\(\int \frac {(a+b x)^2 \sqrt {c+d x} (A+B x+C x^2)}{\sqrt {e+f x}} \, dx\) [69]

Optimal result

Integrand size = 36, antiderivative size = 1035 \[ \int \frac {(a+b x)^2 \sqrt {c+d x} \left (A+B x+C x^2\right )}{\sqrt {e+f x}} \, dx=-\frac {\left (16 a^2 d^2 f^2 \left (2 d f (3 B d e+B c f-4 A d f)-C \left (5 d^2 e^2+2 c d e f+c^2 f^2\right )\right )+4 a b d f \left (C \left (35 d^3 e^3+15 c d^2 e^2 f+9 c^2 d e f^2+5 c^3 f^3\right )+8 d f \left (2 A d f (3 d e+c f)-B \left (5 d^2 e^2+2 c d e f+c^2 f^2\right )\right )\right )-b^2 \left (C \left (63 d^4 e^4+28 c d^3 e^3 f+18 c^2 d^2 e^2 f^2+12 c^3 d e f^3+7 c^4 f^4\right )+2 d f \left (8 A d f \left (5 d^2 e^2+2 c d e f+c^2 f^2\right )-B \left (35 d^3 e^3+15 c d^2 e^2 f+9 c^2 d e f^2+5 c^3 f^3\right )\right )\right )\right ) \sqrt {c+d x} \sqrt {e+f x}}{128 d^4 f^5}-\frac {\left (16 a^2 d^2 f^2 (5 C d e+7 c C f-6 B d f)+4 a b d f \left (8 d f (5 B d e+7 B c f-6 A d f)-C \left (35 d^2 e^2+50 c d e f+59 c^2 f^2\right )\right )+b^2 \left (C \left (63 d^3 e^3+91 c d^2 e^2 f+109 c^2 d e f^2+121 c^3 f^3\right )+2 d f \left (8 A d f (5 d e+7 c f)-B \left (35 d^2 e^2+50 c d e f+59 c^2 f^2\right )\right )\right )\right ) (c+d x)^{3/2} \sqrt {e+f x}}{192 d^4 f^4}+\frac {\left (80 a^2 C d^2 f^2-20 a b d f (7 C d e+17 c C f-8 B d f)-b^2 \left (10 d f (7 B d e+17 B c f-8 A d f)-C \left (63 d^2 e^2+154 c d e f+263 c^2 f^2\right )\right )\right ) (c+d x)^{5/2} \sqrt {e+f x}}{240 d^4 f^3}+\frac {b (20 a C d f-b (9 C d e+31 c C f-10 B d f)) (c+d x)^{7/2} \sqrt {e+f x}}{40 d^4 f^2}+\frac {b^2 C (c+d x)^{9/2} \sqrt {e+f x}}{5 d^4 f}+\frac {(d e-c f) \left (16 a^2 d^2 f^2 \left (2 d f (3 B d e+B c f-4 A d f)-C \left (5 d^2 e^2+2 c d e f+c^2 f^2\right )\right )+4 a b d f \left (C \left (35 d^3 e^3+15 c d^2 e^2 f+9 c^2 d e f^2+5 c^3 f^3\right )+8 d f \left (2 A d f (3 d e+c f)-B \left (5 d^2 e^2+2 c d e f+c^2 f^2\right )\right )\right )-b^2 \left (C \left (63 d^4 e^4+28 c d^3 e^3 f+18 c^2 d^2 e^2 f^2+12 c^3 d e f^3+7 c^4 f^4\right )+2 d f \left (8 A d f \left (5 d^2 e^2+2 c d e f+c^2 f^2\right )-B \left (35 d^3 e^3+15 c d^2 e^2 f+9 c^2 d e f^2+5 c^3 f^3\right )\right )\right )\right ) \text {arctanh}\left (\frac {\sqrt {f} \sqrt {c+d x}}{\sqrt {d} \sqrt {e+f x}}\right )}{128 d^{9/2} f^{11/2}} \] Output:

-1/128*(16*a^2*d^2*f^2*(2*d*f*(-4*A*d*f+B*c*f+3*B*d*e)-C*(c^2*f^2+2*c*d*e* 
f+5*d^2*e^2))+4*a*b*d*f*(C*(5*c^3*f^3+9*c^2*d*e*f^2+15*c*d^2*e^2*f+35*d^3* 
e^3)+8*d*f*(2*A*d*f*(c*f+3*d*e)-B*(c^2*f^2+2*c*d*e*f+5*d^2*e^2)))-b^2*(C*( 
7*c^4*f^4+12*c^3*d*e*f^3+18*c^2*d^2*e^2*f^2+28*c*d^3*e^3*f+63*d^4*e^4)+2*d 
*f*(8*A*d*f*(c^2*f^2+2*c*d*e*f+5*d^2*e^2)-B*(5*c^3*f^3+9*c^2*d*e*f^2+15*c* 
d^2*e^2*f+35*d^3*e^3))))*(d*x+c)^(1/2)*(f*x+e)^(1/2)/d^4/f^5-1/192*(16*a^2 
*d^2*f^2*(-6*B*d*f+7*C*c*f+5*C*d*e)+4*a*b*d*f*(8*d*f*(-6*A*d*f+7*B*c*f+5*B 
*d*e)-C*(59*c^2*f^2+50*c*d*e*f+35*d^2*e^2))+b^2*(C*(121*c^3*f^3+109*c^2*d* 
e*f^2+91*c*d^2*e^2*f+63*d^3*e^3)+2*d*f*(8*A*d*f*(7*c*f+5*d*e)-B*(59*c^2*f^ 
2+50*c*d*e*f+35*d^2*e^2))))*(d*x+c)^(3/2)*(f*x+e)^(1/2)/d^4/f^4+1/240*(80* 
a^2*C*d^2*f^2-20*a*b*d*f*(-8*B*d*f+17*C*c*f+7*C*d*e)-b^2*(10*d*f*(-8*A*d*f 
+17*B*c*f+7*B*d*e)-C*(263*c^2*f^2+154*c*d*e*f+63*d^2*e^2)))*(d*x+c)^(5/2)* 
(f*x+e)^(1/2)/d^4/f^3+1/40*b*(20*a*C*d*f-b*(-10*B*d*f+31*C*c*f+9*C*d*e))*( 
d*x+c)^(7/2)*(f*x+e)^(1/2)/d^4/f^2+1/5*b^2*C*(d*x+c)^(9/2)*(f*x+e)^(1/2)/d 
^4/f+1/128*(-c*f+d*e)*(16*a^2*d^2*f^2*(2*d*f*(-4*A*d*f+B*c*f+3*B*d*e)-C*(c 
^2*f^2+2*c*d*e*f+5*d^2*e^2))+4*a*b*d*f*(C*(5*c^3*f^3+9*c^2*d*e*f^2+15*c*d^ 
2*e^2*f+35*d^3*e^3)+8*d*f*(2*A*d*f*(c*f+3*d*e)-B*(c^2*f^2+2*c*d*e*f+5*d^2* 
e^2)))-b^2*(C*(7*c^4*f^4+12*c^3*d*e*f^3+18*c^2*d^2*e^2*f^2+28*c*d^3*e^3*f+ 
63*d^4*e^4)+2*d*f*(8*A*d*f*(c^2*f^2+2*c*d*e*f+5*d^2*e^2)-B*(5*c^3*f^3+9*c^ 
2*d*e*f^2+15*c*d^2*e^2*f+35*d^3*e^3))))*arctanh(f^(1/2)*(d*x+c)^(1/2)/d...
 

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(3220\) vs. \(2(1035)=2070\).

Time = 16.91 (sec) , antiderivative size = 3220, normalized size of antiderivative = 3.11 \[ \int \frac {(a+b x)^2 \sqrt {c+d x} \left (A+B x+C x^2\right )}{\sqrt {e+f x}} \, dx=\text {Result too large to show} \] Input:

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

Output:

((-(b*e) + a*f)^2*(d*e - c*f)^2*(C*e^2 - B*e*f + A*f^2)*Sqrt[d/((d^2*e)/(d 
*e - c*f) - (c*d*f)/(d*e - c*f))]*((d^2*e)/(d*e - c*f) - (c*d*f)/(d*e - c* 
f))^2*Sqrt[(d*(e + f*x))/(d*e - c*f)]*Sqrt[1 + (d*f*(c + d*x))/((d*e - c*f 
)*((d^2*e)/(d*e - c*f) - (c*d*f)/(d*e - c*f)))]*((2*d*f*(c + d*x))/((d*e - 
 c*f)*((d^2*e)/(d*e - c*f) - (c*d*f)/(d*e - c*f))) - (2*Sqrt[d]*Sqrt[f]*Sq 
rt[c + d*x]*ArcSinh[(Sqrt[d]*Sqrt[f]*Sqrt[c + d*x])/(Sqrt[d*e - c*f]*Sqrt[ 
(d^2*e)/(d*e - c*f) - (c*d*f)/(d*e - c*f)])])/(Sqrt[d*e - c*f]*Sqrt[(d^2*e 
)/(d*e - c*f) - (c*d*f)/(d*e - c*f)]*Sqrt[1 + (d*f*(c + d*x))/((d*e - c*f) 
*((d^2*e)/(d*e - c*f) - (c*d*f)/(d*e - c*f)))])))/(2*d^3*f^6*Sqrt[c + d*x] 
*Sqrt[e + f*x]) + (2*b^2*C*(d*e - c*f)^3*(c + d*x)^(3/2)*Sqrt[e + f*x]*(1 
+ (d*f*(c + d*x))/((d*e - c*f)*((d^2*e)/(d*e - c*f) - (c*d*f)/(d*e - c*f)) 
))^(9/2)*((3*(35/(64*(1 + (d*f*(c + d*x))/((d*e - c*f)*((d^2*e)/(d*e - c*f 
) - (c*d*f)/(d*e - c*f))))^4) + 35/(48*(1 + (d*f*(c + d*x))/((d*e - c*f)*( 
(d^2*e)/(d*e - c*f) - (c*d*f)/(d*e - c*f))))^3) + 7/(8*(1 + (d*f*(c + d*x) 
)/((d*e - c*f)*((d^2*e)/(d*e - c*f) - (c*d*f)/(d*e - c*f))))^2) + (1 + (d* 
f*(c + d*x))/((d*e - c*f)*((d^2*e)/(d*e - c*f) - (c*d*f)/(d*e - c*f))))^(- 
1)))/10 + (21*(d*e - c*f)^2*((d^2*e)/(d*e - c*f) - (c*d*f)/(d*e - c*f))^2* 
((2*d*f*(c + d*x))/((d*e - c*f)*((d^2*e)/(d*e - c*f) - (c*d*f)/(d*e - c*f) 
)) - (2*Sqrt[d]*Sqrt[f]*Sqrt[c + d*x]*ArcSinh[(Sqrt[d]*Sqrt[f]*Sqrt[c + d* 
x])/(Sqrt[d*e - c*f]*Sqrt[(d^2*e)/(d*e - c*f) - (c*d*f)/(d*e - c*f)])])...
 

