3} x +15 a^{3} A \,d^{3}-63 A \,a^{2} b c \,d^{2}+105 A a \,b^{2} c^{2} d -105 A \,b^{3} c^{3}+6 B \,a^{3} c \,d^{2}-28 B \,a^{2} b \,c^{2} d +70 B a \,b^{2} c^{3}+8 a^{3} c^{2} C d -56 C \,a^{2} b \,c^{3}+48 D a^{3} c^{3}\right )}{105 \left (x d +c \right )^{\frac {7}{2}} \left (a d -b c \right )^{4}}\)	\(550\)
gosper	\(-\frac {2 \sqrt {b x +a}\, \left (-48 A \,b^{3} d^{3} x^{3}+56 B a \,b^{2} d^{3} x^{3}-8 B \,b^{3} c \,d^{2} x^{3}-70 C \,a^{2} b \,d^{3} x^{3}+28 C a \,b^{2} c \,d^{2} x^{3}-6 C \,b^{3} c^{2} d \,x^{3}+105 D a^{3} d^{3} x^{3}-105 D a^{2} b c \,d^{2} x^{3}+63 D a \,b^{2} c^{2} d \,x^{3}-15 D b^{3} c^{3} x^{3}+24 A a \,b^{2} d^{3} x^{2}-168 A \,b^{3} c \,d^{2} x^{2}-28 B \,a^{2} b \,d^{3} x^{2}+200 B a \,b^{2} c \,d^{2} x^{2}-28 B \,b^{3} c^{2} d \,x^{2}+35 C \,a^{3} d^{3} x^{2}-259 C \,a^{2} b c \,d^{2} x^{2}+101 C a \,b^{2} c^{2} d \,x^{2}-21 C \,b^{3} c^{3} x^{2}+210 D a^{3} c \,d^{2} x^{2}-84 D a^{2} b \,c^{2} d \,x^{2}+18 D a \,b^{2} c^{3} x^{2}-18 A \,a^{2} b \,d^{3} x +84 A a \,b^{2} c \,d^{2} x -210 A \,b^{3} c^{2} d x +21 B \,a^{3} d^{3} x -101 B \,a^{2} b c \,d^{2} x +259 B a \,b^{2} c^{2} d x -35 B \,b^{3} c^{3} x +28 C \,a^{3} c \,d^{2} x -200 C \,a^{2} b \,c^{2} d x +28 C a \,b^{2} c^{3} x +168 D a^{3} c^{2} d x -24 D a^{2} b \,c^{3} x +15 a^{3} A \,d^{3}-63 A \,a^{2} b c \,d^{2}+105 A a \,b^{2} c^{2} d -105 A \,b^{3} c^{3}+6 B \,a^{3} c \,d^{2}-28 B \,a^{2} b \,c^{2} d +70 B a \,b^{2} c^{3}+8 a^{3} c^{2} C d -56 C \,a^{2} b \,c^{3}+48 D a^{3} c^{3}\right )}{105 \left (x d +c \right )^{\frac {7}{2}} \left (d^{4} a^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +c^{4} b^{4}\right )}\)	\(591\)
orering	\(-\frac {2 \sqrt {b x +a}\, \left (-48 A \,b^{3} d^{3} x^{3}+56 B a \,b^{2} d^{3} x^{3}-8 B \,b^{3} c \,d^{2} x^{3}-70 C \,a^{2} b \,d^{3} x^{3}+28 C a \,b^{2} c \,d^{2} x^{3}-6 C \,b^{3} c^{2} d \,x^{3}+105 D a^{3} d^{3} x^{3}-105 D a^{2} b c \,d^{2} x^{3}+63 D a \,b^{2} c^{2} d \,x^{3}-15 D b^{3} c^{3} x^{3}+24 A a \,b^{2} d^{3} x^{2}-168 A \,b^{3} c \,d^{2} x^{2}-28 B \,a^{2} b \,d^{3} x^{2}+200 B a \,b^{2} c \,d^{2} x^{2}-28 B \,b^{3} c^{2} d \,x^{2}+35 C \,a^{3} d^{3} x^{2}-259 C \,a^{2} b c \,d^{2} x^{2}+101 C a \,b^{2} c^{2} d \,x^{2}-21 C \,b^{3} c^{3} x^{2}+210 D a^{3} c \,d^{2} x^{2}-84 D a^{2} b \,c^{2} d \,x^{2}+18 D a \,b^{2} c^{3} x^{2}-18 A \,a^{2} b \,d^{3} x +84 A a \,b^{2} c \,d^{2} x -210 A \,b^{3} c^{2} d x +21 B \,a^{3} d^{3} x -101 B \,a^{2} b c \,d^{2} x +259 B a \,b^{2} c^{2} d x -35 B \,b^{3} c^{3} x +28 C \,a^{3} c \,d^{2} x -200 C \,a^{2} b \,c^{2} d x +28 C a \,b^{2} c^{3} x +168 D a^{3} c^{2} d x -24 D a^{2} b \,c^{3} x +15 a^{3} A \,d^{3}-63 A \,a^{2} b c \,d^{2}+105 A a \,b^{2} c^{2} d -105 A \,b^{3} c^{3}+6 B \,a^{3} c \,d^{2}-28 B \,a^{2} b \,c^{2} d +70 B a \,b^{2} c^{3}+8 a^{3} c^{2} C d -56 C \,a^{2} b \,c^{3}+48 D a^{3} c^{3}\right )}{105 \left (x d +c \right )^{\frac {7}{2}} \left (d^{4} a^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +c^{4} b^{4}\right )}\)	\(591\)

