\(\int \frac {A+B x+C x^2+D x^3}{\sqrt {a+b x} (c+d x)^{3/2}} \, dx\) [134]

Optimal result

Integrand size = 34, antiderivative size = 223 \[ \int \frac {A+B x+C x^2+D x^3}{\sqrt {a+b x} (c+d x)^{3/2}} \, dx=\frac {2 \left (c^2 C d-B c d^2+A d^3-c^3 D\right ) \sqrt {a+b x}}{d^3 (b c-a d) \sqrt {c+d x}}+\frac {(4 b C d-9 b c D-3 a d D) \sqrt {a+b x} \sqrt {c+d x}}{4 b^2 d^3}+\frac {D \sqrt {a+b x} (c+d x)^{3/2}}{2 b d^3}+\frac {\left (3 a^2 d^2 D-2 a b d (2 C d-3 c D)-b^2 \left (12 c C d-8 B d^2-15 c^2 D\right )\right ) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{4 b^{5/2} d^{7/2}} \] Output:

2*(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)^(1/2) 
+1/4*(4*C*b*d-3*D*a*d-9*D*b*c)*(b*x+a)^(1/2)*(d*x+c)^(1/2)/b^2/d^3+1/2*D*( 
b*x+a)^(1/2)*(d*x+c)^(3/2)/b/d^3+1/4*(3*a^2*d^2*D-2*a*b*d*(2*C*d-3*D*c)-b^ 
2*(-8*B*d^2+12*C*c*d-15*D*c^2))*arctanh(d^(1/2)*(b*x+a)^(1/2)/b^(1/2)/(d*x 
+c)^(1/2))/b^(5/2)/d^(7/2)
 

Mathematica [A] (verified)

Time = 0.63 (sec) , antiderivative size = 237, normalized size of antiderivative = 1.06 \[ \int \frac {A+B x+C x^2+D x^3}{\sqrt {a+b x} (c+d x)^{3/2}} \, dx=\frac {-\sqrt {b} \sqrt {d} \sqrt {a+b x} \left (3 a^2 d^2 D (c+d x)-2 a b d (c+d x) (2 C d-2 c D+d D x)+b^2 \left (8 A d^3-15 c^3 D+c^2 d (12 C-5 D x)+2 c d^2 \left (-4 B+2 C x+D x^2\right )\right )\right )-(b c-a d) \left (3 a^2 d^2 D+2 a b d (-2 C d+3 c D)+b^2 \left (-12 c C d+8 B d^2+15 c^2 D\right )\right ) \sqrt {c+d x} \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{4 b^{5/2} d^{7/2} (-b c+a d) \sqrt {c+d x}} \] Input:

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

Output:

(-(Sqrt[b]*Sqrt[d]*Sqrt[a + b*x]*(3*a^2*d^2*D*(c + d*x) - 2*a*b*d*(c + d*x 
)*(2*C*d - 2*c*D + d*D*x) + b^2*(8*A*d^3 - 15*c^3*D + c^2*d*(12*C - 5*D*x) 
 + 2*c*d^2*(-4*B + 2*C*x + D*x^2)))) - (b*c - a*d)*(3*a^2*d^2*D + 2*a*b*d* 
(-2*C*d + 3*c*D) + b^2*(-12*c*C*d + 8*B*d^2 + 15*c^2*D))*Sqrt[c + d*x]*Arc 
Tanh[(Sqrt[d]*Sqrt[a + b*x])/(Sqrt[b]*Sqrt[c + d*x])])/(4*b^(5/2)*d^(7/2)* 
(-(b*c) + a*d)*Sqrt[c + d*x])
 

Rubi [A] (verified)

Time = 0.58 (sec) , antiderivative size = 253, normalized size of antiderivative = 1.13, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.206, Rules used = {2124, 27, 1194, 27, 90, 66, 221}

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.