Rubi [A] (verified)

Time = 1.33 (sec) , antiderivative size = 778, normalized size of antiderivative = 0.75, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {2118, 27, 170, 27, 164, 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.

\(\displaystyle \int \frac {(a+b x)^2 \sqrt {c+d x} \left (A+B x+C x^2\right )}{\sqrt {e+f x}} \, dx\)

\(\Big \downarrow \) 2118

\(\displaystyle \frac {\int -\frac {b (a+b x)^2 \sqrt {c+d x} (6 b c C e+3 a C d e+a c C f-10 A b d f+(4 a C d f+b (9 C d e+7 c C f-10 B d f)) x)}{2 \sqrt {e+f x}}dx}{5 b^2 d f}+\frac {C (a+b x)^3 (c+d x)^{3/2} \sqrt {e+f x}}{5 b d f}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {C (a+b x)^3 (c+d x)^{3/2} \sqrt {e+f x}}{5 b d f}-\frac {\int \frac {(a+b x)^2 \sqrt {c+d x} (6 b c C e+3 a C d e+a c C f-10 A b d f+(4 a C d f+b (9 C d e+7 c C f-10 B d f)) x)}{\sqrt {e+f x}}dx}{10 b d f}\)

\(\Big \downarrow \) 170

\(\displaystyle \frac {C (a+b x)^3 (c+d x)^{3/2} \sqrt {e+f x}}{5 b d f}-\frac {\frac {\int \frac {(a+b x) \sqrt {c+d x} (8 a d f (6 b c C e+3 a C d e+a c C f-10 A b d f)-(4 b c e+3 a d e+a c f) (4 a C d f+b (9 C d e+7 c C f-10 B d f))+(8 b d f (6 b c C e+3 a C d e+a c C f-10 A b d f)-(7 b d e+5 b c f-4 a d f) (4 a C d f+b (9 C d e+7 c C f-10 B d f))) x)}{2 \sqrt {e+f x}}dx}{4 d f}+\frac {(a+b x)^2 (c+d x)^{3/2} \sqrt {e+f x} (4 a C d f+b (-10 B d f+7 c C f+9 C d e))}{4 d f}}{10 b d f}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {C (a+b x)^3 (c+d x)^{3/2} \sqrt {e+f x}}{5 b d f}-\frac {\frac {\int \frac {(a+b x) \sqrt {c+d x} (8 a d f (6 b c C e+3 a C d e+a c C f-10 A b d f)-(4 b c e+3 a d e+a c f) (4 a C d f+b (9 C d e+7 c C f-10 B d f))+(8 b d f (6 b c C e+3 a C d e+a c C f-10 A b d f)-(7 b d e+5 b c f-4 a d f) (4 a C d f+b (9 C d e+7 c C f-10 B d f))) x)}{\sqrt {e+f x}}dx}{8 d f}+\frac {(a+b x)^2 (c+d x)^{3/2} \sqrt {e+f x} (4 a C d f+b (-10 B d f+7 c C f+9 C d e))}{4 d f}}{10 b d f}\)

\(\Big \downarrow \) 164

\(\displaystyle \frac {C (a+b x)^3 (c+d x)^{3/2} \sqrt {e+f x}}{5 b d f}-\frac {\frac {\frac {5 b \left (16 a^2 d^2 f^2 \left (2 d f (-4 A d f+B c f+3 B d e)-C \left (c^2 f^2+2 c d e f+5 d^2 e^2\right )\right )+4 a b d f \left (8 d f \left (2 A d f (c f+3 d e)-B \left (c^2 f^2+2 c d e f+5 d^2 e^2\right )\right )+C \left (5 c^3 f^3+9 c^2 d e f^2+15 c d^2 e^2 f+35 d^3 e^3\right )\right )-\left (b^2 \left (2 d f \left (8 A d f \left (c^2 f^2+2 c d e f+5 d^2 e^2\right )-B \left (5 c^3 f^3+9 c^2 d e f^2+15 c d^2 e^2 f+35 d^3 e^3\right )\right )+C \left (7 c^4 f^4+12 c^3 d e f^3+18 c^2 d^2 e^2 f^2+28 c d^3 e^3 f+63 d^4 e^4\right )\right )\right )\right ) \int \frac {\sqrt {c+d x}}{\sqrt {e+f x}}dx}{8 d^2 f^2}+\frac {(c+d x)^{3/2} \sqrt {e+f x} \left (96 a^3 C d^3 f^3+8 a^2 b d^2 f^2 (-30 B d f+9 c C f+23 C d e)+20 a b^2 d f \left (8 d f (-6 A d f+3 B c f+5 B d e)-C \left (15 c^2 f^2+22 c d e f+35 d^2 e^2\right )\right )+4 b d f x (8 b d f (a c C f+3 a C d e-10 A b d f+6 b c C e)-(-4 a d f+5 b c f+7 b d e) (4 a C d f+b (-10 B d f+7 c C f+9 C d e)))+b^3 \left (10 d f \left (8 A d f (3 c f+5 d e)-B \left (15 c^2 f^2+22 c d e f+35 d^2 e^2\right )\right )+C \left (105 c^3 f^3+145 c^2 d e f^2+203 c d^2 e^2 f+315 d^3 e^3\right )\right )\right )}{12 d^2 f^2}}{8 d f}+\frac {(a+b x)^2 (c+d x)^{3/2} \sqrt {e+f x} (4 a C d f+b (-10 B d f+7 c C f+9 C d e))}{4 d f}}{10 b d f}\)

