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

Optimal. Leaf size=1348 \[ \text{result too large to display} \]

[Out]

((d*e - c*f)*(8*a^2*d^2*f^2*(C*(5*d^2*e^2 + 6*c*d*e*f + 5*c^2*f^2) + 8*d*f*(2*A*
d*f - B*(d*e + c*f))) - 8*a*b*d*f*(C*(7*d^3*e^3 + 9*c*d^2*e^2*f + 9*c^2*d*e*f^2
+ 7*c^3*f^3) + 2*d*f*(8*A*d*f*(d*e + c*f) - B*(5*d^2*e^2 + 6*c*d*e*f + 5*c^2*f^2
))) + b^2*(C*(21*d^4*e^4 + 28*c*d^3*e^3*f + 30*c^2*d^2*e^2*f^2 + 28*c^3*d*e*f^3
+ 21*c^4*f^4) + 4*d*f*(2*A*d*f*(5*d^2*e^2 + 6*c*d*e*f + 5*c^2*f^2) - B*(7*d^3*e^
3 + 9*c*d^2*e^2*f + 9*c^2*d*e*f^2 + 7*c^3*f^3))))*Sqrt[c + d*x]*Sqrt[e + f*x])/(
512*d^5*f^5) + ((8*a^2*d^2*f^2*(C*(5*d^2*e^2 + 6*c*d*e*f + 5*c^2*f^2) + 8*d*f*(2
*A*d*f - B*(d*e + c*f))) - 8*a*b*d*f*(C*(7*d^3*e^3 + 9*c*d^2*e^2*f + 9*c^2*d*e*f
^2 + 7*c^3*f^3) + 2*d*f*(8*A*d*f*(d*e + c*f) - B*(5*d^2*e^2 + 6*c*d*e*f + 5*c^2*
f^2))) + b^2*(C*(21*d^4*e^4 + 28*c*d^3*e^3*f + 30*c^2*d^2*e^2*f^2 + 28*c^3*d*e*f
^3 + 21*c^4*f^4) + 4*d*f*(2*A*d*f*(5*d^2*e^2 + 6*c*d*e*f + 5*c^2*f^2) - B*(7*d^3
*e^3 + 9*c*d^2*e^2*f + 9*c^2*d*e*f^2 + 7*c^3*f^3))))*(c + d*x)^(3/2)*Sqrt[e + f*
x])/(256*d^5*f^4) - ((2*a*C*d*f - b*(4*B*d*f - 3*C*(d*e + c*f)))*(a + b*x)^2*(c
+ d*x)^(3/2)*(e + f*x)^(3/2))/(20*b*d^2*f^2) + (C*(a + b*x)^3*(c + d*x)^(3/2)*(e
+ f*x)^(3/2))/(6*b*d*f) - ((c + d*x)^(3/2)*(e + f*x)^(3/2)*(64*a^3*C*d^3*f^3 -
8*a^2*b*d^2*f^2*(16*B*d*f - 7*C*(d*e + c*f)) - 8*a*b^2*d*f*(C*(35*d^2*e^2 + 38*c
*d*e*f + 35*c^2*f^2) + 10*d*f*(8*A*d*f - 5*B*(d*e + c*f))) + b^3*(7*C*(15*d^3*e^
3 + 17*c*d^2*e^2*f + 17*c^2*d*e*f^2 + 15*c^3*f^3) + 4*d*f*(50*A*d*f*(d*e + c*f)
- B*(35*d^2*e^2 + 38*c*d*e*f + 35*c^2*f^2))) + 6*b*d*f*(10*b*d*f*(2*b*c*C*e + a*
C*d*e + a*c*C*f - 4*A*b*d*f) + (4*a*d*f - 7*b*(d*e + c*f))*(2*a*C*d*f - b*(4*B*d
*f - 3*C*(d*e + c*f))))*x))/(960*b*d^4*f^4) - ((d*e - c*f)^2*(8*a^2*d^2*f^2*(C*(
5*d^2*e^2 + 6*c*d*e*f + 5*c^2*f^2) + 8*d*f*(2*A*d*f - B*(d*e + c*f))) - 8*a*b*d*
f*(C*(7*d^3*e^3 + 9*c*d^2*e^2*f + 9*c^2*d*e*f^2 + 7*c^3*f^3) + 2*d*f*(8*A*d*f*(d
*e + c*f) - B*(5*d^2*e^2 + 6*c*d*e*f + 5*c^2*f^2))) + b^2*(C*(21*d^4*e^4 + 28*c*
d^3*e^3*f + 30*c^2*d^2*e^2*f^2 + 28*c^3*d*e*f^3 + 21*c^4*f^4) + 4*d*f*(2*A*d*f*(
5*d^2*e^2 + 6*c*d*e*f + 5*c^2*f^2) - B*(7*d^3*e^3 + 9*c*d^2*e^2*f + 9*c^2*d*e*f^
2 + 7*c^3*f^3))))*ArcTanh[(Sqrt[f]*Sqrt[c + d*x])/(Sqrt[d]*Sqrt[e + f*x])])/(512
*d^(11/2)*f^(11/2))

