\(\int \frac {(c+d x)^{5/2} (A+B x+C x^2+D x^3)}{(a+b x)^{5/2}} \, dx\) [145]

Optimal result

Integrand size = 34, antiderivative size = 520 \[ \int \frac {(c+d x)^{5/2} \left (A+B x+C x^2+D x^3\right )}{(a+b x)^{5/2}} \, dx=-\frac {5 \left (231 a^3 d^3 D-21 a^2 b d^2 (8 C d+9 c D)+7 a b^2 d \left (16 c C d+16 B d^2+3 c^2 D\right )-b^3 \left (8 c^2 C d+48 B c d^2+64 A d^3-c^3 D\right )\right ) \sqrt {a+b x} \sqrt {c+d x}}{64 b^6 d}-\frac {2 \left (b^3 (3 B c+5 A d)-2 a b^2 (3 c C+4 B d)-14 a^3 d D+a^2 b (11 C d+9 c D)\right ) (c+d x)^{3/2}}{3 b^5 \sqrt {a+b x}}+\frac {\left (117 a^2 d^2 D-2 a b d (28 C d+33 c D)+b^2 \left (24 c C d+16 B d^2-3 c^2 D\right )\right ) \sqrt {a+b x} (c+d x)^{3/2}}{32 b^5 d}+\frac {(8 b C d-b c D-23 a d D) (a+b x)^{3/2} (c+d x)^{3/2}}{24 b^5}-\frac {2 \left (A b^3-a \left (b^2 B-a b C+a^2 D\right )\right ) (c+d x)^{5/2}}{3 b^4 (a+b x)^{3/2}}+\frac {D \sqrt {a+b x} (c+d x)^{7/2}}{4 b^3 d}-\frac {5 (b c-a d) \left (231 a^3 d^3 D-21 a^2 b d^2 (8 C d+9 c D)+7 a b^2 d \left (16 c C d+16 B d^2+3 c^2 D\right )-b^3 \left (8 c^2 C d+48 B c d^2+64 A d^3-c^3 D\right )\right ) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{64 b^{13/2} d^{3/2}} \] Output:

-5/64*(231*a^3*d^3*D-21*a^2*b*d^2*(8*C*d+9*D*c)+7*a*b^2*d*(16*B*d^2+16*C*c 
*d+3*D*c^2)-b^3*(64*A*d^3+48*B*c*d^2+8*C*c^2*d-D*c^3))*(b*x+a)^(1/2)*(d*x+ 
c)^(1/2)/b^6/d-2/3*(b^3*(5*A*d+3*B*c)-2*a*b^2*(4*B*d+3*C*c)-14*a^3*d*D+a^2 
*b*(11*C*d+9*D*c))*(d*x+c)^(3/2)/b^5/(b*x+a)^(1/2)+1/32*(117*a^2*d^2*D-2*a 
*b*d*(28*C*d+33*D*c)+b^2*(16*B*d^2+24*C*c*d-3*D*c^2))*(b*x+a)^(1/2)*(d*x+c 
)^(3/2)/b^5/d+1/24*(8*C*b*d-23*D*a*d-D*b*c)*(b*x+a)^(3/2)*(d*x+c)^(3/2)/b^ 
5-2/3*(A*b^3-a*(B*b^2-C*a*b+D*a^2))*(d*x+c)^(5/2)/b^4/(b*x+a)^(3/2)+1/4*D* 
(b*x+a)^(1/2)*(d*x+c)^(7/2)/b^3/d-5/64*(-a*d+b*c)*(231*a^3*d^3*D-21*a^2*b* 
d^2*(8*C*d+9*D*c)+7*a*b^2*d*(16*B*d^2+16*C*c*d+3*D*c^2)-b^3*(64*A*d^3+48*B 
*c*d^2+8*C*c^2*d-D*c^3))*arctanh(d^(1/2)*(b*x+a)^(1/2)/b^(1/2)/(d*x+c)^(1/ 
2))/b^(13/2)/d^(3/2)
 

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.