\(\Big \downarrow \) 60

\(\displaystyle \frac {C (a+b x)^3 (c+d x)^{3/2} \sqrt {e+f x}}{5 b d f}-\frac {\frac {\frac {5 b \left (16 a^2 d^2 f^2 \left (2 d f (-4 A d f+B c f+3 B d e)-C \left (c^2 f^2+2 c d e f+5 d^2 e^2\right )\right )+4 a b d f \left (8 d f \left (2 A d f (c f+3 d e)-B \left (c^2 f^2+2 c d e f+5 d^2 e^2\right )\right )+C \left (5 c^3 f^3+9 c^2 d e f^2+15 c d^2 e^2 f+35 d^3 e^3\right )\right )-\left (b^2 \left (2 d f \left (8 A d f \left (c^2 f^2+2 c d e f+5 d^2 e^2\right )-B \left (5 c^3 f^3+9 c^2 d e f^2+15 c d^2 e^2 f+35 d^3 e^3\right )\right )+C \left (7 c^4 f^4+12 c^3 d e f^3+18 c^2 d^2 e^2 f^2+28 c d^3 e^3 f+63 d^4 e^4\right )\right )\right )\right ) \left (\frac {\sqrt {c+d x} \sqrt {e+f x}}{f}-\frac {(d e-c f) \int \frac {1}{\sqrt {c+d x} \sqrt {e+f x}}dx}{2 f}\right )}{8 d^2 f^2}+\frac {(c+d x)^{3/2} \sqrt {e+f x} \left (96 a^3 C d^3 f^3+8 a^2 b d^2 f^2 (-30 B d f+9 c C f+23 C d e)+20 a b^2 d f \left (8 d f (-6 A d f+3 B c f+5 B d e)-C \left (15 c^2 f^2+22 c d e f+35 d^2 e^2\right )\right )+4 b d f x (8 b d f (a c C f+3 a C d e-10 A b d f+6 b c C e)-(-4 a d f+5 b c f+7 b d e) (4 a C d f+b (-10 B d f+7 c C f+9 C d e)))+b^3 \left (10 d f \left (8 A d f (3 c f+5 d e)-B \left (15 c^2 f^2+22 c d e f+35 d^2 e^2\right )\right )+C \left (105 c^3 f^3+145 c^2 d e f^2+203 c d^2 e^2 f+315 d^3 e^3\right )\right )\right )}{12 d^2 f^2}}{8 d f}+\frac {(a+b x)^2 (c+d x)^{3/2} \sqrt {e+f x} (4 a C d f+b (-10 B d f+7 c C f+9 C d e))}{4 d f}}{10 b d f}\)

\(\Big \downarrow \) 66

\(\displaystyle \frac {C (a+b x)^3 (c+d x)^{3/2} \sqrt {e+f x}}{5 b d f}-\frac {\frac {\frac {5 b \left (16 a^2 d^2 f^2 \left (2 d f (-4 A d f+B c f+3 B d e)-C \left (c^2 f^2+2 c d e f+5 d^2 e^2\right )\right )+4 a b d f \left (8 d f \left (2 A d f (c f+3 d e)-B \left (c^2 f^2+2 c d e f+5 d^2 e^2\right )\right )+C \left (5 c^3 f^3+9 c^2 d e f^2+15 c d^2 e^2 f+35 d^3 e^3\right )\right )-\left (b^2 \left (2 d f \left (8 A d f \left (c^2 f^2+2 c d e f+5 d^2 e^2\right )-B \left (5 c^3 f^3+9 c^2 d e f^2+15 c d^2 e^2 f+35 d^3 e^3\right )\right )+C \left (7 c^4 f^4+12 c^3 d e f^3+18 c^2 d^2 e^2 f^2+28 c d^3 e^3 f+63 d^4 e^4\right )\right )\right )\right ) \left (\frac {\sqrt {c+d x} \sqrt {e+f x}}{f}-\frac {(d e-c f) \int \frac {1}{d-\frac {f (c+d x)}{e+f x}}d\frac {\sqrt {c+d x}}{\sqrt {e+f x}}}{f}\right )}{8 d^2 f^2}+\frac {(c+d x)^{3/2} \sqrt {e+f x} \left (96 a^3 C d^3 f^3+8 a^2 b d^2 f^2 (-30 B d f+9 c C f+23 C d e)+20 a b^2 d f \left (8 d f (-6 A d f+3 B c f+5 B d e)-C \left (15 c^2 f^2+22 c d e f+35 d^2 e^2\right )\right )+4 b d f x (8 b d f (a c C f+3 a C d e-10 A b d f+6 b c C e)-(-4 a d f+5 b c f+7 b d e) (4 a C d f+b (-10 B d f+7 c C f+9 C d e)))+b^3 \left (10 d f \left (8 A d f (3 c f+5 d e)-B \left (15 c^2 f^2+22 c d e f+35 d^2 e^2\right )\right )+C \left (105 c^3 f^3+145 c^2 d e f^2+203 c d^2 e^2 f+315 d^3 e^3\right )\right )\right )}{12 d^2 f^2}}{8 d f}+\frac {(a+b x)^2 (c+d x)^{3/2} \sqrt {e+f x} (4 a C d f+b (-10 B d f+7 c C f+9 C d e))}{4 d f}}{10 b d f}\)

