\(\int \frac {(a+b x)^{3/2} (A+B x+C x^2+D x^3)}{(c+d x)^{5/2}} \, dx\) [114]

Optimal result

Integrand size = 34, antiderivative size = 399 \[ \int \frac {(a+b x)^{3/2} \left (A+B x+C x^2+D x^3\right )}{(c+d x)^{5/2}} \, dx=-\frac {2 \left (c^2 C d-B c d^2+A d^3-c^3 D\right ) (a+b x)^{3/2}}{3 d^4 (c+d x)^{3/2}}+\frac {2 \left (a d \left (2 c C d-B d^2-3 c^2 D\right )-b \left (3 c^2 C d-2 B c d^2+A d^3-4 c^3 D\right )\right ) \sqrt {a+b x}}{d^5 \sqrt {c+d x}}-\frac {\left (a^2 d^2 D-2 a b d (3 C d-8 c D)+b^2 \left (22 c C d-8 B d^2-41 c^2 D\right )\right ) \sqrt {a+b x} \sqrt {c+d x}}{8 b d^5}+\frac {(6 b C d-17 b c D-a d D) (a+b x)^{3/2} \sqrt {c+d x}}{12 b d^4}+\frac {D (a+b x)^{5/2} \sqrt {c+d x}}{3 b d^3}-\frac {\left (a^3 d^3 D-3 a^2 b d^2 (2 C d-5 c D)+3 a b^2 d \left (20 c C d-8 B d^2-35 c^2 D\right )-b^3 \left (70 c^2 C d-40 B c d^2+16 A d^3-105 c^3 D\right )\right ) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{8 b^{3/2} d^{11/2}} \] Output:

-2/3*(A*d^3-B*c*d^2+C*c^2*d-D*c^3)*(b*x+a)^(3/2)/d^4/(d*x+c)^(3/2)+2*(a*d* 
(-B*d^2+2*C*c*d-3*D*c^2)-b*(A*d^3-2*B*c*d^2+3*C*c^2*d-4*D*c^3))*(b*x+a)^(1 
/2)/d^5/(d*x+c)^(1/2)-1/8*(a^2*d^2*D-2*a*b*d*(3*C*d-8*D*c)+b^2*(-8*B*d^2+2 
2*C*c*d-41*D*c^2))*(b*x+a)^(1/2)*(d*x+c)^(1/2)/b/d^5+1/12*(6*C*b*d-D*a*d-1 
7*D*b*c)*(b*x+a)^(3/2)*(d*x+c)^(1/2)/b/d^4+1/3*D*(b*x+a)^(5/2)*(d*x+c)^(1/ 
2)/b/d^3-1/8*(a^3*d^3*D-3*a^2*b*d^2*(2*C*d-5*D*c)+3*a*b^2*d*(-8*B*d^2+20*C 
*c*d-35*D*c^2)-b^3*(16*A*d^3-40*B*c*d^2+70*C*c^2*d-105*D*c^3))*arctanh(d^( 
1/2)*(b*x+a)^(1/2)/b^(1/2)/(d*x+c)^(1/2))/b^(3/2)/d^(11/2)
 

Mathematica [A] (verified)

Time = 1.04 (sec) , antiderivative size = 339, normalized size of antiderivative = 0.85 \[ \int \frac {(a+b x)^{3/2} \left (A+B x+C x^2+D x^3\right )}{(c+d x)^{5/2}} \, dx=\frac {\sqrt {a+b x} \left (3 a^2 d^2 D (c+d x)^2-2 a b d \left (105 c^3 D+c^2 (-55 C d+147 d D x)+c d^2 \left (16 B-78 C x+27 D x^2\right )+d^3 \left (8 A+24 B x-15 C x^2-7 D x^3\right )\right )+b^2 \left (315 c^4 D-210 c^3 d (C-2 D x)+c^2 d^2 (120 B+7 x (-40 C+9 D x))+4 d^4 x \left (-16 A+x \left (6 B+3 C x+2 D x^2\right )\right )-2 c d^3 \left (24 A+x \left (-80 B+21 C x+9 D x^2\right )\right )\right )\right )}{24 b d^5 (c+d x)^{3/2}}-\frac {\left (a^3 d^3 D+3 a^2 b d^2 (-2 C d+5 c D)-3 a b^2 d \left (-20 c C d+8 B d^2+35 c^2 D\right )+b^3 \left (-70 c^2 C d+40 B c d^2-16 A d^3+105 c^3 D\right )\right ) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {d} \sqrt {a+b x}}\right )}{8 b^{3/2} d^{11/2}} \] Input:

Integrate[((a + b*x)^(3/2)*(A + B*x + C*x^2 + D*x^3))/(c + d*x)^(5/2),x]
 

Output:

(Sqrt[a + b*x]*(3*a^2*d^2*D*(c + d*x)^2 - 2*a*b*d*(105*c^3*D + c^2*(-55*C* 
d + 147*d*D*x) + c*d^2*(16*B - 78*C*x + 27*D*x^2) + d^3*(8*A + 24*B*x - 15 
*C*x^2 - 7*D*x^3)) + b^2*(315*c^4*D - 210*c^3*d*(C - 2*D*x) + c^2*d^2*(120 
*B + 7*x*(-40*C + 9*D*x)) + 4*d^4*x*(-16*A + x*(6*B + 3*C*x + 2*D*x^2)) - 
2*c*d^3*(24*A + x*(-80*B + 21*C*x + 9*D*x^2)))))/(24*b*d^5*(c + d*x)^(3/2) 
) - ((a^3*d^3*D + 3*a^2*b*d^2*(-2*C*d + 5*c*D) - 3*a*b^2*d*(-20*c*C*d + 8* 
B*d^2 + 35*c^2*D) + b^3*(-70*c^2*C*d + 40*B*c*d^2 - 16*A*d^3 + 105*c^3*D)) 
*ArcTanh[(Sqrt[b]*Sqrt[c + d*x])/(Sqrt[d]*Sqrt[a + b*x])])/(8*b^(3/2)*d^(1 
1/2))
 

Rubi [A] (verified)

Time = 0.75 (sec) , antiderivative size = 411, normalized size of antiderivative = 1.03, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.265, Rules used = {2124, 27, 1193, 27, 90, 60, 60, 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.

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

\(\Big \downarrow \) 2124

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1193

\(\displaystyle \frac {\frac {2 \int \frac {(a+b x)^{3/2} \left (\left (-50 D c^3+35 C d c^2-20 B d^2 c+8 A d^3\right ) b^2-6 a d \left (-8 D c^2+5 C d c-2 B d^2\right ) b+3 a^2 d^2 (C d-2 c D)+3 d (b c-a d)^2 D x\right )}{2 d^3 \sqrt {c+d x}}dx}{b c-a d}+\frac {2 (a+b x)^{5/2} \left (3 a d \left (-B d^2-3 c^2 D+2 c C d\right )-b \left (2 A d^3-5 B c d^2-11 c^3 D+8 c^2 C d\right )\right )}{d^3 \sqrt {c+d x} (b c-a d)}}{3 (b c-a d)}+\frac {2 (a+b x)^{5/2} \left (A+\frac {c \left (-B d^2+c^2 (-D)+c C d\right )}{d^3}\right )}{3 (c+d x)^{3/2} (b c-a d)}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {\int \frac {(a+b x)^{3/2} \left (\left (-50 D c^3+35 C d c^2-20 B d^2 c+8 A d^3\right ) b^2-6 a d \left (-8 D c^2+5 C d c-2 B d^2\right ) b+3 a^2 d^2 (C d-2 c D)+3 d (b c-a d)^2 D x\right )}{\sqrt {c+d x}}dx}{d^3 (b c-a d)}+\frac {2 (a+b x)^{5/2} \left (3 a d \left (-B d^2-3 c^2 D+2 c C d\right )-b \left (2 A d^3-5 B c d^2-11 c^3 D+8 c^2 C d\right )\right )}{d^3 \sqrt {c+d x} (b c-a d)}}{3 (b c-a d)}+\frac {2 (a+b x)^{5/2} \left (A+\frac {c \left (-B d^2+c^2 (-D)+c C d\right )}{d^3}\right )}{3 (c+d x)^{3/2} (b c-a d)}\)