Time = 10.32 (sec) , antiderivative size = 251, normalized size of antiderivative = 0.48 \[ \int \frac {(c+d x)^{5/2} \left (A+B x+C x^2+D x^3\right )}{(a+b x)^{5/2}} \, dx=\frac {2 (c+d x)^{5/2} \left (-(b c-a d)^3 D \operatorname {Hypergeometric2F1}\left (-\frac {11}{2},-\frac {3}{2},-\frac {1}{2},\frac {d (a+b x)}{-b c+a d}\right )+b (b c-a d)^2 (-C d+3 c D) \operatorname {Hypergeometric2F1}\left (-\frac {9}{2},-\frac {3}{2},-\frac {1}{2},\frac {d (a+b x)}{-b c+a d}\right )-b^2 (b c-a d) \left (-2 c C d+B d^2+3 c^2 D\right ) \operatorname {Hypergeometric2F1}\left (-\frac {7}{2},-\frac {3}{2},-\frac {1}{2},\frac {d (a+b x)}{-b c+a d}\right )+b^3 \left (-c^2 C d+B c d^2-A d^3+c^3 D\right ) \operatorname {Hypergeometric2F1}\left (-\frac {5}{2},-\frac {3}{2},-\frac {1}{2},\frac {d (a+b x)}{-b c+a d}\right )\right )}{3 b^4 d^3 (a+b x)^{3/2} \left (\frac {b (c+d x)}{b c-a d}\right )^{5/2}} \] Input:

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

Output:

(2*(c + d*x)^(5/2)*(-((b*c - a*d)^3*D*Hypergeometric2F1[-11/2, -3/2, -1/2, 
 (d*(a + b*x))/(-(b*c) + a*d)]) + b*(b*c - a*d)^2*(-(C*d) + 3*c*D)*Hyperge 
ometric2F1[-9/2, -3/2, -1/2, (d*(a + b*x))/(-(b*c) + a*d)] - b^2*(b*c - a* 
d)*(-2*c*C*d + B*d^2 + 3*c^2*D)*Hypergeometric2F1[-7/2, -3/2, -1/2, (d*(a 
+ b*x))/(-(b*c) + a*d)] + b^3*(-(c^2*C*d) + B*c*d^2 - A*d^3 + c^3*D)*Hyper 
geometric2F1[-5/2, -3/2, -1/2, (d*(a + b*x))/(-(b*c) + a*d)]))/(3*b^4*d^3* 
(a + b*x)^(3/2)*((b*(c + d*x))/(b*c - a*d))^(5/2))
 

Rubi [A] (verified)

Time = 0.83 (sec) , antiderivative size = 457, normalized size of antiderivative = 0.88, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.294, Rules used = {2124, 27, 1193, 27, 90, 60, 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.

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

\(\Big \downarrow \) 2124

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1193

\(\displaystyle \frac {-\frac {2 \int -\frac {3 (c+d x)^{5/2} \left (-28 d^2 D a^3+b d (21 C d+22 c D) a^2-2 b^2 \left (D c^2+7 C d c+7 B d^2\right ) a+b^3 \left (C c^2+6 B d c+8 A d^2\right )+b (b c-a d)^2 D x\right )}{2 b^3 \sqrt {a+b x}}dx}{b c-a d}-\frac {2 (c+d x)^{7/2} \left (-13 a^3 d D+a^2 b (9 c D+10 C d)-a b^2 (7 B d+6 c C)+b^3 (4 A d+3 B c)\right )}{b^3 \sqrt {a+b x} (b c-a d)}}{3 (b c-a d)}-\frac {2 (c+d x)^{7/2} \left (A b^3-a \left (a^2 D-a b C+b^2 B\right )\right )}{3 b^3 (a+b x)^{3/2} (b c-a d)}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {3 \int \frac {(c+d x)^{5/2} \left (-28 d^2 D a^3+b d (21 C d+22 c D) a^2-2 b^2 \left (D c^2+7 C d c+7 B d^2\right ) a+b^3 \left (C c^2+6 B d c+8 A d^2\right )+b (b c-a d)^2 D x\right )}{\sqrt {a+b x}}dx}{b^3 (b c-a d)}-\frac {2 (c+d x)^{7/2} \left (-13 a^3 d D+a^2 b (9 c D+10 C d)-a b^2 (7 B d+6 c C)+b^3 (4 A d+3 B c)\right )}{b^3 \sqrt {a+b x} (b c-a d)}}{3 (b c-a d)}-\frac {2 (c+d x)^{7/2} \left (A b^3-a \left (a^2 D-a b C+b^2 B\right )\right )}{3 b^3 (a+b x)^{3/2} (b c-a d)}\)