\(\Big \downarrow \) 221

\(\displaystyle \frac {C (a+b x)^3 (c+d x)^{3/2} \sqrt {e+f x}}{5 b d f}-\frac {\frac {\frac {5 b \left (\frac {\sqrt {c+d x} \sqrt {e+f x}}{f}-\frac {(d e-c f) \text {arctanh}\left (\frac {\sqrt {f} \sqrt {c+d x}}{\sqrt {d} \sqrt {e+f x}}\right )}{\sqrt {d} f^{3/2}}\right ) \left (16 a^2 d^2 f^2 \left (2 d f (-4 A d f+B c f+3 B d e)-C \left (c^2 f^2+2 c d e f+5 d^2 e^2\right )\right )+4 a b d f \left (8 d f \left (2 A d f (c f+3 d e)-B \left (c^2 f^2+2 c d e f+5 d^2 e^2\right )\right )+C \left (5 c^3 f^3+9 c^2 d e f^2+15 c d^2 e^2 f+35 d^3 e^3\right )\right )-\left (b^2 \left (2 d f \left (8 A d f \left (c^2 f^2+2 c d e f+5 d^2 e^2\right )-B \left (5 c^3 f^3+9 c^2 d e f^2+15 c d^2 e^2 f+35 d^3 e^3\right )\right )+C \left (7 c^4 f^4+12 c^3 d e f^3+18 c^2 d^2 e^2 f^2+28 c d^3 e^3 f+63 d^4 e^4\right )\right )\right )\right )}{8 d^2 f^2}+\frac {(c+d x)^{3/2} \sqrt {e+f x} \left (96 a^3 C d^3 f^3+8 a^2 b d^2 f^2 (-30 B d f+9 c C f+23 C d e)+20 a b^2 d f \left (8 d f (-6 A d f+3 B c f+5 B d e)-C \left (15 c^2 f^2+22 c d e f+35 d^2 e^2\right )\right )+4 b d f x (8 b d f (a c C f+3 a C d e-10 A b d f+6 b c C e)-(-4 a d f+5 b c f+7 b d e) (4 a C d f+b (-10 B d f+7 c C f+9 C d e)))+b^3 \left (10 d f \left (8 A d f (3 c f+5 d e)-B \left (15 c^2 f^2+22 c d e f+35 d^2 e^2\right )\right )+C \left (105 c^3 f^3+145 c^2 d e f^2+203 c d^2 e^2 f+315 d^3 e^3\right )\right )\right )}{12 d^2 f^2}}{8 d f}+\frac {(a+b x)^2 (c+d x)^{3/2} \sqrt {e+f x} (4 a C d f+b (-10 B d f+7 c C f+9 C d e))}{4 d f}}{10 b d f}\)

Input:

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

Output:

(C*(a + b*x)^3*(c + d*x)^(3/2)*Sqrt[e + f*x])/(5*b*d*f) - (((4*a*C*d*f + b 
*(9*C*d*e + 7*c*C*f - 10*B*d*f))*(a + b*x)^2*(c + d*x)^(3/2)*Sqrt[e + f*x] 
)/(4*d*f) + (((c + d*x)^(3/2)*Sqrt[e + f*x]*(96*a^3*C*d^3*f^3 + 8*a^2*b*d^ 
2*f^2*(23*C*d*e + 9*c*C*f - 30*B*d*f) + 20*a*b^2*d*f*(8*d*f*(5*B*d*e + 3*B 
*c*f - 6*A*d*f) - C*(35*d^2*e^2 + 22*c*d*e*f + 15*c^2*f^2)) + b^3*(C*(315* 
d^3*e^3 + 203*c*d^2*e^2*f + 145*c^2*d*e*f^2 + 105*c^3*f^3) + 10*d*f*(8*A*d 
*f*(5*d*e + 3*c*f) - B*(35*d^2*e^2 + 22*c*d*e*f + 15*c^2*f^2))) + 4*b*d*f* 
(8*b*d*f*(6*b*c*C*e + 3*a*C*d*e + a*c*C*f - 10*A*b*d*f) - (7*b*d*e + 5*b*c 
*f - 4*a*d*f)*(4*a*C*d*f + b*(9*C*d*e + 7*c*C*f - 10*B*d*f)))*x))/(12*d^2* 
f^2) + (5*b*(16*a^2*d^2*f^2*(2*d*f*(3*B*d*e + B*c*f - 4*A*d*f) - C*(5*d^2* 
e^2 + 2*c*d*e*f + c^2*f^2)) + 4*a*b*d*f*(C*(35*d^3*e^3 + 15*c*d^2*e^2*f + 
9*c^2*d*e*f^2 + 5*c^3*f^3) + 8*d*f*(2*A*d*f*(3*d*e + c*f) - B*(5*d^2*e^2 + 
 2*c*d*e*f + c^2*f^2))) - b^2*(C*(63*d^4*e^4 + 28*c*d^3*e^3*f + 18*c^2*d^2 
*e^2*f^2 + 12*c^3*d*e*f^3 + 7*c^4*f^4) + 2*d*f*(8*A*d*f*(5*d^2*e^2 + 2*c*d 
*e*f + c^2*f^2) - B*(35*d^3*e^3 + 15*c*d^2*e^2*f + 9*c^2*d*e*f^2 + 5*c^3*f 
^3))))*((Sqrt[c + d*x]*Sqrt[e + f*x])/f - ((d*e - c*f)*ArcTanh[(Sqrt[f]*Sq 
rt[c + d*x])/(Sqrt[d]*Sqrt[e + f*x])])/(Sqrt[d]*f^(3/2))))/(8*d^2*f^2))/(8 
*d*f))/(10*b*d*f)
 