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

\(\Big \downarrow \) 2124

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1194

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 90

\(\displaystyle \frac {2 \sqrt {a+b x} \left (A d^3-B c d^2+c^3 (-D)+c^2 C d\right )}{d^3 \sqrt {c+d x} (b c-a d)}-\frac {\frac {(b c-a d) \left (-\frac {1}{2} \left (3 a^2 d^2 D-2 a b d (2 C d-3 c D)-\left (b^2 \left (-8 B d^2-15 c^2 D+12 c C d\right )\right )\right ) \int \frac {1}{\sqrt {a+b x} \sqrt {c+d x}}dx-\sqrt {a+b x} \sqrt {c+d x} (-5 a d D-7 b c D+4 b C d)\right )}{4 b^2 d^3}-\frac {D (a+b x)^{3/2} \sqrt {c+d x} (b c-a d)}{2 b^2 d^2}}{b c-a d}\)

\(\Big \downarrow \) 66

\(\displaystyle \frac {2 \sqrt {a+b x} \left (A d^3-B c d^2+c^3 (-D)+c^2 C d\right )}{d^3 \sqrt {c+d x} (b c-a d)}-\frac {\frac {(b c-a d) \left (-\left (3 a^2 d^2 D-2 a b d (2 C d-3 c D)-\left (b^2 \left (-8 B d^2-15 c^2 D+12 c C d\right )\right )\right ) \int \frac {1}{b-\frac {d (a+b x)}{c+d x}}d\frac {\sqrt {a+b x}}{\sqrt {c+d x}}-\sqrt {a+b x} \sqrt {c+d x} (-5 a d D-7 b c D+4 b C d)\right )}{4 b^2 d^3}-\frac {D (a+b x)^{3/2} \sqrt {c+d x} (b c-a d)}{2 b^2 d^2}}{b c-a d}\)

\(\Big \downarrow \) 221

\(\displaystyle \frac {2 \sqrt {a+b x} \left (A d^3-B c d^2+c^3 (-D)+c^2 C d\right )}{d^3 \sqrt {c+d x} (b c-a d)}-\frac {\frac {(b c-a d) \left (-\frac {\text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right ) \left (3 a^2 d^2 D-2 a b d (2 C d-3 c D)-\left (b^2 \left (-8 B d^2-15 c^2 D+12 c C d\right )\right )\right )}{\sqrt {b} \sqrt {d}}-\sqrt {a+b x} \sqrt {c+d x} (-5 a d D-7 b c D+4 b C d)\right )}{4 b^2 d^3}-\frac {D (a+b x)^{3/2} \sqrt {c+d x} (b c-a d)}{2 b^2 d^2}}{b c-a d}\)

Input:

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

Output:

(2*(c^2*C*d - B*c*d^2 + A*d^3 - c^3*D)*Sqrt[a + b*x])/(d^3*(b*c - a*d)*Sqr 
t[c + d*x]) - (-1/2*((b*c - a*d)*D*(a + b*x)^(3/2)*Sqrt[c + d*x])/(b^2*d^2 
) + ((b*c - a*d)*(-((4*b*C*d - 7*b*c*D - 5*a*d*D)*Sqrt[a + b*x]*Sqrt[c + d 
*x]) - ((3*a^2*d^2*D - 2*a*b*d*(2*C*d - 3*c*D) - b^2*(12*c*C*d - 8*B*d^2 - 
 15*c^2*D))*ArcTanh[(Sqrt[d]*Sqrt[a + b*x])/(Sqrt[b]*Sqrt[c + d*x])])/(Sqr 
t[b]*Sqrt[d])))/(4*b^2*d^3))/(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 66