\(\Big \downarrow \) 90

\(\displaystyle \frac {\frac {\frac {D (a+b x)^{5/2} \sqrt {c+d x} (b c-a d)^2}{b}-\frac {\left (a^3 d^3 D-3 a^2 b d^2 (2 C d-5 c D)+3 a b^2 d \left (-8 B d^2-35 c^2 D+20 c C d\right )-\left (b^3 \left (16 A d^3-40 B c d^2-105 c^3 D+70 c^2 C d\right )\right )\right ) \int \frac {(a+b x)^{3/2}}{\sqrt {c+d x}}dx}{2 b}}{d^3 (b c-a d)}+\frac {2 (a+b x)^{5/2} \left (3 a d \left (-B d^2-3 c^2 D+2 c C d\right )-b \left (2 A d^3-5 B c d^2-11 c^3 D+8 c^2 C d\right )\right )}{d^3 \sqrt {c+d x} (b c-a d)}}{3 (b c-a d)}+\frac {2 (a+b x)^{5/2} \left (A+\frac {c \left (-B d^2+c^2 (-D)+c C d\right )}{d^3}\right )}{3 (c+d x)^{3/2} (b c-a d)}\)

\(\Big \downarrow \) 60

\(\displaystyle \frac {\frac {\frac {D (a+b x)^{5/2} \sqrt {c+d x} (b c-a d)^2}{b}-\frac {\left (a^3 d^3 D-3 a^2 b d^2 (2 C d-5 c D)+3 a b^2 d \left (-8 B d^2-35 c^2 D+20 c C d\right )-\left (b^3 \left (16 A d^3-40 B c d^2-105 c^3 D+70 c^2 C d\right )\right )\right ) \left (\frac {(a+b x)^{3/2} \sqrt {c+d x}}{2 d}-\frac {3 (b c-a d) \int \frac {\sqrt {a+b x}}{\sqrt {c+d x}}dx}{4 d}\right )}{2 b}}{d^3 (b c-a d)}+\frac {2 (a+b x)^{5/2} \left (3 a d \left (-B d^2-3 c^2 D+2 c C d\right )-b \left (2 A d^3-5 B c d^2-11 c^3 D+8 c^2 C d\right )\right )}{d^3 \sqrt {c+d x} (b c-a d)}}{3 (b c-a d)}+\frac {2 (a+b x)^{5/2} \left (A+\frac {c \left (-B d^2+c^2 (-D)+c C d\right )}{d^3}\right )}{3 (c+d x)^{3/2} (b c-a d)}\)

\(\Big \downarrow \) 60

\(\displaystyle \frac {\frac {\frac {D (a+b x)^{5/2} \sqrt {c+d x} (b c-a d)^2}{b}-\frac {\left (a^3 d^3 D-3 a^2 b d^2 (2 C d-5 c D)+3 a b^2 d \left (-8 B d^2-35 c^2 D+20 c C d\right )-\left (b^3 \left (16 A d^3-40 B c d^2-105 c^3 D+70 c^2 C d\right )\right )\right ) \left (\frac {(a+b x)^{3/2} \sqrt {c+d x}}{2 d}-\frac {3 (b c-a d) \left (\frac {\sqrt {a+b x} \sqrt {c+d x}}{d}-\frac {(b c-a d) \int \frac {1}{\sqrt {a+b x} \sqrt {c+d x}}dx}{2 d}\right )}{4 d}\right )}{2 b}}{d^3 (b c-a d)}+\frac {2 (a+b x)^{5/2} \left (3 a d \left (-B d^2-3 c^2 D+2 c C d\right )-b \left (2 A d^3-5 B c d^2-11 c^3 D+8 c^2 C d\right )\right )}{d^3 \sqrt {c+d x} (b c-a d)}}{3 (b c-a d)}+\frac {2 (a+b x)^{5/2} \left (A+\frac {c \left (-B d^2+c^2 (-D)+c C d\right )}{d^3}\right )}{3 (c+d x)^{3/2} (b c-a d)}\)