\(\Big \downarrow \) 90

\(\displaystyle \frac {\frac {3 \left (\frac {D \sqrt {a+b x} (c+d x)^{7/2} (b c-a d)^2}{4 d}-\frac {\left (231 a^3 d^3 D-21 a^2 b d^2 (9 c D+8 C d)+7 a b^2 d \left (16 B d^2+3 c^2 D+16 c C d\right )-\left (b^3 \left (64 A d^3+48 B c d^2+c^3 (-D)+8 c^2 C d\right )\right )\right ) \int \frac {(c+d x)^{5/2}}{\sqrt {a+b x}}dx}{8 d}\right )}{b^3 (b c-a d)}-\frac {2 (c+d x)^{7/2} \left (-13 a^3 d D+a^2 b (9 c D+10 C d)-a b^2 (7 B d+6 c C)+b^3 (4 A d+3 B c)\right )}{b^3 \sqrt {a+b x} (b c-a d)}}{3 (b c-a d)}-\frac {2 (c+d x)^{7/2} \left (A b^3-a \left (a^2 D-a b C+b^2 B\right )\right )}{3 b^3 (a+b x)^{3/2} (b c-a d)}\)

\(\Big \downarrow \) 60

\(\displaystyle \frac {\frac {3 \left (\frac {D \sqrt {a+b x} (c+d x)^{7/2} (b c-a d)^2}{4 d}-\frac {\left (231 a^3 d^3 D-21 a^2 b d^2 (9 c D+8 C d)+7 a b^2 d \left (16 B d^2+3 c^2 D+16 c C d\right )-\left (b^3 \left (64 A d^3+48 B c d^2+c^3 (-D)+8 c^2 C d\right )\right )\right ) \left (\frac {5 (b c-a d) \int \frac {(c+d x)^{3/2}}{\sqrt {a+b x}}dx}{6 b}+\frac {\sqrt {a+b x} (c+d x)^{5/2}}{3 b}\right )}{8 d}\right )}{b^3 (b c-a d)}-\frac {2 (c+d x)^{7/2} \left (-13 a^3 d D+a^2 b (9 c D+10 C d)-a b^2 (7 B d+6 c C)+b^3 (4 A d+3 B c)\right )}{b^3 \sqrt {a+b x} (b c-a d)}}{3 (b c-a d)}-\frac {2 (c+d x)^{7/2} \left (A b^3-a \left (a^2 D-a b C+b^2 B\right )\right )}{3 b^3 (a+b x)^{3/2} (b c-a d)}\)

\(\Big \downarrow \) 60

\(\displaystyle \frac {\frac {3 \left (\frac {D \sqrt {a+b x} (c+d x)^{7/2} (b c-a d)^2}{4 d}-\frac {\left (231 a^3 d^3 D-21 a^2 b d^2 (9 c D+8 C d)+7 a b^2 d \left (16 B d^2+3 c^2 D+16 c C d\right )-\left (b^3 \left (64 A d^3+48 B c d^2+c^3 (-D)+8 c^2 C d\right )\right )\right ) \left (\frac {5 (b c-a d) \left (\frac {3 (b c-a d) \int \frac {\sqrt {c+d x}}{\sqrt {a+b x}}dx}{4 b}+\frac {\sqrt {a+b x} (c+d x)^{3/2}}{2 b}\right )}{6 b}+\frac {\sqrt {a+b x} (c+d x)^{5/2}}{3 b}\right )}{8 d}\right )}{b^3 (b c-a d)}-\frac {2 (c+d x)^{7/2} \left (-13 a^3 d D+a^2 b (9 c D+10 C d)-a b^2 (7 B d+6 c C)+b^3 (4 A d+3 B c)\right )}{b^3 \sqrt {a+b x} (b c-a d)}}{3 (b c-a d)}-\frac {2 (c+d x)^{7/2} \left (A b^3-a \left (a^2 D-a b C+b^2 B\right )\right )}{3 b^3 (a+b x)^{3/2} (b c-a d)}\)