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Rubi [A]  time = 6.34984, antiderivative size = 1345, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 36, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167 \[ \frac{C (c+d x)^{3/2} (e+f x)^{3/2} (a+b x)^3}{6 b d f}+\frac{(4 b B d f-2 a C d f-3 b C (d e+c f)) (c+d x)^{3/2} (e+f x)^{3/2} (a+b x)^2}{20 b d^2 f^2}-\frac{(c+d x)^{3/2} (e+f x)^{3/2} \left (\left (7 C \left (15 d^3 e^3+17 c d^2 f e^2+17 c^2 d f^2 e+15 c^3 f^3\right )+4 d f \left (50 A d f (d e+c f)-B \left (35 d^2 e^2+38 c d f e+35 c^2 f^2\right )\right )\right ) b^3-8 a d f \left (C \left (35 d^2 e^2+38 c d f e+35 c^2 f^2\right )+10 d f (8 A d f-5 B (d e+c f))\right ) b^2-8 a^2 d^2 f^2 (16 B d f-7 C (d e+c f)) b+6 d f (10 b d f (2 b c C e+a C d e+a c C f-4 A b d f)-(4 a d f-7 b (d e+c f)) (4 b B d f-2 a C d f-3 b C (d e+c f))) x b+64 a^3 C d^3 f^3\right )}{960 b d^4 f^4}-\frac{(d e-c f)^2 \left (\left (C \left (21 d^4 e^4+28 c d^3 f e^3+30 c^2 d^2 f^2 e^2+28 c^3 d f^3 e+21 c^4 f^4\right )+4 d f \left (2 A d f \left (5 d^2 e^2+6 c d f e+5 c^2 f^2\right )-B \left (7 d^3 e^3+9 c d^2 f e^2+9 c^2 d f^2 e+7 c^3 f^3\right )\right )\right ) b^2-8 a d f \left (C \left (7 d^3 e^3+9 c d^2 f e^2+9 c^2 d f^2 e+7 c^3 f^3\right )+2 d f \left (8 A d f (d e+c f)-B \left (5 d^2 e^2+6 c d f e+5 c^2 f^2\right )\right )\right ) b+8 a^2 d^2 f^2 \left (C \left (5 d^2 e^2+6 c d f e+5 c^2 f^2\right )+8 d f (2 A d f-B (d e+c f))\right )\right ) \tanh ^{-1}\left (\frac{\sqrt{f} \sqrt{c+d x}}{\sqrt{d} \sqrt{e+f x}}\right )}{512 d^{11/2} f^{11/2}}+\frac{\left (\left (C \left (21 d^4 e^4+28 c d^3 f e^3+30 c^2 d^2 f^2 e^2+28 c^3 d f^3 e+21 c^4 f^4\right )+4 d f \left (2 A d f \left (5 d^2 e^2+6 c d f e+5 c^2 f^2\right )-B \left (7 d^3 e^3+9 c d^2 f e^2+9 c^2 d f^2 e+7 c^3 f^3\right )\right )\right ) b^2-8 a d f \left (C \left (7 d^3 e^3+9 c d^2 f e^2+9 c^2 d f^2 e+7 c^3 f^3\right )+2 d f \left (8 A d f (d e+c f)-B \left (5 d^2 e^2+6 c d f e+5 c^2 f^2\right )\right )\right ) b+8 a^2 d^2 f^2 \left (C \left (5 d^2 e^2+6 c d f e+5 c^2 f^2\right )+8 d f (2 A d f-B (d e+c f))\right )\right ) (c+d x)^{3/2} \sqrt{e+f x}}{256 d^5 f^4}+\frac{(d e-c f) \left (\left (C \left (21 d^4 e^4+28 c d^3 f e^3+30 c^2 d^2 f^2 e^2+28 c^3 d f^3 e+21 c^4 f^4\right )+4 d f \left (2 A d f \left (5 d^2 e^2+6 c d f e+5 c^2 f^2\right )-B \left (7 d^3 e^3+9 c d^2 f e^2+9 c^2 d f^2 e+7 c^3 f^3\right )\right )\right ) b^2-8 a d f \left (C \left (7 d^3 e^3+9 c d^2 f e^2+9 c^2 d f^2 e+7 c^3 f^3\right )+2 d f \left (8 A d f (d e+c f)-B \left (5 d^2 e^2+6 c d f e+5 c^2 f^2\right )\right )\right ) b+8 a^2 d^2 f^2 \left (C \left (5 d^2 e^2+6 c d f e+5 c^2 f^2\right )+8 d f (2 A d f-B (d e+c f))\right )\right ) \sqrt{c+d x} \sqrt{e+f x}}{512 d^5 f^5} \]