\(\Big \downarrow \) 66

\(\displaystyle \frac {\frac {\frac {D (a+b x)^{5/2} \sqrt {c+d x} (b c-a d)^2}{b}-\frac {\left (a^3 d^3 D-3 a^2 b d^2 (2 C d-5 c D)+3 a b^2 d \left (-8 B d^2-35 c^2 D+20 c C d\right )-\left (b^3 \left (16 A d^3-40 B c d^2-105 c^3 D+70 c^2 C d\right )\right )\right ) \left (\frac {(a+b x)^{3/2} \sqrt {c+d x}}{2 d}-\frac {3 (b c-a d) \left (\frac {\sqrt {a+b x} \sqrt {c+d x}}{d}-\frac {(b c-a d) \int \frac {1}{b-\frac {d (a+b x)}{c+d x}}d\frac {\sqrt {a+b x}}{\sqrt {c+d x}}}{d}\right )}{4 d}\right )}{2 b}}{d^3 (b c-a d)}+\frac {2 (a+b x)^{5/2} \left (3 a d \left (-B d^2-3 c^2 D+2 c C d\right )-b \left (2 A d^3-5 B c d^2-11 c^3 D+8 c^2 C d\right )\right )}{d^3 \sqrt {c+d x} (b c-a d)}}{3 (b c-a d)}+\frac {2 (a+b x)^{5/2} \left (A+\frac {c \left (-B d^2+c^2 (-D)+c C d\right )}{d^3}\right )}{3 (c+d x)^{3/2} (b c-a d)}\)

\(\Big \downarrow \) 221

\(\displaystyle \frac {\frac {\frac {D (a+b x)^{5/2} \sqrt {c+d x} (b c-a d)^2}{b}-\frac {\left (\frac {(a+b x)^{3/2} \sqrt {c+d x}}{2 d}-\frac {3 (b c-a d) \left (\frac {\sqrt {a+b x} \sqrt {c+d x}}{d}-\frac {(b c-a d) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{\sqrt {b} d^{3/2}}\right )}{4 d}\right ) \left (a^3 d^3 D-3 a^2 b d^2 (2 C d-5 c D)+3 a b^2 d \left (-8 B d^2-35 c^2 D+20 c C d\right )-\left (b^3 \left (16 A d^3-40 B c d^2-105 c^3 D+70 c^2 C d\right )\right )\right )}{2 b}}{d^3 (b c-a d)}+\frac {2 (a+b x)^{5/2} \left (3 a d \left (-B d^2-3 c^2 D+2 c C d\right )-b \left (2 A d^3-5 B c d^2-11 c^3 D+8 c^2 C d\right )\right )}{d^3 \sqrt {c+d x} (b c-a d)}}{3 (b c-a d)}+\frac {2 (a+b x)^{5/2} \left (A+\frac {c \left (-B d^2+c^2 (-D)+c C d\right )}{d^3}\right )}{3 (c+d x)^{3/2} (b c-a d)}\)

Input:

Int[((a + b*x)^(3/2)*(A + B*x + C*x^2 + D*x^3))/(c + d*x)^(5/2),x]
 

Output:

(2*(A + (c*(c*C*d - B*d^2 - c^2*D))/d^3)*(a + b*x)^(5/2))/(3*(b*c - a*d)*( 
c + d*x)^(3/2)) + ((2*(3*a*d*(2*c*C*d - B*d^2 - 3*c^2*D) - b*(8*c^2*C*d - 
5*B*c*d^2 + 2*A*d^3 - 11*c^3*D))*(a + b*x)^(5/2))/(d^3*(b*c - a*d)*Sqrt[c 
+ d*x]) + (((b*c - a*d)^2*D*(a + b*x)^(5/2)*Sqrt[c + d*x])/b - ((a^3*d^3*D 
 - 3*a^2*b*d^2*(2*C*d - 5*c*D) + 3*a*b^2*d*(20*c*C*d - 8*B*d^2 - 35*c^2*D) 
 - b^3*(70*c^2*C*d - 40*B*c*d^2 + 16*A*d^3 - 105*c^3*D))*(((a + b*x)^(3/2) 
*Sqrt[c + d*x])/(2*d) - (3*(b*c - a*d)*((Sqrt[a + b*x]*Sqrt[c + d*x])/d - 
((b*c - a*d)*ArcTanh[(Sqrt[d]*Sqrt[a + b*x])/(Sqrt[b]*Sqrt[c + d*x])])/(Sq 
rt[b]*d^(3/2))))/(4*d)))/(2*b))/(d^3*(b*c - a*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 60