Int[1/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]), x_Symbol] :> Simp[ 
2   Subst[Int[1/(b - d*x^2), x], x, Sqrt[a + b*x]/Sqrt[c + d*x]], x] /; Fre 
eQ[{a, b, c, d}, x] &&  !GtQ[c - a*(d/b), 0]
 

rule 90

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

rule 221

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x 
/Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
 

rule 1194

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_)*((a_.) + (b_.)*(x_) 
 + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[c^p*(d + e*x)^(m + 2*p)*((f + g*x 
)^(n + 1)/(g*e^(2*p)*(m + n + 2*p + 1))), x] + Simp[1/(g*e^(2*p)*(m + n + 2 
*p + 1))   Int[(d + e*x)^m*(f + g*x)^n*ExpandToSum[g*(m + n + 2*p + 1)*(e^( 
2*p)*(a + b*x + c*x^2)^p - c^p*(d + e*x)^(2*p)) - c^p*(e*f - d*g)*(m + 2*p) 
*(d + e*x)^(2*p - 1), x], x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && IGtQ 
[p, 0] &&  !IntegerQ[m] &&  !IntegerQ[n] && NeQ[m + n + 2*p + 1, 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 [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1369\) vs. \(2(193)=386\).

Time = 0.61 (sec) , antiderivative size = 1370, normalized size of antiderivative = 6.14

method	result	size
default	\(\text {Expression too large to display}\)	\(1370\)

Input:

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

Output:

-1/8*(b*x+a)^(1/2)*(6*D*a^2*d^3*x*((b*x+a)*(d*x+c))^(1/2)*(d*b)^(1/2)-16*B 
*b^2*c*d^2*((b*x+a)*(d*x+c))^(1/2)*(d*b)^(1/2)+6*D*a^2*c*d^2*((b*x+a)*(d*x 
+c))^(1/2)*(d*b)^(1/2)-8*B*ln(1/2*(2*b*d*x+2*((b*x+a)*(d*x+c))^(1/2)*(d*b) 
^(1/2)+a*d+b*c)/(d*b)^(1/2))*a*b^2*d^4*x+8*B*ln(1/2*(2*b*d*x+2*((b*x+a)*(d 
*x+c))^(1/2)*(d*b)^(1/2)+a*d+b*c)/(d*b)^(1/2))*b^3*c*d^3*x+24*C*b^2*c^2*d* 
((b*x+a)*(d*x+c))^(1/2)*(d*b)^(1/2)-8*C*a*b*d^3*x*((b*x+a)*(d*x+c))^(1/2)* 
(d*b)^(1/2)-30*D*b^2*c^3*((b*x+a)*(d*x+c))^(1/2)*(d*b)^(1/2)-3*D*ln(1/2*(2 
*b*d*x+2*((b*x+a)*(d*x+c))^(1/2)*(d*b)^(1/2)+a*d+b*c)/(d*b)^(1/2))*a^3*d^4 
*x+8*B*ln(1/2*(2*b*d*x+2*((b*x+a)*(d*x+c))^(1/2)*(d*b)^(1/2)+a*d+b*c)/(d*b 
)^(1/2))*b^3*c^2*d^2-12*C*ln(1/2*(2*b*d*x+2*((b*x+a)*(d*x+c))^(1/2)*(d*b)^ 
(1/2)+a*d+b*c)/(d*b)^(1/2))*b^3*c^3*d-3*D*ln(1/2*(2*b*d*x+2*((b*x+a)*(d*x+ 
c))^(1/2)*(d*b)^(1/2)+a*d+b*c)/(d*b)^(1/2))*a^3*c*d^3+16*A*b^2*d^3*((b*x+a 
)*(d*x+c))^(1/2)*(d*b)^(1/2)+4*D*a*b*c*d^2*x*((b*x+a)*(d*x+c))^(1/2)*(d*b) 
^(1/2)-9*D*ln(1/2*(2*b*d*x+2*((b*x+a)*(d*x+c))^(1/2)*(d*b)^(1/2)+a*d+b*c)/ 
(d*b)^(1/2))*a*b^2*c^2*d^2*x-4*D*a*b*d^3*x^2*((b*x+a)*(d*x+c))^(1/2)*(d*b) 
^(1/2)+4*D*b^2*c*d^2*x^2*((b*x+a)*(d*x+c))^(1/2)*(d*b)^(1/2)+8*C*b^2*c*d^2 
*x*((b*x+a)*(d*x+c))^(1/2)*(d*b)^(1/2)-10*D*b^2*c^2*d*x*((b*x+a)*(d*x+c))^ 
(1/2)*(d*b)^(1/2)-8*C*a*b*c*d^2*((b*x+a)*(d*x+c))^(1/2)*(d*b)^(1/2)+8*D*a* 
b*c^2*d*((b*x+a)*(d*x+c))^(1/2)*(d*b)^(1/2)+8*C*ln(1/2*(2*b*d*x+2*((b*x+a) 
*(d*x+c))^(1/2)*(d*b)^(1/2)+a*d+b*c)/(d*b)^(1/2))*a*b^2*c*d^3*x-3*D*ln(...
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 458 vs. \(2 (192) = 384\).

