\(\int \frac {A+B x+C x^2+D x^3}{(a+b x)^{3/2} (c+d x)^{7/2}} \, dx\) [144]

Optimal result

Integrand size = 34, antiderivative size = 355 \[ \int \frac {A+B x+C x^2+D x^3}{(a+b x)^{3/2} (c+d x)^{7/2}} \, dx=-\frac {2 \left (c^2 C d-B c d^2+A d^3-c^3 D\right ) \sqrt {a+b x}}{5 d^2 (b c-a d)^2 (c+d x)^{5/2}}-\frac {2 \left (A b^3-a \left (b^2 B-a b C+a^2 D\right )\right )}{b^2 (b c-a d)^2 \sqrt {a+b x} (c+d x)^{3/2}}-\frac {2 \left (15 a^2 b C d^3-15 a^3 d^3 D+5 a b^2 d \left (2 c C d-4 B d^2-3 c^2 D\right )-b^3 \left (c^2 C d+4 B c d^2-24 A d^3-6 c^3 D\right )\right ) \sqrt {a+b x}}{15 b^2 d^2 (b c-a d)^3 (c+d x)^{3/2}}+\frac {2 \left (15 a^3 d^3 D-15 a^2 b d^2 (2 C d-3 c D)-5 a b^2 d \left (4 c C d-8 B d^2+3 c^2 D\right )+b^3 \left (2 c^2 C d+8 B c d^2-48 A d^3+3 c^3 D\right )\right ) \sqrt {a+b x}}{15 b d^2 (b c-a d)^4 \sqrt {c+d x}} \] Output:

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

Mathematica [A] (verified)

Time = 0.36 (sec) , antiderivative size = 377, normalized size of antiderivative = 1.06 \[ \int \frac {A+B x+C x^2+D x^3}{(a+b x)^{3/2} (c+d x)^{7/2}} \, dx=-\frac {2 (a+b x)^{5/2} \left (3 c^2 C d-3 B c d^2+3 A d^3-3 c^3 D-\frac {5 b c^2 C (c+d x)}{a+b x}+\frac {10 b B c d (c+d x)}{a+b x}-\frac {10 a c C d (c+d x)}{a+b x}-\frac {15 A b d^2 (c+d x)}{a+b x}+\frac {5 a B d^2 (c+d x)}{a+b x}+\frac {15 a c^2 D (c+d x)}{a+b x}-\frac {15 b^2 B c (c+d x)^2}{(a+b x)^2}+\frac {30 a b c C (c+d x)^2}{(a+b x)^2}+\frac {45 A b^2 d (c+d x)^2}{(a+b x)^2}-\frac {30 a b B d (c+d x)^2}{(a+b x)^2}+\frac {15 a^2 C d (c+d x)^2}{(a+b x)^2}-\frac {45 a^2 c D (c+d x)^2}{(a+b x)^2}+\frac {15 A b^3 (c+d x)^3}{(a+b x)^3}-\frac {15 a b^2 B (c+d x)^3}{(a+b x)^3}+\frac {15 a^2 b C (c+d x)^3}{(a+b x)^3}-\frac {15 a^3 D (c+d x)^3}{(a+b x)^3}\right )}{15 (b c-a d)^4 (c+d x)^{5/2}} \] Input:

Integrate[(A + B*x + C*x^2 + D*x^3)/((a + b*x)^(3/2)*(c + d*x)^(7/2)),x]
 

Output:

(-2*(a + b*x)^(5/2)*(3*c^2*C*d - 3*B*c*d^2 + 3*A*d^3 - 3*c^3*D - (5*b*c^2* 
C*(c + d*x))/(a + b*x) + (10*b*B*c*d*(c + d*x))/(a + b*x) - (10*a*c*C*d*(c 
 + d*x))/(a + b*x) - (15*A*b*d^2*(c + d*x))/(a + b*x) + (5*a*B*d^2*(c + d* 
x))/(a + b*x) + (15*a*c^2*D*(c + d*x))/(a + b*x) - (15*b^2*B*c*(c + d*x)^2 
)/(a + b*x)^2 + (30*a*b*c*C*(c + d*x)^2)/(a + b*x)^2 + (45*A*b^2*d*(c + d* 
x)^2)/(a + b*x)^2 - (30*a*b*B*d*(c + d*x)^2)/(a + b*x)^2 + (15*a^2*C*d*(c 
+ d*x)^2)/(a + b*x)^2 - (45*a^2*c*D*(c + d*x)^2)/(a + b*x)^2 + (15*A*b^3*( 
c + d*x)^3)/(a + b*x)^3 - (15*a*b^2*B*(c + d*x)^3)/(a + b*x)^3 + (15*a^2*b 
*C*(c + d*x)^3)/(a + b*x)^3 - (15*a^3*D*(c + d*x)^3)/(a + b*x)^3))/(15*(b* 
c - a*d)^4*(c + d*x)^(5/2))
 

Rubi [A] (verified)

Time = 0.75 (sec) , antiderivative size = 396, normalized size of antiderivative = 1.12, 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.

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

\(\Big \downarrow \) 2124

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1193

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 87

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

\(\Big \downarrow \) 48

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

Input:

Int[(A + B*x + C*x^2 + D*x^3)/((a + b*x)^(3/2)*(c + d*x)^(7/2)),x]
 

Output:

(-2*(A - (a*(b^2*B - a*b*C + a^2*D))/b^3))/((b*c - a*d)*Sqrt[a + b*x]*(c + 
 d*x)^(5/2)) + ((-2*((c^2*C)/d + 6*A*d - B*(c + (5*a*d)/b) - (c^3*D)/d^2 + 
 (5*a^2*d*(b*C - a*D))/b^3)*Sqrt[a + b*x])/(5*(b*c - a*d)*(c + d*x)^(5/2)) 
 + ((-2*(15*a^2*b*C*d^3 - 15*a^3*d^3*D + 5*a*b^2*d*(2*c*C*d - 4*B*d^2 - 3* 
c^2*D) - b^3*(c^2*C*d + 4*B*c*d^2 - 24*A*d^3 - 6*c^3*D))*Sqrt[a + b*x])/(3 
*(b*c - a*d)*(c + d*x)^(3/2)) + (2*b*(15*a^3*d^3*D - 15*a^2*b*d^2*(2*C*d - 
 3*c*D) - 5*a*b^2*d*(4*c*C*d - 8*B*d^2 + 3*c^2*D) + b^3*(2*c^2*C*d + 8*B*c 
*d^2 - 48*A*d^3 + 3*c^3*D))*Sqrt[a + b*x])/(3*(b*c - a*d)^2*Sqrt[c + d*x]) 
)/(5*b^2*d^2*(b*c - a*d)))/(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 = 0.63 (sec) , antiderivative size = 550, normalized size of antiderivative = 1.55