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ 
(a + b*x)^(m + 1)*((c + d*x)^n/(b*(m + n + 1))), x] + Simp[n*((b*c - a*d)/( 
b*(m + n + 1)))   Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, 
 c, d}, x] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !Integer 
Q[n] || (GtQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 0] && IntLinear 
Q[a, b, c, d, m, n, 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 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 [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(2407\) vs. \(2(357)=714\).

Time = 0.53 (sec) , antiderivative size = 2408, normalized size of antiderivative = 6.04

method	result	size
default	\(\text {Expression too large to display}\)	\(2408\)

Input:

int((b*x+a)^(3/2)*(D*x^3+C*x^2+B*x+A)/(d*x+c)^(5/2),x,method=_RETURNVERBOS 
E)
 

Output:

1/48*(b*x+a)^(1/2)*(-6*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^4*x+18*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^2*b*c^2*d^3-180*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^3*d^2-45*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^2*b*c^3*d^2+28*D*a*b*d^4*x^3*(d*b)^(1/2)*((b*x+a)*(d*x+c) 
)^(1/2)-36*D*b^2*c*d^3*x^3*(d*b)^(1/2)*((b*x+a)*(d*x+c))^(1/2)-420*D*a*b*c 
^3*d*(d*b)^(1/2)*((b*x+a)*(d*x+c))^(1/2)-180*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^4*x^2-45*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^ 
2*b*c*d^4*x^2+315*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^3*x^2+144*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*c*d^4*x+48*A*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*d 
^5*x^2+48*A*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^3+240*B*b^2*c^2*d^2*(d*b)^(1/2)*((b*x+a)*(d*x+c))^ 
(1/2)-420*C*b^2*c^3*d*(d*b)^(1/2)*((b*x+a)*(d*x+c))^(1/2)+48*B*b^2*d^4*x^2 
*((b*x+a)*(d*x+c))^(1/2)*(d*b)^(1/2)+96*A*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^4*x+6*D*a^2*c^2*d^2*(d 
*b)^(1/2)*((b*x+a)*(d*x+c))^(1/2)+72*B*ln(1/2*(2*b*d*x+2*((b*x+a)*(d*x+...
 

Fricas [A] (verification not implemented)

Time = 16.85 (sec) , antiderivative size = 1400, normalized size of antiderivative = 3.51 \[ \int \frac {(a+b x)^{3/2} \left (A+B x+C x^2+D x^3\right )}{(c+d x)^{5/2}} \, dx=\text {Too large to display} \] Input:

integrate((b*x+a)^(3/2)*(D*x^3+C*x^2+B*x+A)/(d*x+c)^(5/2),x, algorithm="fr 
icas")
 

Output:

[-1/96*(3*(105*D*b^3*c^5 - 35*(3*D*a*b^2 + 2*C*b^3)*c^4*d + 5*(3*D*a^2*b + 
 12*C*a*b^2 + 8*B*b^3)*c^3*d^2 + (D*a^3 - 6*C*a^2*b - 24*B*a*b^2 - 16*A*b^ 
3)*c^2*d^3 + (105*D*b^3*c^3*d^2 - 35*(3*D*a*b^2 + 2*C*b^3)*c^2*d^3 + 5*(3* 
D*a^2*b + 12*C*a*b^2 + 8*B*b^3)*c*d^4 + (D*a^3 - 6*C*a^2*b - 24*B*a*b^2 - 
16*A*b^3)*d^5)*x^2 + 2*(105*D*b^3*c^4*d - 35*(3*D*a*b^2 + 2*C*b^3)*c^3*d^2 
 + 5*(3*D*a^2*b + 12*C*a*b^2 + 8*B*b^3)*c^2*d^3 + (D*a^3 - 6*C*a^2*b - 24* 
B*a*b^2 - 16*A*b^3)*c*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*(8*D*b^3*d^5*x^4 + 315*D*b^3*c^4*d - 
16*A*a*b^2*d^5 - 210*(D*a*b^2 + C*b^3)*c^3*d^2 + (3*D*a^2*b + 110*C*a*b^2 
+ 120*B*b^3)*c^2*d^3 - 16*(2*B*a*b^2 + 3*A*b^3)*c*d^4 - 2*(9*D*b^3*c*d^4 - 
 (7*D*a*b^2 + 6*C*b^3)*d^5)*x^3 + 3*(21*D*b^3*c^2*d^3 - 2*(9*D*a*b^2 + 7*C 
*b^3)*c*d^4 + (D*a^2*b + 10*C*a*b^2 + 8*B*b^3)*d^5)*x^2 + 2*(210*D*b^3*c^3 
*d^2 - 7*(21*D*a*b^2 + 20*C*b^3)*c^2*d^3 + (3*D*a^2*b + 78*C*a*b^2 + 80*B* 
b^3)*c*d^4 - 8*(3*B*a*b^2 + 4*A*b^3)*d^5)*x)*sqrt(b*x + a)*sqrt(d*x + c))/ 
(b^2*d^8*x^2 + 2*b^2*c*d^7*x + b^2*c^2*d^6), 1/48*(3*(105*D*b^3*c^5 - 35*( 
3*D*a*b^2 + 2*C*b^3)*c^4*d + 5*(3*D*a^2*b + 12*C*a*b^2 + 8*B*b^3)*c^3*d^2 
+ (D*a^3 - 6*C*a^2*b - 24*B*a*b^2 - 16*A*b^3)*c^2*d^3 + (105*D*b^3*c^3*d^2 
 - 35*(3*D*a*b^2 + 2*C*b^3)*c^2*d^3 + 5*(3*D*a^2*b + 12*C*a*b^2 + 8*B*b^3) 
*c*d^4 + (D*a^3 - 6*C*a^2*b - 24*B*a*b^2 - 16*A*b^3)*d^5)*x^2 + 2*(105*...
 

Sympy [F]

\[ \int \frac {(a+b x)^{3/2} \left (A+B x+C x^2+D x^3\right )}{(c+d x)^{5/2}} \, dx=\int \frac {\left (a + b x\right )^{\frac {3}{2}} \left (A + B x + C x^{2} + D x^{3}\right )}{\left (c + d x\right )^{\frac {5}{2}}}\, dx \] Input:

integrate((b*x+a)**(3/2)*(D*x**3+C*x**2+B*x+A)/(d*x+c)**(5/2),x)
 

Output:

Integral((a + b*x)**(3/2)*(A + B*x + C*x**2 + D*x**3)/(c + d*x)**(5/2), x)
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {(a+b x)^{3/2} \left (A+B x+C x^2+D x^3\right )}{(c+d x)^{5/2}} \, dx=\text {Exception raised: ValueError} \] Input:

integrate((b*x+a)^(3/2)*(D*x^3+C*x^2+B*x+A)/(d*x+c)^(5/2),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. 1006 vs. \(2 (356) = 712\).

Time = 0.31 (sec) , antiderivative size = 1006, normalized size of antiderivative = 2.52 \[ \int \frac {(a+b x)^{3/2} \left (A+B x+C x^2+D x^3\right )}{(c+d x)^{5/2}} \, dx=\text {Too large to display} \] Input:

integrate((b*x+a)^(3/2)*(D*x^3+C*x^2+B*x+A)/(d*x+c)^(5/2),x, algorithm="gi 
ac")
 

Output:

1/24*(((2*(b*x + a)*(4*(D*b^5*c*d^8*abs(b) - D*a*b^4*d^9*abs(b))*(b*x + a) 
/(b^6*c*d^9 - a*b^5*d^10) - 3*(3*D*b^6*c^2*d^7*abs(b) - 2*C*b^6*c*d^8*abs( 
b) - 3*D*a^2*b^4*d^9*abs(b) + 2*C*a*b^5*d^9*abs(b))/(b^6*c*d^9 - a*b^5*d^1 
0)) + 3*(21*D*b^7*c^3*d^6*abs(b) - 21*D*a*b^6*c^2*d^7*abs(b) - 14*C*b^7*c^ 
2*d^7*abs(b) + 3*D*a^2*b^5*c*d^8*abs(b) + 12*C*a*b^6*c*d^8*abs(b) + 8*B*b^ 
7*c*d^8*abs(b) - 3*D*a^3*b^4*d^9*abs(b) + 2*C*a^2*b^5*d^9*abs(b) - 8*B*a*b 
^6*d^9*abs(b))/(b^6*c*d^9 - a*b^5*d^10))*(b*x + a) + 4*(105*D*b^8*c^4*d^5* 
abs(b) - 210*D*a*b^7*c^3*d^6*abs(b) - 70*C*b^8*c^3*d^6*abs(b) + 120*D*a^2* 
b^6*c^2*d^7*abs(b) + 130*C*a*b^7*c^2*d^7*abs(b) + 40*B*b^8*c^2*d^7*abs(b) 
- 14*D*a^3*b^5*c*d^8*abs(b) - 66*C*a^2*b^6*c*d^8*abs(b) - 64*B*a*b^7*c*d^8 
*abs(b) - 16*A*b^8*c*d^8*abs(b) - D*a^4*b^4*d^9*abs(b) + 6*C*a^3*b^5*d^9*a 
bs(b) + 24*B*a^2*b^6*d^9*abs(b) + 16*A*a*b^7*d^9*abs(b))/(b^6*c*d^9 - a*b^ 
5*d^10))*(b*x + a) + 3*(105*D*b^9*c^5*d^4*abs(b) - 315*D*a*b^8*c^4*d^5*abs 
(b) - 70*C*b^9*c^4*d^5*abs(b) + 330*D*a^2*b^7*c^3*d^6*abs(b) + 200*C*a*b^8 
*c^3*d^6*abs(b) + 40*B*b^9*c^3*d^6*abs(b) - 134*D*a^3*b^6*c^2*d^7*abs(b) - 
 196*C*a^2*b^7*c^2*d^7*abs(b) - 104*B*a*b^8*c^2*d^7*abs(b) - 16*A*b^9*c^2* 
d^7*abs(b) + 13*D*a^4*b^5*c*d^8*abs(b) + 72*C*a^3*b^6*c*d^8*abs(b) + 88*B* 
a^2*b^7*c*d^8*abs(b) + 32*A*a*b^8*c*d^8*abs(b) + D*a^5*b^4*d^9*abs(b) - 6* 
C*a^4*b^5*d^9*abs(b) - 24*B*a^3*b^6*d^9*abs(b) - 16*A*a^2*b^7*d^9*abs(b))/ 
(b^6*c*d^9 - a*b^5*d^10))*sqrt(b*x + a)/(b^2*c + (b*x + a)*b*d - a*b*d)...
 

Mupad [F(-1)]

Timed out. \[ \int \frac {(a+b x)^{3/2} \left (A+B x+C x^2+D x^3\right )}{(c+d x)^{5/2}} \, dx=\int \frac {{\left (a+b\,x\right )}^{3/2}\,\left (A+B\,x+C\,x^2+x^3\,D\right )}{{\left (c+d\,x\right )}^{5/2}} \,d x \] Input:

int(((a + b*x)^(3/2)*(A + B*x + C*x^2 + x^3*D))/(c + d*x)^(5/2),x)
                                                                                    
                                                                                    
 

Output:

int(((a + b*x)^(3/2)*(A + B*x + C*x^2 + x^3*D))/(c + d*x)^(5/2), x)
 

Reduce [F]

\[ \int \frac {(a+b x)^{3/2} \left (A+B x+C x^2+D x^3\right )}{(c+d x)^{5/2}} \, dx=\int \frac {\left (b x +a \right )^{\frac {3}{2}} \left (D x^{3}+C \,x^{2}+B x +A \right )}{\left (d x +c \right )^{\frac {5}{2}}}d x \] Input:

int((b*x+a)^(3/2)*(D*x^3+C*x^2+B*x+A)/(d*x+c)^(5/2),x)
 

Output:

int((b*x+a)^(3/2)*(D*x^3+C*x^2+B*x+A)/(d*x+c)^(5/2),x)