Time = 3.12 (sec) , antiderivative size = 930, normalized size of antiderivative = 4.17 \[ \int \frac {A+B x+C x^2+D x^3}{\sqrt {a+b x} (c+d x)^{3/2}} \, dx =\text {Too large to display} \] Input:

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

Output:

[1/16*((15*D*b^3*c^4 - 3*(3*D*a*b^2 + 4*C*b^3)*c^3*d - (3*D*a^2*b - 8*C*a* 
b^2 - 8*B*b^3)*c^2*d^2 - (3*D*a^3 - 4*C*a^2*b + 8*B*a*b^2)*c*d^3 + (15*D*b 
^3*c^3*d - 3*(3*D*a*b^2 + 4*C*b^3)*c^2*d^2 - (3*D*a^2*b - 8*C*a*b^2 - 8*B* 
b^3)*c*d^3 - (3*D*a^3 - 4*C*a^2*b + 8*B*a*b^2)*d^4)*x)*sqrt(b*d)*log(8*b^2 
*d^2*x^2 + b^2*c^2 + 6*a*b*c*d + a^2*d^2 + 4*(2*b*d*x + b*c + a*d)*sqrt(b* 
d)*sqrt(b*x + a)*sqrt(d*x + c) + 8*(b^2*c*d + a*b*d^2)*x) - 4*(15*D*b^3*c^ 
3*d - 8*A*b^3*d^4 - 4*(D*a*b^2 + 3*C*b^3)*c^2*d^2 - (3*D*a^2*b - 4*C*a*b^2 
 - 8*B*b^3)*c*d^3 - 2*(D*b^3*c*d^3 - D*a*b^2*d^4)*x^2 + (5*D*b^3*c^2*d^2 - 
 2*(D*a*b^2 + 2*C*b^3)*c*d^3 - (3*D*a^2*b - 4*C*a*b^2)*d^4)*x)*sqrt(b*x + 
a)*sqrt(d*x + c))/(b^4*c^2*d^4 - a*b^3*c*d^5 + (b^4*c*d^5 - a*b^3*d^6)*x), 
 -1/8*((15*D*b^3*c^4 - 3*(3*D*a*b^2 + 4*C*b^3)*c^3*d - (3*D*a^2*b - 8*C*a* 
b^2 - 8*B*b^3)*c^2*d^2 - (3*D*a^3 - 4*C*a^2*b + 8*B*a*b^2)*c*d^3 + (15*D*b 
^3*c^3*d - 3*(3*D*a*b^2 + 4*C*b^3)*c^2*d^2 - (3*D*a^2*b - 8*C*a*b^2 - 8*B* 
b^3)*c*d^3 - (3*D*a^3 - 4*C*a^2*b + 8*B*a*b^2)*d^4)*x)*sqrt(-b*d)*arctan(1 
/2*(2*b*d*x + b*c + a*d)*sqrt(-b*d)*sqrt(b*x + a)*sqrt(d*x + c)/(b^2*d^2*x 
^2 + a*b*c*d + (b^2*c*d + a*b*d^2)*x)) + 2*(15*D*b^3*c^3*d - 8*A*b^3*d^4 - 
 4*(D*a*b^2 + 3*C*b^3)*c^2*d^2 - (3*D*a^2*b - 4*C*a*b^2 - 8*B*b^3)*c*d^3 - 
 2*(D*b^3*c*d^3 - D*a*b^2*d^4)*x^2 + (5*D*b^3*c^2*d^2 - 2*(D*a*b^2 + 2*C*b 
^3)*c*d^3 - (3*D*a^2*b - 4*C*a*b^2)*d^4)*x)*sqrt(b*x + a)*sqrt(d*x + c))/( 
b^4*c^2*d^4 - a*b^3*c*d^5 + (b^4*c*d^5 - a*b^3*d^6)*x)]
 