\(\Big \downarrow \) 60

\(\displaystyle \frac {\frac {3 \left (\frac {D \sqrt {a+b x} (c+d x)^{7/2} (b c-a d)^2}{4 d}-\frac {\left (231 a^3 d^3 D-21 a^2 b d^2 (9 c D+8 C d)+7 a b^2 d \left (16 B d^2+3 c^2 D+16 c C d\right )-\left (b^3 \left (64 A d^3+48 B c d^2+c^3 (-D)+8 c^2 C d\right )\right )\right ) \left (\frac {5 (b c-a d) \left (\frac {3 (b c-a d) \left (\frac {(b c-a d) \int \frac {1}{\sqrt {a+b x} \sqrt {c+d x}}dx}{2 b}+\frac {\sqrt {a+b x} \sqrt {c+d x}}{b}\right )}{4 b}+\frac {\sqrt {a+b x} (c+d x)^{3/2}}{2 b}\right )}{6 b}+\frac {\sqrt {a+b x} (c+d x)^{5/2}}{3 b}\right )}{8 d}\right )}{b^3 (b c-a d)}-\frac {2 (c+d x)^{7/2} \left (-13 a^3 d D+a^2 b (9 c D+10 C d)-a b^2 (7 B d+6 c C)+b^3 (4 A d+3 B c)\right )}{b^3 \sqrt {a+b x} (b c-a d)}}{3 (b c-a d)}-\frac {2 (c+d x)^{7/2} \left (A b^3-a \left (a^2 D-a b C+b^2 B\right )\right )}{3 b^3 (a+b x)^{3/2} (b c-a d)}\)

\(\Big \downarrow \) 66

\(\displaystyle \frac {\frac {3 \left (\frac {D \sqrt {a+b x} (c+d x)^{7/2} (b c-a d)^2}{4 d}-\frac {\left (231 a^3 d^3 D-21 a^2 b d^2 (9 c D+8 C d)+7 a b^2 d \left (16 B d^2+3 c^2 D+16 c C d\right )-\left (b^3 \left (64 A d^3+48 B c d^2+c^3 (-D)+8 c^2 C d\right )\right )\right ) \left (\frac {5 (b c-a d) \left (\frac {3 (b c-a d) \left (\frac {(b c-a d) \int \frac {1}{b-\frac {d (a+b x)}{c+d x}}d\frac {\sqrt {a+b x}}{\sqrt {c+d x}}}{b}+\frac {\sqrt {a+b x} \sqrt {c+d x}}{b}\right )}{4 b}+\frac {\sqrt {a+b x} (c+d x)^{3/2}}{2 b}\right )}{6 b}+\frac {\sqrt {a+b x} (c+d x)^{5/2}}{3 b}\right )}{8 d}\right )}{b^3 (b c-a d)}-\frac {2 (c+d x)^{7/2} \left (-13 a^3 d D+a^2 b (9 c D+10 C d)-a b^2 (7 B d+6 c C)+b^3 (4 A d+3 B c)\right )}{b^3 \sqrt {a+b x} (b c-a d)}}{3 (b c-a d)}-\frac {2 (c+d x)^{7/2} \left (A b^3-a \left (a^2 D-a b C+b^2 B\right )\right )}{3 b^3 (a+b x)^{3/2} (b c-a d)}\)