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 164

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_) + (f_.)*(x_ 
))*((g_.) + (h_.)*(x_)), x_] :> Simp[(-(a*d*f*h*(n + 2) + b*c*f*h*(m + 2) - 
 b*d*(f*g + e*h)*(m + n + 3) - b*d*f*h*(m + n + 2)*x))*(a + b*x)^(m + 1)*(( 
c + d*x)^(n + 1)/(b^2*d^2*(m + n + 2)*(m + n + 3))), x] + Simp[(a^2*d^2*f*h 
*(n + 1)*(n + 2) + a*b*d*(n + 1)*(2*c*f*h*(m + 1) - d*(f*g + e*h)*(m + n + 
3)) + b^2*(c^2*f*h*(m + 1)*(m + 2) - c*d*(f*g + e*h)*(m + 1)*(m + n + 3) + 
d^2*e*g*(m + n + 2)*(m + n + 3)))/(b^2*d^2*(m + n + 2)*(m + n + 3))   Int[( 
a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, e, f, g, h, m, n}, x] 
&& NeQ[m + n + 2, 0] && NeQ[m + n + 3, 0]
 

rule 170

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) 
)^(p_)*((g_.) + (h_.)*(x_)), x_] :> Simp[h*(a + b*x)^m*(c + d*x)^(n + 1)*(( 
e + f*x)^(p + 1)/(d*f*(m + n + p + 2))), x] + Simp[1/(d*f*(m + n + p + 2)) 
  Int[(a + b*x)^(m - 1)*(c + d*x)^n*(e + f*x)^p*Simp[a*d*f*g*(m + n + p + 2 
) - h*(b*c*e*m + a*(d*e*(n + 1) + c*f*(p + 1))) + (b*d*f*g*(m + n + p + 2) 
+ h*(a*d*f*m - b*(d*e*(m + n + 1) + c*f*(m + p + 1))))*x, x], x], x] /; Fre 
eQ[{a, b, c, d, e, f, g, h, n, p}, x] && GtQ[m, 0] && NeQ[m + n + p + 2, 0] 
 && IntegerQ[m]
 

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 2118

Int[(Px_)*((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f 
_.)*(x_))^(p_.), x_Symbol] :> With[{q = Expon[Px, x], k = Coeff[Px, x, Expo 
n[Px, x]]}, Simp[k*(a + b*x)^(m + q - 1)*(c + d*x)^(n + 1)*((e + f*x)^(p + 
1)/(d*f*b^(q - 1)*(m + n + p + q + 1))), x] + Simp[1/(d*f*b^q*(m + n + p + 
q + 1))   Int[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p*ExpandToSum[d*f*b^q*(m + 
n + p + q + 1)*Px - d*f*k*(m + n + p + q + 1)*(a + b*x)^q + k*(a + b*x)^(q 
- 2)*(a^2*d*f*(m + n + p + q + 1) - b*(b*c*e*(m + q - 1) + a*(d*e*(n + 1) + 
 c*f*(p + 1))) + b*(a*d*f*(2*(m + q) + n + p) - b*(d*e*(m + q + n) + c*f*(m 
 + q + p)))*x), x], x], x] /; NeQ[m + n + p + q + 1, 0]] /; FreeQ[{a, b, c, 
 d, e, f, m, n, p}, x] && PolyQ[Px, x]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(3957\) vs. \(2(991)=1982\).

Time = 0.64 (sec) , antiderivative size = 3958, normalized size of antiderivative = 3.82

method	result	size
default	\(\text {Expression too large to display}\)	\(3958\)

Input:

int((b*x+a)^2*(d*x+c)^(1/2)*(C*x^2+B*x+A)/(f*x+e)^(1/2),x,method=_RETURNVE 
RBOSE)
 

Output:

1/3840*(d*x+c)^(1/2)*(f*x+e)^(1/2)*(90*C*ln(1/2*(2*d*f*x+2*((f*x+e)*(d*x+c 
))^(1/2)*(d*f)^(1/2)+c*f+d*e)/(d*f)^(1/2))*b^2*c^3*d^2*e^2*f^3+150*C*ln(1/ 
2*(2*d*f*x+2*((f*x+e)*(d*x+c))^(1/2)*(d*f)^(1/2)+c*f+d*e)/(d*f)^(1/2))*b^2 
*c^2*d^3*e^3*f^2-1120*B*b^2*d^4*e*f^3*x^2*((f*x+e)*(d*x+c))^(1/2)*(d*f)^(1 
/2)-112*C*b^2*c^2*d^2*f^4*x^2*((f*x+e)*(d*x+c))^(1/2)*(d*f)^(1/2)+1008*C*b 
^2*d^4*e^2*f^2*x^2*((f*x+e)*(d*x+c))^(1/2)*(d*f)^(1/2)+2800*C*((f*x+e)*(d* 
x+c))^(1/2)*(d*f)^(1/2)*a*b*d^4*e^2*f^2*x-210*C*((f*x+e)*(d*x+c))^(1/2)*(d 
*f)^(1/2)*b^2*c^4*f^4+240*A*ln(1/2*(2*d*f*x+2*((f*x+e)*(d*x+c))^(1/2)*(d*f 
)^(1/2)+c*f+d*e)/(d*f)^(1/2))*b^2*c^3*d^2*f^5-960*B*ln(1/2*(2*d*f*x+2*((f* 
x+e)*(d*x+c))^(1/2)*(d*f)^(1/2)+c*f+d*e)/(d*f)^(1/2))*a^2*c*d^4*e*f^4-2400 
*B*ln(1/2*(2*d*f*x+2*((f*x+e)*(d*x+c))^(1/2)*(d*f)^(1/2)+c*f+d*e)/(d*f)^(1 
/2))*a*b*d^5*e^3*f^2-600*B*ln(1/2*(2*d*f*x+2*((f*x+e)*(d*x+c))^(1/2)*(d*f) 
^(1/2)+c*f+d*e)/(d*f)^(1/2))*b^2*c*d^4*e^3*f^2-960*B*((f*x+e)*(d*x+c))^(1/ 
2)*(d*f)^(1/2)*a*b*c^2*d^2*f^4+2400*A*(d*f)^(1/2)*((f*x+e)*(d*x+c))^(1/2)* 
b^2*d^4*e^2*f^2-3200*B*((f*x+e)*(d*x+c))^(1/2)*(d*f)^(1/2)*a*b*d^4*e*f^3*x 
+140*C*((f*x+e)*(d*x+c))^(1/2)*(d*f)^(1/2)*b^2*c^3*d*f^4*x-1260*C*((f*x+e) 
*(d*x+c))^(1/2)*(d*f)^(1/2)*b^2*d^4*e^3*f*x-2880*B*(d*f)^(1/2)*((f*x+e)*(d 
*x+c))^(1/2)*a^2*d^4*e*f^3-2100*B*(d*f)^(1/2)*((f*x+e)*(d*x+c))^(1/2)*b^2* 
d^4*e^3*f+2400*C*(d*f)^(1/2)*((f*x+e)*(d*x+c))^(1/2)*a^2*d^4*e^2*f^2+340*B 
*((f*x+e)*(d*x+c))^(1/2)*(d*f)^(1/2)*b^2*c^2*d^2*e*f^3+500*B*((f*x+e)*(...
 

Fricas [A] (verification not implemented)

Time = 2.49 (sec) , antiderivative size = 2176, normalized size of antiderivative = 2.10 \[ \int \frac {(a+b x)^2 \sqrt {c+d x} \left (A+B x+C x^2\right )}{\sqrt {e+f x}} \, dx=\text {Too large to display} \] Input:

integrate((b*x+a)^2*(d*x+c)^(1/2)*(C*x^2+B*x+A)/(f*x+e)^(1/2),x, algorithm 
="fricas")
 

Output:

[-1/7680*(15*(63*C*b^2*d^5*e^5 - 35*(C*b^2*c*d^4 + 2*(2*C*a*b + B*b^2)*d^5 
)*e^4*f - 10*(C*b^2*c^2*d^3 - 4*(2*C*a*b + B*b^2)*c*d^4 - 8*(C*a^2 + 2*B*a 
*b + A*b^2)*d^5)*e^3*f^2 - 6*(C*b^2*c^3*d^2 - 2*(2*C*a*b + B*b^2)*c^2*d^3 
+ 8*(C*a^2 + 2*B*a*b + A*b^2)*c*d^4 + 16*(B*a^2 + 2*A*a*b)*d^5)*e^2*f^3 - 
(5*C*b^2*c^4*d - 128*A*a^2*d^5 - 8*(2*C*a*b + B*b^2)*c^3*d^2 + 16*(C*a^2 + 
 2*B*a*b + A*b^2)*c^2*d^3 - 64*(B*a^2 + 2*A*a*b)*c*d^4)*e*f^4 - (7*C*b^2*c 
^5 + 128*A*a^2*c*d^4 - 10*(2*C*a*b + B*b^2)*c^4*d + 16*(C*a^2 + 2*B*a*b + 
A*b^2)*c^3*d^2 - 32*(B*a^2 + 2*A*a*b)*c^2*d^3)*f^5)*sqrt(d*f)*log(8*d^2*f^ 
2*x^2 + d^2*e^2 + 6*c*d*e*f + c^2*f^2 + 4*(2*d*f*x + d*e + c*f)*sqrt(d*f)* 
sqrt(d*x + c)*sqrt(f*x + e) + 8*(d^2*e*f + c*d*f^2)*x) - 4*(384*C*b^2*d^5* 
f^5*x^4 + 945*C*b^2*d^5*e^4*f - 210*(C*b^2*c*d^4 + 5*(2*C*a*b + B*b^2)*d^5 
)*e^3*f^2 - 2*(68*C*b^2*c^2*d^3 - 125*(2*C*a*b + B*b^2)*c*d^4 - 600*(C*a^2 
 + 2*B*a*b + A*b^2)*d^5)*e^2*f^3 - 10*(11*C*b^2*c^3*d^2 - 17*(2*C*a*b + B* 
b^2)*c^2*d^3 + 32*(C*a^2 + 2*B*a*b + A*b^2)*c*d^4 + 144*(B*a^2 + 2*A*a*b)* 
d^5)*e*f^4 - 15*(7*C*b^2*c^4*d - 128*A*a^2*d^5 - 10*(2*C*a*b + B*b^2)*c^3* 
d^2 + 16*(C*a^2 + 2*B*a*b + A*b^2)*c^2*d^3 - 32*(B*a^2 + 2*A*a*b)*c*d^4)*f 
^5 - 48*(9*C*b^2*d^5*e*f^4 - (C*b^2*c*d^4 + 10*(2*C*a*b + B*b^2)*d^5)*f^5) 
*x^3 + 8*(63*C*b^2*d^5*e^2*f^3 - 2*(4*C*b^2*c*d^4 + 35*(2*C*a*b + B*b^2)*d 
^5)*e*f^4 - (7*C*b^2*c^2*d^3 - 10*(2*C*a*b + B*b^2)*c*d^4 - 80*(C*a^2 + 2* 
B*a*b + A*b^2)*d^5)*f^5)*x^2 - 2*(315*C*b^2*d^5*e^3*f^2 - 7*(7*C*b^2*c*...
 