Sympy [F]

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

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

Output:

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

Maxima [F(-2)]

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

integrate((D*x^3+C*x^2+B*x+A)/(b*x+a)^(1/2)/(d*x+c)^(3/2),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. 437 vs. \(2 (192) = 384\).

Time = 0.19 (sec) , antiderivative size = 437, normalized size of antiderivative = 1.96 \[ \int \frac {A+B x+C x^2+D x^3}{\sqrt {a+b x} (c+d x)^{3/2}} \, dx=\frac {{\left ({\left (b x + a\right )} {\left (\frac {2 \, {\left (D b^{6} c d^{4} {\left | b \right |} - D a b^{5} d^{5} {\left | b \right |}\right )} {\left (b x + a\right )}}{b^{9} c d^{5} - a b^{8} d^{6}} - \frac {5 \, D b^{7} c^{2} d^{3} {\left | b \right |} + 2 \, D a b^{6} c d^{4} {\left | b \right |} - 4 \, C b^{7} c d^{4} {\left | b \right |} - 7 \, D a^{2} b^{5} d^{5} {\left | b \right |} + 4 \, C a b^{6} d^{5} {\left | b \right |}}{b^{9} c d^{5} - a b^{8} d^{6}}\right )} - \frac {15 \, D b^{8} c^{3} d^{2} {\left | b \right |} - 9 \, D a b^{7} c^{2} d^{3} {\left | b \right |} - 12 \, C b^{8} c^{2} d^{3} {\left | b \right |} - 3 \, D a^{2} b^{6} c d^{4} {\left | b \right |} + 8 \, C a b^{7} c d^{4} {\left | b \right |} + 8 \, B b^{8} c d^{4} {\left | b \right |} + 5 \, D a^{3} b^{5} d^{5} {\left | b \right |} - 4 \, C a^{2} b^{6} d^{5} {\left | b \right |} - 8 \, A b^{8} d^{5} {\left | b \right |}}{b^{9} c d^{5} - a b^{8} d^{6}}\right )} \sqrt {b x + a}}{4 \, \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}} - \frac {{\left (15 \, D b^{2} c^{2} {\left | b \right |} + 6 \, D a b c d {\left | b \right |} - 12 \, C b^{2} c d {\left | b \right |} + 3 \, D a^{2} d^{2} {\left | b \right |} - 4 \, C a b d^{2} {\left | b \right |} + 8 \, B b^{2} d^{2} {\left | b \right |}\right )} \log \left ({\left | -\sqrt {b d} \sqrt {b x + a} + \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d} \right |}\right )}{4 \, \sqrt {b d} b^{3} d^{3}} \] Input:

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

Output:

1/4*((b*x + a)*(2*(D*b^6*c*d^4*abs(b) - D*a*b^5*d^5*abs(b))*(b*x + a)/(b^9 
*c*d^5 - a*b^8*d^6) - (5*D*b^7*c^2*d^3*abs(b) + 2*D*a*b^6*c*d^4*abs(b) - 4 
*C*b^7*c*d^4*abs(b) - 7*D*a^2*b^5*d^5*abs(b) + 4*C*a*b^6*d^5*abs(b))/(b^9* 
c*d^5 - a*b^8*d^6)) - (15*D*b^8*c^3*d^2*abs(b) - 9*D*a*b^7*c^2*d^3*abs(b) 
- 12*C*b^8*c^2*d^3*abs(b) - 3*D*a^2*b^6*c*d^4*abs(b) + 8*C*a*b^7*c*d^4*abs 
(b) + 8*B*b^8*c*d^4*abs(b) + 5*D*a^3*b^5*d^5*abs(b) - 4*C*a^2*b^6*d^5*abs( 
b) - 8*A*b^8*d^5*abs(b))/(b^9*c*d^5 - a*b^8*d^6))*sqrt(b*x + a)/sqrt(b^2*c 
 + (b*x + a)*b*d - a*b*d) - 1/4*(15*D*b^2*c^2*abs(b) + 6*D*a*b*c*d*abs(b) 
- 12*C*b^2*c*d*abs(b) + 3*D*a^2*d^2*abs(b) - 4*C*a*b*d^2*abs(b) + 8*B*b^2* 
d^2*abs(b))*log(abs(-sqrt(b*d)*sqrt(b*x + a) + sqrt(b^2*c + (b*x + a)*b*d 
- a*b*d)))/(sqrt(b*d)*b^3*d^3)
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.17 (sec) , antiderivative size = 573, normalized size of antiderivative = 2.57 \[ \int \frac {A+B x+C x^2+D x^3}{\sqrt {a+b x} (c+d x)^{3/2}} \, dx=\frac {-3 \sqrt {d x +c}\, \sqrt {b x +a}\, a b c \,d^{2}-3 \sqrt {d x +c}\, \sqrt {b x +a}\, a b \,d^{3} x -8 \sqrt {d x +c}\, \sqrt {b x +a}\, b^{3} d^{2}-3 \sqrt {d x +c}\, \sqrt {b x +a}\, b^{2} c^{2} d -\sqrt {d x +c}\, \sqrt {b x +a}\, b^{2} c \,d^{2} x +2 \sqrt {d x +c}\, \sqrt {b x +a}\, b^{2} d^{3} x^{2}+3 \sqrt {d}\, \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {d}\, \sqrt {b x +a}+\sqrt {b}\, \sqrt {d x +c}}{\sqrt {a d -b c}}\right ) a^{2} c \,d^{2}+3 \sqrt {d}\, \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {d}\, \sqrt {b x +a}+\sqrt {b}\, \sqrt {d x +c}}{\sqrt {a d -b c}}\right ) a^{2} d^{3} x +2 \sqrt {d}\, \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {d}\, \sqrt {b x +a}+\sqrt {b}\, \sqrt {d x +c}}{\sqrt {a d -b c}}\right ) a b \,c^{2} d +2 \sqrt {d}\, \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {d}\, \sqrt {b x +a}+\sqrt {b}\, \sqrt {d x +c}}{\sqrt {a d -b c}}\right ) a b c \,d^{2} x +8 \sqrt {d}\, \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {d}\, \sqrt {b x +a}+\sqrt {b}\, \sqrt {d x +c}}{\sqrt {a d -b c}}\right ) b^{3} c d +8 \sqrt {d}\, \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {d}\, \sqrt {b x +a}+\sqrt {b}\, \sqrt {d x +c}}{\sqrt {a d -b c}}\right ) b^{3} d^{2} x +3 \sqrt {d}\, \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {d}\, \sqrt {b x +a}+\sqrt {b}\, \sqrt {d x +c}}{\sqrt {a d -b c}}\right ) b^{2} c^{3}+3 \sqrt {d}\, \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {d}\, \sqrt {b x +a}+\sqrt {b}\, \sqrt {d x +c}}{\sqrt {a d -b c}}\right ) b^{2} c^{2} d x +\sqrt {d}\, \sqrt {b}\, a^{2} c \,d^{2}+\sqrt {d}\, \sqrt {b}\, a^{2} d^{3} x -8 \sqrt {d}\, \sqrt {b}\, b^{3} c d -8 \sqrt {d}\, \sqrt {b}\, b^{3} d^{2} x -\sqrt {d}\, \sqrt {b}\, b^{2} c^{3}-\sqrt {d}\, \sqrt {b}\, b^{2} c^{2} d x}{4 b^{3} d^{3} \left (d x +c \right )} \] Input:

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

Output:

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