\(\Big \downarrow \) 221

\(\displaystyle \frac {\frac {3 \left (\frac {D \sqrt {a+b x} (c+d x)^{7/2} (b c-a d)^2}{4 d}-\frac {\left (\frac {5 (b c-a d) \left (\frac {3 (b c-a d) \left (\frac {(b c-a d) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{b^{3/2} \sqrt {d}}+\frac {\sqrt {a+b x} \sqrt {c+d x}}{b}\right )}{4 b}+\frac {\sqrt {a+b x} (c+d x)^{3/2}}{2 b}\right )}{6 b}+\frac {\sqrt {a+b x} (c+d x)^{5/2}}{3 b}\right ) \left (231 a^3 d^3 D-21 a^2 b d^2 (9 c D+8 C d)+7 a b^2 d \left (16 B d^2+3 c^2 D+16 c C d\right )-\left (b^3 \left (64 A d^3+48 B c d^2+c^3 (-D)+8 c^2 C d\right )\right )\right )}{8 d}\right )}{b^3 (b c-a d)}-\frac {2 (c+d x)^{7/2} \left (-13 a^3 d D+a^2 b (9 c D+10 C d)-a b^2 (7 B d+6 c C)+b^3 (4 A d+3 B c)\right )}{b^3 \sqrt {a+b x} (b c-a d)}}{3 (b c-a d)}-\frac {2 (c+d x)^{7/2} \left (A b^3-a \left (a^2 D-a b C+b^2 B\right )\right )}{3 b^3 (a+b x)^{3/2} (b c-a d)}\)

Input:

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

Output:

(-2*(A*b^3 - a*(b^2*B - a*b*C + a^2*D))*(c + d*x)^(7/2))/(3*b^3*(b*c - a*d 
)*(a + b*x)^(3/2)) + ((-2*(b^3*(3*B*c + 4*A*d) - a*b^2*(6*c*C + 7*B*d) - 1 
3*a^3*d*D + a^2*b*(10*C*d + 9*c*D))*(c + d*x)^(7/2))/(b^3*(b*c - a*d)*Sqrt 
[a + b*x]) + (3*(((b*c - a*d)^2*D*Sqrt[a + b*x]*(c + d*x)^(7/2))/(4*d) - ( 
(231*a^3*d^3*D - 21*a^2*b*d^2*(8*C*d + 9*c*D) + 7*a*b^2*d*(16*c*C*d + 16*B 
*d^2 + 3*c^2*D) - b^3*(8*c^2*C*d + 48*B*c*d^2 + 64*A*d^3 - c^3*D))*((Sqrt[ 
a + b*x]*(c + d*x)^(5/2))/(3*b) + (5*(b*c - a*d)*((Sqrt[a + b*x]*(c + d*x) 
^(3/2))/(2*b) + (3*(b*c - a*d)*((Sqrt[a + b*x]*Sqrt[c + d*x])/b + ((b*c - 
a*d)*ArcTanh[(Sqrt[d]*Sqrt[a + b*x])/(Sqrt[b]*Sqrt[c + d*x])])/(b^(3/2)*Sq 
rt[d])))/(4*b)))/(6*b)))/(8*d)))/(b^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. \(3663\) vs. \(2(470)=940\).

Time = 0.57 (sec) , antiderivative size = 3664, normalized size of antiderivative = 7.05

method	result	size
default	\(\text {Expression too large to display}\)	\(3664\)

Input:

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

Output:

-1/384*(d*x+c)^(1/2)*(5040*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^4*b^2*d^4*x-6930*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^5*b*d^4*x-960*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))* 
a^2*b^4*c*d^3-3150*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^4*b^2*c^2*d^2-1920*A*(d*b)^(1/2)*((b*x+a)*(d*x+c) 
)^(1/2)*a^2*b^3*d^3-1808*C*((b*x+a)*(d*x+c))^(1/2)*(d*b)^(1/2)*a^2*b^3*c^2 
*d+288*C*((b*x+a)*(d*x+c))^(1/2)*(d*b)^(1/2)*a*b^4*d^3*x^3-396*D*((b*x+a)* 
(d*x+c))^(1/2)*(d*b)^(1/2)*a^2*b^3*d^3*x^3-236*D*((b*x+a)*(d*x+c))^(1/2)*( 
d*b)^(1/2)*b^5*c^2*d*x^3-96*D*b^5*d^3*x^5*(d*b)^(1/2)*((b*x+a)*(d*x+c))^(1 
/2)+2400*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^5*c*d^3*x^2-10290*D*(d*b)^(1/2)*((b*x+a)*(d*x+c))^(1/2)*a 
^4*b*c*d^2-528*C*((b*x+a)*(d*x+c))^(1/2)*(d*b)^(1/2)*b^5*c^2*d*x^2-30*D*(( 
b*x+a)*(d*x+c))^(1/2)*(d*b)^(1/2)*a^2*b^3*c^3-120*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^4*c^3*d+300*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*b^3*c^3*d-128*C*b^5*d^3*x^4*(d*b)^(1/2)*((b*x+a)*(d*x+c))^(1/2)+256* 
A*b^5*c^2*d*(d*b)^(1/2)*((b*x+a)*(d*x+c))^(1/2)+966*D*((b*x+a)*(d*x+c))^(1 
/2)*(d*b)^(1/2)*a*b^4*c^2*d*x^2+15*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))*b^6*c^4*x^2+15*D*ln(1/2*(2*b*d*x...
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1046 vs. \(2 (468) = 936\).