method	result	size
default	\(-\frac {2 \left (48 A \,b^{3} d^{3} x^{3}-40 B a \,b^{2} d^{3} x^{3}-8 B \,b^{3} c \,d^{2} x^{3}+30 C \,a^{2} b \,d^{3} x^{3}+20 C a \,b^{2} c \,d^{2} x^{3}-2 C \,b^{3} c^{2} d \,x^{3}-15 D a^{3} d^{3} x^{3}-45 D a^{2} b c \,d^{2} x^{3}+15 D a \,b^{2} c^{2} d \,x^{3}-3 D b^{3} c^{3} x^{3}+24 A a \,b^{2} d^{3} x^{2}+120 A \,b^{3} c \,d^{2} x^{2}-20 B \,a^{2} b \,d^{3} x^{2}-104 B a \,b^{2} c \,d^{2} x^{2}-20 B \,b^{3} c^{2} d \,x^{2}+15 C \,a^{3} d^{3} x^{2}+85 C \,a^{2} b c \,d^{2} x^{2}+49 C a \,b^{2} c^{2} d \,x^{2}-5 C \,b^{3} c^{3} x^{2}-90 D a^{3} c \,d^{2} x^{2}-60 D a^{2} b \,c^{2} d \,x^{2}+6 D a \,b^{2} c^{3} x^{2}-6 A \,a^{2} b \,d^{3} x +60 A a \,b^{2} c \,d^{2} x +90 A \,b^{3} c^{2} d x +5 B \,a^{3} d^{3} x -49 B \,a^{2} b c \,d^{2} x -85 B a \,b^{2} c^{2} d x -15 B \,b^{3} c^{3} x +20 C \,a^{3} c \,d^{2} x +104 C \,a^{2} b \,c^{2} d x +20 C a \,b^{2} c^{3} x -120 D a^{3} c^{2} d x -24 D a^{2} b \,c^{3} x +3 a^{3} A \,d^{3}-15 A \,a^{2} b c \,d^{2}+45 A a \,b^{2} c^{2} d +15 A \,b^{3} c^{3}+2 B \,a^{3} c \,d^{2}-20 B \,a^{2} b \,c^{2} d -30 B a \,b^{2} c^{3}+8 a^{3} c^{2} C d +40 C \,a^{2} b \,c^{3}-48 D a^{3} c^{3}\right )}{15 \left (x d +c \right )^{\frac {5}{2}} \sqrt {b x +a}\, \left (a d -b c \right )^{4}}\)	\(550\)
gosper	\(-\frac {2 \left (48 A \,b^{3} d^{3} x^{3}-40 B a \,b^{2} d^{3} x^{3}-8 B \,b^{3} c \,d^{2} x^{3}+30 C \,a^{2} b \,d^{3} x^{3}+20 C a \,b^{2} c \,d^{2} x^{3}-2 C \,b^{3} c^{2} d \,x^{3}-15 D a^{3} d^{3} x^{3}-45 D a^{2} b c \,d^{2} x^{3}+15 D a \,b^{2} c^{2} d \,x^{3}-3 D b^{3} c^{3} x^{3}+24 A a \,b^{2} d^{3} x^{2}+120 A \,b^{3} c \,d^{2} x^{2}-20 B \,a^{2} b \,d^{3} x^{2}-104 B a \,b^{2} c \,d^{2} x^{2}-20 B \,b^{3} c^{2} d \,x^{2}+15 C \,a^{3} d^{3} x^{2}+85 C \,a^{2} b c \,d^{2} x^{2}+49 C a \,b^{2} c^{2} d \,x^{2}-5 C \,b^{3} c^{3} x^{2}-90 D a^{3} c \,d^{2} x^{2}-60 D a^{2} b \,c^{2} d \,x^{2}+6 D a \,b^{2} c^{3} x^{2}-6 A \,a^{2} b \,d^{3} x +60 A a \,b^{2} c \,d^{2} x +90 A \,b^{3} c^{2} d x +5 B \,a^{3} d^{3} x -49 B \,a^{2} b c \,d^{2} x -85 B a \,b^{2} c^{2} d x -15 B \,b^{3} c^{3} x +20 C \,a^{3} c \,d^{2} x +104 C \,a^{2} b \,c^{2} d x +20 C a \,b^{2} c^{3} x -120 D a^{3} c^{2} d x -24 D a^{2} b \,c^{3} x +3 a^{3} A \,d^{3}-15 A \,a^{2} b c \,d^{2}+45 A a \,b^{2} c^{2} d +15 A \,b^{3} c^{3}+2 B \,a^{3} c \,d^{2}-20 B \,a^{2} b \,c^{2} d -30 B a \,b^{2} c^{3}+8 a^{3} c^{2} C d +40 C \,a^{2} b \,c^{3}-48 D a^{3} c^{3}\right )}{15 \sqrt {b x +a}\, \left (x d +c \right )^{\frac {5}{2}} \left (d^{4} a^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +c^{4} b^{4}\right )}\)	\(591\)
orering	\(-\frac {2 \left (48 A \,b^{3} d^{3} x^{3}-40 B a \,b^{2} d^{3} x^{3}-8 B \,b^{3} c \,d^{2} x^{3}+30 C \,a^{2} b \,d^{3} x^{3}+20 C a \,b^{2} c \,d^{2} x^{3}-2 C \,b^{3} c^{2} d \,x^{3}-15 D a^{3} d^{3} x^{3}-45 D a^{2} b c \,d^{2} x^{3}+15 D a \,b^{2} c^{2} d \,x^{3}-3 D b^{3} c^{3} x^{3}+24 A a \,b^{2} d^{3} x^{2}+120 A \,b^{3} c \,d^{2} x^{2}-20 B \,a^{2} b \,d^{3} x^{2}-104 B a \,b^{2} c \,d^{2} x^{2}-20 B \,b^{3} c^{2} d \,x^{2}+15 C \,a^{3} d^{3} x^{2}+85 C \,a^{2} b c \,d^{2} x^{2}+49 C a \,b^{2} c^{2} d \,x^{2}-5 C \,b^{3} c^{3} x^{2}-90 D a^{3} c \,d^{2} x^{2}-60 D a^{2} b \,c^{2} d \,x^{2}+6 D a \,b^{2} c^{3} x^{2}-6 A \,a^{2} b \,d^{3} x +60 A a \,b^{2} c \,d^{2} x +90 A \,b^{3} c^{2} d x +5 B \,a^{3} d^{3} x -49 B \,a^{2} b c \,d^{2} x -85 B a \,b^{2} c^{2} d x -15 B \,b^{3} c^{3} x +20 C \,a^{3} c \,d^{2} x +104 C \,a^{2} b \,c^{2} d x +20 C a \,b^{2} c^{3} x -120 D a^{3} c^{2} d x -24 D a^{2} b \,c^{3} x +3 a^{3} A \,d^{3}-15 A \,a^{2} b c \,d^{2}+45 A a \,b^{2} c^{2} d +15 A \,b^{3} c^{3}+2 B \,a^{3} c \,d^{2}-20 B \,a^{2} b \,c^{2} d -30 B a \,b^{2} c^{3}+8 a^{3} c^{2} C d +40 C \,a^{2} b \,c^{3}-48 D a^{3} c^{3}\right )}{15 \sqrt {b x +a}\, \left (x d +c \right )^{\frac {5}{2}} \left (d^{4} a^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +c^{4} b^{4}\right )}\)	\(591\)