Antiderivative was successfully verified.

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

[Out]

((d*e - c*f)*(8*a^2*d^2*f^2*(C*(5*d^2*e^2 + 6*c*d*e*f + 5*c^2*f^2) + 8*d*f*(2*A*
d*f - B*(d*e + c*f))) - 8*a*b*d*f*(C*(7*d^3*e^3 + 9*c*d^2*e^2*f + 9*c^2*d*e*f^2
+ 7*c^3*f^3) + 2*d*f*(8*A*d*f*(d*e + c*f) - B*(5*d^2*e^2 + 6*c*d*e*f + 5*c^2*f^2
))) + b^2*(C*(21*d^4*e^4 + 28*c*d^3*e^3*f + 30*c^2*d^2*e^2*f^2 + 28*c^3*d*e*f^3
+ 21*c^4*f^4) + 4*d*f*(2*A*d*f*(5*d^2*e^2 + 6*c*d*e*f + 5*c^2*f^2) - B*(7*d^3*e^
3 + 9*c*d^2*e^2*f + 9*c^2*d*e*f^2 + 7*c^3*f^3))))*Sqrt[c + d*x]*Sqrt[e + f*x])/(
512*d^5*f^5) + ((8*a^2*d^2*f^2*(C*(5*d^2*e^2 + 6*c*d*e*f + 5*c^2*f^2) + 8*d*f*(2
*A*d*f - B*(d*e + c*f))) - 8*a*b*d*f*(C*(7*d^3*e^3 + 9*c*d^2*e^2*f + 9*c^2*d*e*f
^2 + 7*c^3*f^3) + 2*d*f*(8*A*d*f*(d*e + c*f) - B*(5*d^2*e^2 + 6*c*d*e*f + 5*c^2*
f^2))) + b^2*(C*(21*d^4*e^4 + 28*c*d^3*e^3*f + 30*c^2*d^2*e^2*f^2 + 28*c^3*d*e*f
^3 + 21*c^4*f^4) + 4*d*f*(2*A*d*f*(5*d^2*e^2 + 6*c*d*e*f + 5*c^2*f^2) - B*(7*d^3
*e^3 + 9*c*d^2*e^2*f + 9*c^2*d*e*f^2 + 7*c^3*f^3))))*(c + d*x)^(3/2)*Sqrt[e + f*
x])/(256*d^5*f^4) + ((4*b*B*d*f - 2*a*C*d*f - 3*b*C*(d*e + c*f))*(a + b*x)^2*(c
+ d*x)^(3/2)*(e + f*x)^(3/2))/(20*b*d^2*f^2) + (C*(a + b*x)^3*(c + d*x)^(3/2)*(e
+ f*x)^(3/2))/(6*b*d*f) - ((c + d*x)^(3/2)*(e + f*x)^(3/2)*(64*a^3*C*d^3*f^3 -
8*a^2*b*d^2*f^2*(16*B*d*f - 7*C*(d*e + c*f)) - 8*a*b^2*d*f*(C*(35*d^2*e^2 + 38*c
*d*e*f + 35*c^2*f^2) + 10*d*f*(8*A*d*f - 5*B*(d*e + c*f))) + b^3*(7*C*(15*d^3*e^
3 + 17*c*d^2*e^2*f + 17*c^2*d*e*f^2 + 15*c^3*f^3) + 4*d*f*(50*A*d*f*(d*e + c*f)
- B*(35*d^2*e^2 + 38*c*d*e*f + 35*c^2*f^2))) + 6*b*d*f*(10*b*d*f*(2*b*c*C*e + a*
C*d*e + a*c*C*f - 4*A*b*d*f) - (4*a*d*f - 7*b*(d*e + c*f))*(4*b*B*d*f - 2*a*C*d*
f - 3*b*C*(d*e + c*f)))*x))/(960*b*d^4*f^4) - ((d*e - c*f)^2*(8*a^2*d^2*f^2*(C*(
5*d^2*e^2 + 6*c*d*e*f + 5*c^2*f^2) + 8*d*f*(2*A*d*f - B*(d*e + c*f))) - 8*a*b*d*
f*(C*(7*d^3*e^3 + 9*c*d^2*e^2*f + 9*c^2*d*e*f^2 + 7*c^3*f^3) + 2*d*f*(8*A*d*f*(d
*e + c*f) - B*(5*d^2*e^2 + 6*c*d*e*f + 5*c^2*f^2))) + b^2*(C*(21*d^4*e^4 + 28*c*
d^3*e^3*f + 30*c^2*d^2*e^2*f^2 + 28*c^3*d*e*f^3 + 21*c^4*f^4) + 4*d*f*(2*A*d*f*(
5*d^2*e^2 + 6*c*d*e*f + 5*c^2*f^2) - B*(7*d^3*e^3 + 9*c*d^2*e^2*f + 9*c^2*d*e*f^
2 + 7*c^3*f^3))))*ArcTanh[(Sqrt[f]*Sqrt[c + d*x])/(Sqrt[d]*Sqrt[e + f*x])])/(512
*d^(11/2)*f^(11/2))