Sympy [F]

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

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

Output:

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

Maxima [F(-2)]

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

integrate((b*x+a)^2*(d*x+c)^(1/2)*(C*x^2+B*x+A)/(f*x+e)^(1/2),x, algorithm 
="maxima")
 

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(c*f-d*e>0)', see `assume?` for m 
ore detail
 

Giac [A] (verification not implemented)

Time = 0.27 (sec) , antiderivative size = 1509, normalized size of antiderivative = 1.46 \[ \int \frac {(a+b x)^2 \sqrt {c+d x} \left (A+B x+C x^2\right )}{\sqrt {e+f x}} \, dx=\text {Too large to display} \] Input:

integrate((b*x+a)^2*(d*x+c)^(1/2)*(C*x^2+B*x+A)/(f*x+e)^(1/2),x, algorithm 
="giac")
 

Output:

1/1920*(sqrt(d^2*e + (d*x + c)*d*f - c*d*f)*(2*(4*(d*x + c)*(6*(d*x + c)*( 
8*(d*x + c)*C*b^2/(d^5*f) - (9*C*b^2*d^21*e*f^7 + 31*C*b^2*c*d^20*f^8 - 20 
*C*a*b*d^21*f^8 - 10*B*b^2*d^21*f^8)/(d^25*f^9)) + (63*C*b^2*d^22*e^2*f^6 
+ 154*C*b^2*c*d^21*e*f^7 - 140*C*a*b*d^22*e*f^7 - 70*B*b^2*d^22*e*f^7 + 26 
3*C*b^2*c^2*d^20*f^8 - 340*C*a*b*c*d^21*f^8 - 170*B*b^2*c*d^21*f^8 + 80*C* 
a^2*d^22*f^8 + 160*B*a*b*d^22*f^8 + 80*A*b^2*d^22*f^8)/(d^25*f^9)) - 5*(63 
*C*b^2*d^23*e^3*f^5 + 91*C*b^2*c*d^22*e^2*f^6 - 140*C*a*b*d^23*e^2*f^6 - 7 
0*B*b^2*d^23*e^2*f^6 + 109*C*b^2*c^2*d^21*e*f^7 - 200*C*a*b*c*d^22*e*f^7 - 
 100*B*b^2*c*d^22*e*f^7 + 80*C*a^2*d^23*e*f^7 + 160*B*a*b*d^23*e*f^7 + 80* 
A*b^2*d^23*e*f^7 + 121*C*b^2*c^3*d^20*f^8 - 236*C*a*b*c^2*d^21*f^8 - 118*B 
*b^2*c^2*d^21*f^8 + 112*C*a^2*c*d^22*f^8 + 224*B*a*b*c*d^22*f^8 + 112*A*b^ 
2*c*d^22*f^8 - 96*B*a^2*d^23*f^8 - 192*A*a*b*d^23*f^8)/(d^25*f^9))*(d*x + 
c) + 15*(63*C*b^2*d^24*e^4*f^4 + 28*C*b^2*c*d^23*e^3*f^5 - 140*C*a*b*d^24* 
e^3*f^5 - 70*B*b^2*d^24*e^3*f^5 + 18*C*b^2*c^2*d^22*e^2*f^6 - 60*C*a*b*c*d 
^23*e^2*f^6 - 30*B*b^2*c*d^23*e^2*f^6 + 80*C*a^2*d^24*e^2*f^6 + 160*B*a*b* 
d^24*e^2*f^6 + 80*A*b^2*d^24*e^2*f^6 + 12*C*b^2*c^3*d^21*e*f^7 - 36*C*a*b* 
c^2*d^22*e*f^7 - 18*B*b^2*c^2*d^22*e*f^7 + 32*C*a^2*c*d^23*e*f^7 + 64*B*a* 
b*c*d^23*e*f^7 + 32*A*b^2*c*d^23*e*f^7 - 96*B*a^2*d^24*e*f^7 - 192*A*a*b*d 
^24*e*f^7 + 7*C*b^2*c^4*d^20*f^8 - 20*C*a*b*c^3*d^21*f^8 - 10*B*b^2*c^3*d^ 
21*f^8 + 16*C*a^2*c^2*d^22*f^8 + 32*B*a*b*c^2*d^22*f^8 + 16*A*b^2*c^2*d...
 

Mupad [F(-1)]

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

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

Output:

\text{Hanged}
 

Reduce [F]

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

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

Output:

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