Input:

int((D*x^3+C*x^2+B*x+A)/(b*x+a)^(3/2)/(d*x+c)^(7/2),x,method=_RETURNVERBOS 
E)
 

Output:

-2/15*(48*A*b^3*d^3*x^3-40*B*a*b^2*d^3*x^3-8*B*b^3*c*d^2*x^3+30*C*a^2*b*d^ 
3*x^3+20*C*a*b^2*c*d^2*x^3-2*C*b^3*c^2*d*x^3-15*D*a^3*d^3*x^3-45*D*a^2*b*c 
*d^2*x^3+15*D*a*b^2*c^2*d*x^3-3*D*b^3*c^3*x^3+24*A*a*b^2*d^3*x^2+120*A*b^3 
*c*d^2*x^2-20*B*a^2*b*d^3*x^2-104*B*a*b^2*c*d^2*x^2-20*B*b^3*c^2*d*x^2+15* 
C*a^3*d^3*x^2+85*C*a^2*b*c*d^2*x^2+49*C*a*b^2*c^2*d*x^2-5*C*b^3*c^3*x^2-90 
*D*a^3*c*d^2*x^2-60*D*a^2*b*c^2*d*x^2+6*D*a*b^2*c^3*x^2-6*A*a^2*b*d^3*x+60 
*A*a*b^2*c*d^2*x+90*A*b^3*c^2*d*x+5*B*a^3*d^3*x-49*B*a^2*b*c*d^2*x-85*B*a* 
b^2*c^2*d*x-15*B*b^3*c^3*x+20*C*a^3*c*d^2*x+104*C*a^2*b*c^2*d*x+20*C*a*b^2 
*c^3*x-120*D*a^3*c^2*d*x-24*D*a^2*b*c^3*x+3*A*a^3*d^3-15*A*a^2*b*c*d^2+45* 
A*a*b^2*c^2*d+15*A*b^3*c^3+2*B*a^3*c*d^2-20*B*a^2*b*c^2*d-30*B*a*b^2*c^3+8 
*C*a^3*c^2*d+40*C*a^2*b*c^3-48*D*a^3*c^3)/(d*x+c)^(5/2)/(b*x+a)^(1/2)/(a*d 
-b*c)^4
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 748 vs. \(2 (334) = 668\).