Time = 18.86 (sec) , antiderivative size = 2106, normalized size of antiderivative = 4.05 \[ \int \frac {(c+d x)^{5/2} \left (A+B x+C x^2+D x^3\right )}{(a+b x)^{5/2}} \, dx=\text {Too large to display} \] Input:

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

Output:

[1/768*(15*(D*a^2*b^4*c^4 + 4*(5*D*a^3*b^3 - 2*C*a^2*b^4)*c^3*d - 6*(35*D* 
a^4*b^2 - 20*C*a^3*b^3 + 8*B*a^2*b^4)*c^2*d^2 + 4*(105*D*a^5*b - 70*C*a^4* 
b^2 + 40*B*a^3*b^3 - 16*A*a^2*b^4)*c*d^3 - (231*D*a^6 - 168*C*a^5*b + 112* 
B*a^4*b^2 - 64*A*a^3*b^3)*d^4 + (D*b^6*c^4 + 4*(5*D*a*b^5 - 2*C*b^6)*c^3*d 
 - 6*(35*D*a^2*b^4 - 20*C*a*b^5 + 8*B*b^6)*c^2*d^2 + 4*(105*D*a^3*b^3 - 70 
*C*a^2*b^4 + 40*B*a*b^5 - 16*A*b^6)*c*d^3 - (231*D*a^4*b^2 - 168*C*a^3*b^3 
 + 112*B*a^2*b^4 - 64*A*a*b^5)*d^4)*x^2 + 2*(D*a*b^5*c^4 + 4*(5*D*a^2*b^4 
- 2*C*a*b^5)*c^3*d - 6*(35*D*a^3*b^3 - 20*C*a^2*b^4 + 8*B*a*b^5)*c^2*d^2 + 
 4*(105*D*a^4*b^2 - 70*C*a^3*b^3 + 40*B*a^2*b^4 - 16*A*a*b^5)*c*d^3 - (231 
*D*a^5*b - 168*C*a^4*b^2 + 112*B*a^3*b^3 - 64*A*a^2*b^4)*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*(4 
8*D*b^6*d^4*x^5 + 15*D*a^2*b^4*c^3*d - (1743*D*a^3*b^3 - 904*C*a^2*b^4 + 2 
56*B*a*b^5 + 128*A*b^6)*c^2*d^2 + 5*(1029*D*a^4*b^2 - 672*C*a^3*b^3 + 368* 
B*a^2*b^4 - 128*A*a*b^5)*c*d^3 - 15*(231*D*a^5*b - 168*C*a^4*b^2 + 112*B*a 
^3*b^3 - 64*A*a^2*b^4)*d^4 + 8*(17*D*b^6*c*d^3 - (11*D*a*b^5 - 8*C*b^6)*d^ 
4)*x^4 + 2*(59*D*b^6*c^2*d^2 - 2*(79*D*a*b^5 - 52*C*b^6)*c*d^3 + 3*(33*D*a 
^2*b^4 - 24*C*a*b^5 + 16*B*b^6)*d^4)*x^3 + 3*(5*D*b^6*c^3*d - (161*D*a*b^5 
 - 88*C*b^6)*c^2*d^2 + (387*D*a^2*b^4 - 256*C*a*b^5 + 144*B*b^6)*c*d^3 - ( 
231*D*a^3*b^3 - 168*C*a^2*b^4 + 112*B*a*b^5 - 64*A*b^6)*d^4)*x^2 + 2*(1...
 