Input:

int((D*x^3+C*x^2+B*x+A)/(b*x+a)^(1/2)/(d*x+c)^(9/2),x,method=_RETURNVERBOS 
E)
 

Output:

-2/105*(b*x+a)^(1/2)*(-48*A*b^3*d^3*x^3+56*B*a*b^2*d^3*x^3-8*B*b^3*c*d^2*x 
^3-70*C*a^2*b*d^3*x^3+28*C*a*b^2*c*d^2*x^3-6*C*b^3*c^2*d*x^3+105*D*a^3*d^3 
*x^3-105*D*a^2*b*c*d^2*x^3+63*D*a*b^2*c^2*d*x^3-15*D*b^3*c^3*x^3+24*A*a*b^ 
2*d^3*x^2-168*A*b^3*c*d^2*x^2-28*B*a^2*b*d^3*x^2+200*B*a*b^2*c*d^2*x^2-28* 
B*b^3*c^2*d*x^2+35*C*a^3*d^3*x^2-259*C*a^2*b*c*d^2*x^2+101*C*a*b^2*c^2*d*x 
^2-21*C*b^3*c^3*x^2+210*D*a^3*c*d^2*x^2-84*D*a^2*b*c^2*d*x^2+18*D*a*b^2*c^ 
3*x^2-18*A*a^2*b*d^3*x+84*A*a*b^2*c*d^2*x-210*A*b^3*c^2*d*x+21*B*a^3*d^3*x 
-101*B*a^2*b*c*d^2*x+259*B*a*b^2*c^2*d*x-35*B*b^3*c^3*x+28*C*a^3*c*d^2*x-2 
00*C*a^2*b*c^2*d*x+28*C*a*b^2*c^3*x+168*D*a^3*c^2*d*x-24*D*a^2*b*c^3*x+15* 
A*a^3*d^3-63*A*a^2*b*c*d^2+105*A*a*b^2*c^2*d-105*A*b^3*c^3+6*B*a^3*c*d^2-2 
8*B*a^2*b*c^2*d+70*B*a*b^2*c^3+8*C*a^3*c^2*d-56*C*a^2*b*c^3+48*D*a^3*c^3)/ 
(d*x+c)^(7/2)/(a*d-b*c)^4
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 712 vs. \(2 (350) = 700\).