Time = 33.65 (sec) , antiderivative size = 748, normalized size of antiderivative = 2.11 \[ \int \frac {A+B x+C x^2+D x^3}{(a+b x)^{3/2} (c+d x)^{7/2}} \, dx=-\frac {2 \, {\left (3 \, A a^{3} d^{3} - {\left (48 \, D a^{3} - 40 \, C a^{2} b + 30 \, B a b^{2} - 15 \, A b^{3}\right )} c^{3} + {\left (8 \, C a^{3} - 20 \, B a^{2} b + 45 \, A a b^{2}\right )} c^{2} d + {\left (2 \, B a^{3} - 15 \, A a^{2} b\right )} c d^{2} - {\left (3 \, D b^{3} c^{3} - {\left (15 \, D a b^{2} - 2 \, C b^{3}\right )} c^{2} d + {\left (45 \, D a^{2} b - 20 \, C a b^{2} + 8 \, B b^{3}\right )} c d^{2} + {\left (15 \, D a^{3} - 30 \, C a^{2} b + 40 \, B a b^{2} - 48 \, A b^{3}\right )} d^{3}\right )} x^{3} + {\left ({\left (6 \, D a b^{2} - 5 \, C b^{3}\right )} c^{3} - {\left (60 \, D a^{2} b - 49 \, C a b^{2} + 20 \, B b^{3}\right )} c^{2} d - {\left (90 \, D a^{3} - 85 \, C a^{2} b + 104 \, B a b^{2} - 120 \, A b^{3}\right )} c d^{2} + {\left (15 \, C a^{3} - 20 \, B a^{2} b + 24 \, A a b^{2}\right )} d^{3}\right )} x^{2} - {\left ({\left (24 \, D a^{2} b - 20 \, C a b^{2} + 15 \, B b^{3}\right )} c^{3} + {\left (120 \, D a^{3} - 104 \, C a^{2} b + 85 \, B a b^{2} - 90 \, A b^{3}\right )} c^{2} d - {\left (20 \, C a^{3} - 49 \, B a^{2} b + 60 \, A a b^{2}\right )} c d^{2} - {\left (5 \, B a^{3} - 6 \, A a^{2} b\right )} d^{3}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{15 \, {\left (a b^{4} c^{7} - 4 \, a^{2} b^{3} c^{6} d + 6 \, a^{3} b^{2} c^{5} d^{2} - 4 \, a^{4} b c^{4} d^{3} + a^{5} c^{3} d^{4} + {\left (b^{5} c^{4} d^{3} - 4 \, a b^{4} c^{3} d^{4} + 6 \, a^{2} b^{3} c^{2} d^{5} - 4 \, a^{3} b^{2} c d^{6} + a^{4} b d^{7}\right )} x^{4} + {\left (3 \, b^{5} c^{5} d^{2} - 11 \, a b^{4} c^{4} d^{3} + 14 \, a^{2} b^{3} c^{3} d^{4} - 6 \, a^{3} b^{2} c^{2} d^{5} - a^{4} b c d^{6} + a^{5} d^{7}\right )} x^{3} + 3 \, {\left (b^{5} c^{6} d - 3 \, a b^{4} c^{5} d^{2} + 2 \, a^{2} b^{3} c^{4} d^{3} + 2 \, a^{3} b^{2} c^{3} d^{4} - 3 \, a^{4} b c^{2} d^{5} + a^{5} c d^{6}\right )} x^{2} + {\left (b^{5} c^{7} - a b^{4} c^{6} d - 6 \, a^{2} b^{3} c^{5} d^{2} + 14 \, a^{3} b^{2} c^{4} d^{3} - 11 \, a^{4} b c^{3} d^{4} + 3 \, a^{5} c^{2} d^{5}\right )} x\right )}} \] Input:

integrate((D*x^3+C*x^2+B*x+A)/(b*x+a)^(3/2)/(d*x+c)^(7/2),x, algorithm="fr 
icas")
 

Output:

-2/15*(3*A*a^3*d^3 - (48*D*a^3 - 40*C*a^2*b + 30*B*a*b^2 - 15*A*b^3)*c^3 + 
 (8*C*a^3 - 20*B*a^2*b + 45*A*a*b^2)*c^2*d + (2*B*a^3 - 15*A*a^2*b)*c*d^2 
- (3*D*b^3*c^3 - (15*D*a*b^2 - 2*C*b^3)*c^2*d + (45*D*a^2*b - 20*C*a*b^2 + 
 8*B*b^3)*c*d^2 + (15*D*a^3 - 30*C*a^2*b + 40*B*a*b^2 - 48*A*b^3)*d^3)*x^3 
 + ((6*D*a*b^2 - 5*C*b^3)*c^3 - (60*D*a^2*b - 49*C*a*b^2 + 20*B*b^3)*c^2*d 
 - (90*D*a^3 - 85*C*a^2*b + 104*B*a*b^2 - 120*A*b^3)*c*d^2 + (15*C*a^3 - 2 
0*B*a^2*b + 24*A*a*b^2)*d^3)*x^2 - ((24*D*a^2*b - 20*C*a*b^2 + 15*B*b^3)*c 
^3 + (120*D*a^3 - 104*C*a^2*b + 85*B*a*b^2 - 90*A*b^3)*c^2*d - (20*C*a^3 - 
 49*B*a^2*b + 60*A*a*b^2)*c*d^2 - (5*B*a^3 - 6*A*a^2*b)*d^3)*x)*sqrt(b*x + 
 a)*sqrt(d*x + c)/(a*b^4*c^7 - 4*a^2*b^3*c^6*d + 6*a^3*b^2*c^5*d^2 - 4*a^4 
*b*c^4*d^3 + a^5*c^3*d^4 + (b^5*c^4*d^3 - 4*a*b^4*c^3*d^4 + 6*a^2*b^3*c^2* 
d^5 - 4*a^3*b^2*c*d^6 + a^4*b*d^7)*x^4 + (3*b^5*c^5*d^2 - 11*a*b^4*c^4*d^3 
 + 14*a^2*b^3*c^3*d^4 - 6*a^3*b^2*c^2*d^5 - a^4*b*c*d^6 + a^5*d^7)*x^3 + 3 
*(b^5*c^6*d - 3*a*b^4*c^5*d^2 + 2*a^2*b^3*c^4*d^3 + 2*a^3*b^2*c^3*d^4 - 3* 
a^4*b*c^2*d^5 + a^5*c*d^6)*x^2 + (b^5*c^7 - a*b^4*c^6*d - 6*a^2*b^3*c^5*d^ 
2 + 14*a^3*b^2*c^4*d^3 - 11*a^4*b*c^3*d^4 + 3*a^5*c^2*d^5)*x)
 

Sympy [F]

\[ \int \frac {A+B x+C x^2+D x^3}{(a+b x)^{3/2} (c+d x)^{7/2}} \, dx=\int \frac {A + B x + C x^{2} + D x^{3}}{\left (a + b x\right )^{\frac {3}{2}} \left (c + d x\right )^{\frac {7}{2}}}\, dx \] Input:

integrate((D*x**3+C*x**2+B*x+A)/(b*x+a)**(3/2)/(d*x+c)**(7/2),x)
 

Output:

Integral((A + B*x + C*x**2 + D*x**3)/((a + b*x)**(3/2)*(c + d*x)**(7/2)), 
x)
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {A+B x+C x^2+D x^3}{(a+b x)^{3/2} (c+d x)^{7/2}} \, dx=\text {Exception raised: ValueError} \] Input:

integrate((D*x^3+C*x^2+B*x+A)/(b*x+a)^(3/2)/(d*x+c)^(7/2),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. 2300 vs. \(2 (334) = 668\).