Sympy [F]

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

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

Output:

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

Maxima [F(-2)]

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

integrate((d*x+c)^(5/2)*(D*x^3+C*x^2+B*x+A)/(b*x+a)^(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. 2808 vs. \(2 (468) = 936\).

Time = 1.04 (sec) , antiderivative size = 2808, normalized size of antiderivative = 5.40 \[ \int \frac {(c+d x)^{5/2} \left (A+B x+C x^2+D x^3\right )}{(a+b x)^{5/2}} \, dx=\text {Too large to display} \] Input:

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

Output:

1/192*sqrt(b^2*c + (b*x + a)*b*d - a*b*d)*(2*(b*x + a)*(4*(b*x + a)*(6*(b* 
x + a)*D*d^2*abs(b)/b^8 + (17*D*b^32*c*d^7*abs(b) - 41*D*a*b^31*d^8*abs(b) 
 + 8*C*b^32*d^8*abs(b))/(b^39*d^6)) + (59*D*b^33*c^2*d^6*abs(b) - 430*D*a* 
b^32*c*d^7*abs(b) + 104*C*b^33*c*d^7*abs(b) + 515*D*a^2*b^31*d^8*abs(b) - 
200*C*a*b^32*d^8*abs(b) + 48*B*b^33*d^8*abs(b))/(b^39*d^6)) + 3*(5*D*b^34* 
c^3*d^5*abs(b) - 279*D*a*b^33*c^2*d^6*abs(b) + 88*C*b^34*c^2*d^6*abs(b) + 
975*D*a^2*b^32*c*d^7*abs(b) - 464*C*a*b^33*c*d^7*abs(b) + 144*B*b^34*c*d^7 
*abs(b) - 765*D*a^3*b^31*d^8*abs(b) + 440*C*a^2*b^32*d^8*abs(b) - 208*B*a* 
b^33*d^8*abs(b) + 64*A*b^34*d^8*abs(b))/(b^39*d^6))*sqrt(b*x + a) + 5/128* 
(D*b^4*c^4*abs(b) + 20*D*a*b^3*c^3*d*abs(b) - 8*C*b^4*c^3*d*abs(b) - 210*D 
*a^2*b^2*c^2*d^2*abs(b) + 120*C*a*b^3*c^2*d^2*abs(b) - 48*B*b^4*c^2*d^2*ab 
s(b) + 420*D*a^3*b*c*d^3*abs(b) - 280*C*a^2*b^2*c*d^3*abs(b) + 160*B*a*b^3 
*c*d^3*abs(b) - 64*A*b^4*c*d^3*abs(b) - 231*D*a^4*d^4*abs(b) + 168*C*a^3*b 
*d^4*abs(b) - 112*B*a^2*b^2*d^4*abs(b) + 64*A*a*b^3*d^4*abs(b))*log((sqrt( 
b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2)/(sqrt(b*d)*b^ 
7*d) - 4/3*(9*D*a^2*b^7*c^5*d*abs(b) - 6*C*a*b^8*c^5*d*abs(b) + 3*B*b^9*c^ 
5*d*abs(b) - 52*D*a^3*b^6*c^4*d^2*abs(b) + 37*C*a^2*b^7*c^4*d^2*abs(b) - 2 
2*B*a*b^8*c^4*d^2*abs(b) + 7*A*b^9*c^4*d^2*abs(b) + 118*D*a^4*b^5*c^3*d^3* 
abs(b) - 88*C*a^3*b^6*c^3*d^3*abs(b) + 58*B*a^2*b^7*c^3*d^3*abs(b) - 28*A* 
a*b^8*c^3*d^3*abs(b) - 132*D*a^5*b^4*c^2*d^4*abs(b) + 102*C*a^4*b^5*c^2...
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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