Time = 73.91 (sec) , antiderivative size = 712, normalized size of antiderivative = 1.90 \[ \int \frac {A+B x+C x^2+D x^3}{\sqrt {a+b x} (c+d x)^{9/2}} \, dx=-\frac {2 \, {\left (15 \, A a^{3} d^{3} + {\left (48 \, D a^{3} - 56 \, C a^{2} b + 70 \, B a b^{2} - 105 \, A b^{3}\right )} c^{3} + {\left (8 \, C a^{3} - 28 \, B a^{2} b + 105 \, A a b^{2}\right )} c^{2} d + 3 \, {\left (2 \, B a^{3} - 21 \, A a^{2} b\right )} c d^{2} - {\left (15 \, D b^{3} c^{3} - 3 \, {\left (21 \, D a b^{2} - 2 \, C b^{3}\right )} c^{2} d + {\left (105 \, D a^{2} b - 28 \, C a b^{2} + 8 \, B b^{3}\right )} c d^{2} - {\left (105 \, D a^{3} - 70 \, C a^{2} b + 56 \, B a b^{2} - 48 \, A b^{3}\right )} d^{3}\right )} x^{3} + {\left (3 \, {\left (6 \, D a b^{2} - 7 \, C b^{3}\right )} c^{3} - {\left (84 \, D a^{2} b - 101 \, C a b^{2} + 28 \, B b^{3}\right )} c^{2} d + {\left (210 \, D a^{3} - 259 \, C a^{2} b + 200 \, B a b^{2} - 168 \, A b^{3}\right )} c d^{2} + {\left (35 \, C a^{3} - 28 \, B a^{2} b + 24 \, A a b^{2}\right )} d^{3}\right )} x^{2} - {\left ({\left (24 \, D a^{2} b - 28 \, C a b^{2} + 35 \, B b^{3}\right )} c^{3} - {\left (168 \, D a^{3} - 200 \, C a^{2} b + 259 \, B a b^{2} - 210 \, A b^{3}\right )} c^{2} d - {\left (28 \, C a^{3} - 101 \, B a^{2} b + 84 \, A a b^{2}\right )} c d^{2} - 3 \, {\left (7 \, B a^{3} - 6 \, A a^{2} b\right )} d^{3}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{105 \, {\left (b^{4} c^{8} - 4 \, a b^{3} c^{7} d + 6 \, a^{2} b^{2} c^{6} d^{2} - 4 \, a^{3} b c^{5} d^{3} + a^{4} c^{4} d^{4} + {\left (b^{4} c^{4} d^{4} - 4 \, a b^{3} c^{3} d^{5} + 6 \, a^{2} b^{2} c^{2} d^{6} - 4 \, a^{3} b c d^{7} + a^{4} d^{8}\right )} x^{4} + 4 \, {\left (b^{4} c^{5} d^{3} - 4 \, a b^{3} c^{4} d^{4} + 6 \, a^{2} b^{2} c^{3} d^{5} - 4 \, a^{3} b c^{2} d^{6} + a^{4} c d^{7}\right )} x^{3} + 6 \, {\left (b^{4} c^{6} d^{2} - 4 \, a b^{3} c^{5} d^{3} + 6 \, a^{2} b^{2} c^{4} d^{4} - 4 \, a^{3} b c^{3} d^{5} + a^{4} c^{2} d^{6}\right )} x^{2} + 4 \, {\left (b^{4} c^{7} d - 4 \, a b^{3} c^{6} d^{2} + 6 \, a^{2} b^{2} c^{5} d^{3} - 4 \, a^{3} b c^{4} d^{4} + a^{4} c^{3} d^{5}\right )} x\right )}} \] Input:

integrate((D*x^3+C*x^2+B*x+A)/(b*x+a)^(1/2)/(d*x+c)^(9/2),x, algorithm="fr 
icas")
 

Output:

-2/105*(15*A*a^3*d^3 + (48*D*a^3 - 56*C*a^2*b + 70*B*a*b^2 - 105*A*b^3)*c^ 
3 + (8*C*a^3 - 28*B*a^2*b + 105*A*a*b^2)*c^2*d + 3*(2*B*a^3 - 21*A*a^2*b)* 
c*d^2 - (15*D*b^3*c^3 - 3*(21*D*a*b^2 - 2*C*b^3)*c^2*d + (105*D*a^2*b - 28 
*C*a*b^2 + 8*B*b^3)*c*d^2 - (105*D*a^3 - 70*C*a^2*b + 56*B*a*b^2 - 48*A*b^ 
3)*d^3)*x^3 + (3*(6*D*a*b^2 - 7*C*b^3)*c^3 - (84*D*a^2*b - 101*C*a*b^2 + 2 
8*B*b^3)*c^2*d + (210*D*a^3 - 259*C*a^2*b + 200*B*a*b^2 - 168*A*b^3)*c*d^2 
 + (35*C*a^3 - 28*B*a^2*b + 24*A*a*b^2)*d^3)*x^2 - ((24*D*a^2*b - 28*C*a*b 
^2 + 35*B*b^3)*c^3 - (168*D*a^3 - 200*C*a^2*b + 259*B*a*b^2 - 210*A*b^3)*c 
^2*d - (28*C*a^3 - 101*B*a^2*b + 84*A*a*b^2)*c*d^2 - 3*(7*B*a^3 - 6*A*a^2* 
b)*d^3)*x)*sqrt(b*x + a)*sqrt(d*x + c)/(b^4*c^8 - 4*a*b^3*c^7*d + 6*a^2*b^ 
2*c^6*d^2 - 4*a^3*b*c^5*d^3 + a^4*c^4*d^4 + (b^4*c^4*d^4 - 4*a*b^3*c^3*d^5 
 + 6*a^2*b^2*c^2*d^6 - 4*a^3*b*c*d^7 + a^4*d^8)*x^4 + 4*(b^4*c^5*d^3 - 4*a 
*b^3*c^4*d^4 + 6*a^2*b^2*c^3*d^5 - 4*a^3*b*c^2*d^6 + a^4*c*d^7)*x^3 + 6*(b 
^4*c^6*d^2 - 4*a*b^3*c^5*d^3 + 6*a^2*b^2*c^4*d^4 - 4*a^3*b*c^3*d^5 + a^4*c 
^2*d^6)*x^2 + 4*(b^4*c^7*d - 4*a*b^3*c^6*d^2 + 6*a^2*b^2*c^5*d^3 - 4*a^3*b 
*c^4*d^4 + a^4*c^3*d^5)*x)
 

Sympy [F]

\[ \int \frac {A+B x+C x^2+D x^3}{\sqrt {a+b x} (c+d x)^{9/2}} \, dx=\int \frac {A + B x + C x^{2} + D x^{3}}{\sqrt {a + b x} \left (c + d x\right )^{\frac {9}{2}}}\, dx \] Input:

integrate((D*x**3+C*x**2+B*x+A)/(b*x+a)**(1/2)/(d*x+c)**(9/2),x)
 

Output:

Integral((A + B*x + C*x**2 + D*x**3)/(sqrt(a + b*x)*(c + d*x)**(9/2)), x)
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {A+B x+C x^2+D x^3}{\sqrt {a+b x} (c+d x)^{9/2}} \, dx=\text {Exception raised: ValueError} \] Input:

integrate((D*x^3+C*x^2+B*x+A)/(b*x+a)^(1/2)/(d*x+c)^(9/2),x, algorithm="ma 
xima")
 

Output:

Exception raised: ValueError >> Computation failed since Maxima requested 
additional constraints; using the 'assume' command before evaluation *may* 
 help (example of legal syntax is 'assume(a*d-b*c>0)', see `assume?` for m 
ore detail
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1086 vs. \(2 (350) = 700\).

Time = 0.33 (sec) , antiderivative size = 1086, normalized size of antiderivative = 2.90 \[ \int \frac {A+B x+C x^2+D x^3}{\sqrt {a+b x} (c+d x)^{9/2}} \, dx=\text {Too large to display} \] Input:

integrate((D*x^3+C*x^2+B*x+A)/(b*x+a)^(1/2)/(d*x+c)^(9/2),x, algorithm="gi 
ac")
 

Output:

2/105*(((b*x + a)*((15*D*b^10*c^3*d^3*abs(b) - 63*D*a*b^9*c^2*d^4*abs(b) + 
 6*C*b^10*c^2*d^4*abs(b) + 105*D*a^2*b^8*c*d^5*abs(b) - 28*C*a*b^9*c*d^5*a 
bs(b) + 8*B*b^10*c*d^5*abs(b) - 105*D*a^3*b^7*d^6*abs(b) + 70*C*a^2*b^8*d^ 
6*abs(b) - 56*B*a*b^9*d^6*abs(b) + 48*A*b^10*d^6*abs(b))*(b*x + a)/(b^8*c^ 
4*d^3 - 4*a*b^7*c^3*d^4 + 6*a^2*b^6*c^2*d^5 - 4*a^3*b^5*c*d^6 + a^4*b^4*d^ 
7) - 7*(9*D*a*b^10*c^3*d^3*abs(b) - 3*C*b^11*c^3*d^3*abs(b) - 39*D*a^2*b^9 
*c^2*d^4*abs(b) + 17*C*a*b^10*c^2*d^4*abs(b) - 4*B*b^11*c^2*d^4*abs(b) + 7 
5*D*a^3*b^8*c*d^5*abs(b) - 49*C*a^2*b^9*c*d^5*abs(b) + 32*B*a*b^10*c*d^5*a 
bs(b) - 24*A*b^11*c*d^5*abs(b) - 45*D*a^4*b^7*d^6*abs(b) + 35*C*a^3*b^8*d^ 
6*abs(b) - 28*B*a^2*b^9*d^6*abs(b) + 24*A*a*b^10*d^6*abs(b))/(b^8*c^4*d^3 
- 4*a*b^7*c^3*d^4 + 6*a^2*b^6*c^2*d^5 - 4*a^3*b^5*c*d^6 + a^4*b^4*d^7)) + 
35*(3*D*a^2*b^10*c^3*d^3*abs(b) - 2*C*a*b^11*c^3*d^3*abs(b) + B*b^12*c^3*d 
^3*abs(b) - 15*D*a^3*b^9*c^2*d^4*abs(b) + 12*C*a^2*b^10*c^2*d^4*abs(b) - 9 
*B*a*b^11*c^2*d^4*abs(b) + 6*A*b^12*c^2*d^4*abs(b) + 21*D*a^4*b^8*c*d^5*ab 
s(b) - 18*C*a^3*b^9*c*d^5*abs(b) + 15*B*a^2*b^10*c*d^5*abs(b) - 12*A*a*b^1 
1*c*d^5*abs(b) - 9*D*a^5*b^7*d^6*abs(b) + 8*C*a^4*b^8*d^6*abs(b) - 7*B*a^3 
*b^9*d^6*abs(b) + 6*A*a^2*b^10*d^6*abs(b))/(b^8*c^4*d^3 - 4*a*b^7*c^3*d^4 
+ 6*a^2*b^6*c^2*d^5 - 4*a^3*b^5*c*d^6 + a^4*b^4*d^7))*(b*x + a) - 105*(D*a 
^3*b^10*c^3*d^3*abs(b) - C*a^2*b^11*c^3*d^3*abs(b) + B*a*b^12*c^3*d^3*abs( 
b) - A*b^13*c^3*d^3*abs(b) - 3*D*a^4*b^9*c^2*d^4*abs(b) + 3*C*a^3*b^10*...
 

Mupad [F(-1)]

Timed out. \[ \int \frac {A+B x+C x^2+D x^3}{\sqrt {a+b x} (c+d x)^{9/2}} \, dx=\int \frac {A+B\,x+C\,x^2+x^3\,D}{\sqrt {a+b\,x}\,{\left (c+d\,x\right )}^{9/2}} \,d x \] Input:

int((A + B*x + C*x^2 + x^3*D)/((a + b*x)^(1/2)*(c + d*x)^(9/2)),x)
 

Output:

int((A + B*x + C*x^2 + x^3*D)/((a + b*x)^(1/2)*(c + d*x)^(9/2)), x)
 

Reduce [F]

\[ \int \frac {A+B x+C x^2+D x^3}{\sqrt {a+b x} (c+d x)^{9/2}} \, dx=\int \frac {D x^{3}+C \,x^{2}+B x +A}{\sqrt {b x +a}\, \left (d x +c \right )^{\frac {9}{2}}}d x \] Input:

int((D*x^3+C*x^2+B*x+A)/(b*x+a)^(1/2)/(d*x+c)^(9/2),x)
 

Output:

int((D*x^3+C*x^2+B*x+A)/(b*x+a)^(1/2)/(d*x+c)^(9/2),x)