Time = 0.53 (sec) , antiderivative size = 2300, normalized size of antiderivative = 6.48 \[ \int \frac {A+B x+C x^2+D x^3}{(a+b x)^{3/2} (c+d x)^{7/2}} \, dx=\text {Too large to display} \] Input:

integrate((D*x^3+C*x^2+B*x+A)/(b*x+a)^(3/2)/(d*x+c)^(7/2),x, algorithm="gi 
ac")
 

Output:

4*(D^2*a^6*b^3*d - 2*C*D*a^5*b^4*d + C^2*a^4*b^5*d + 2*B*D*a^4*b^5*d - 2*B 
*C*a^3*b^6*d - 2*A*D*a^3*b^6*d + B^2*a^2*b^7*d + 2*A*C*a^2*b^7*d - 2*A*B*a 
*b^8*d + A^2*b^9*d)/((sqrt(b*d)*D*a^3*b^3*c - sqrt(b*d)*C*a^2*b^4*c + sqrt 
(b*d)*B*a*b^5*c - sqrt(b*d)*A*b^6*c - sqrt(b*d)*D*a^4*b^2*d + sqrt(b*d)*C* 
a^3*b^3*d - sqrt(b*d)*B*a^2*b^4*d + sqrt(b*d)*A*a*b^5*d - sqrt(b*d)*(sqrt( 
b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2*D*a^3*b + sqrt 
(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2*C* 
a^2*b^2 - sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d 
- a*b*d))^2*B*a*b^3 + sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b 
*x + a)*b*d - a*b*d))^2*A*b^4)*(b^3*c^3*abs(b) - 3*a*b^2*c^2*d*abs(b) + 3* 
a^2*b*c*d^2*abs(b) - a^3*d^3*abs(b))) + 2/15*((b*x + a)*((3*D*b^13*c^8*d^2 
 - 30*D*a*b^12*c^7*d^3 + 2*C*b^13*c^7*d^3 + 150*D*a^2*b^11*c^6*d^4 - 30*C* 
a*b^12*c^6*d^4 + 8*B*b^13*c^6*d^4 - 405*D*a^3*b^10*c^5*d^5 + 105*C*a^2*b^1 
1*c^5*d^5 - 15*B*a*b^12*c^5*d^5 - 33*A*b^13*c^5*d^5 + 615*D*a^4*b^9*c^4*d^ 
6 - 145*C*a^3*b^10*c^4*d^6 - 45*B*a^2*b^11*c^4*d^6 + 165*A*a*b^12*c^4*d^6 
- 528*D*a^5*b^8*c^3*d^7 + 60*C*a^4*b^9*c^3*d^7 + 170*B*a^3*b^10*c^3*d^7 - 
330*A*a^2*b^11*c^3*d^7 + 240*D*a^6*b^7*c^2*d^8 + 48*C*a^5*b^8*c^2*d^8 - 21 
0*B*a^4*b^9*c^2*d^8 + 330*A*a^3*b^10*c^2*d^8 - 45*D*a^7*b^6*c*d^9 - 55*C*a 
^6*b^7*c*d^9 + 117*B*a^5*b^8*c*d^9 - 165*A*a^4*b^9*c*d^9 + 15*C*a^7*b^6*d^ 
10 - 25*B*a^6*b^7*d^10 + 33*A*a^5*b^8*d^10)*(b*x + a)/(b^11*c^9*d^2*abs...
 

Mupad [F(-1)]

Timed out. \[ \int \frac {A+B x+C x^2+D x^3}{(a+b x)^{3/2} (c+d x)^{7/2}} \, dx=\int \frac {A+B\,x+C\,x^2+x^3\,D}{{\left (a+b\,x\right )}^{3/2}\,{\left (c+d\,x\right )}^{7/2}} \,d x \] Input:

int((A + B*x + C*x^2 + x^3*D)/((a + b*x)^(3/2)*(c + d*x)^(7/2)),x)
 

Output:

int((A + B*x + C*x^2 + x^3*D)/((a + b*x)^(3/2)*(c + d*x)^(7/2)), x)
 

Reduce [F]

\[ \int \frac {A+B x+C x^2+D x^3}{(a+b x)^{3/2} (c+d x)^{7/2}} \, dx=\int \frac {D x^{3}+C \,x^{2}+B x +A}{\left (b x +a \right )^{\frac {3}{2}} \left (d x +c \right )^{\frac {7}{2}}}d x \] Input:

int((D*x^3+C*x^2+B*x+A)/(b*x+a)^(3/2)/(d*x+c)^(7/2),x)
 

Output:

int((D*x^3+C*x^2+B*x+A)/(b*x+a)^(3/2)/(d*x+c)^(7/2),x)