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Rubi in Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)**2*(C*x**2+B*x+A)*(d*x+c)**(1/2)*(f*x+e)**(1/2),x)

[Out]

Timed out

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Mathematica [A]  time = 6.53181, size = 1801, normalized size = 1.34 \[ \sqrt{c+d x} \sqrt{e+f x} \left (\frac{1}{6} b^2 C x^5+\frac{b (b C d e+b c C f+12 b B d f+24 a C d f) x^4}{60 d f}+\frac{\left (-9 C d^2 e^2 b^2+120 A d^2 f^2 b^2-9 c^2 C f^2 b^2+12 B c d f^2 b^2+12 B d^2 e f b^2+2 c C d e f b^2+240 a B d^2 f^2 b+24 a c C d f^2 b+24 a C d^2 e f b+120 a^2 C d^2 f^2\right ) x^3}{480 d^2 f^2}+\frac{\left (21 b^2 C e^3 d^3+640 a A b f^3 d^3+320 a^2 B f^3 d^3+40 A b^2 e f^2 d^3+80 a b B e f^2 d^3+40 a^2 C e f^2 d^3-28 b^2 B e^2 f d^3-56 a b C e^2 f d^3+40 A b^2 c f^3 d^2+80 a b B c f^3 d^2+40 a^2 c C f^3 d^2+8 b^2 B c e f^2 d^2+16 a b c C e f^2 d^2-5 b^2 c C e^2 f d^2-28 b^2 B c^2 f^3 d-56 a b c^2 C f^3 d-5 b^2 c^2 C e f^2 d+21 b^2 c^3 C f^3\right ) x^2}{960 d^3 f^3}+\frac{\left (-105 b^2 C e^4 d^4+1920 a^2 A f^4 d^4+640 a A b e f^3 d^4+320 a^2 B e f^3 d^4-200 A b^2 e^2 f^2 d^4-400 a b B e^2 f^2 d^4-200 a^2 C e^2 f^2 d^4+140 b^2 B e^3 f d^4+280 a b C e^3 f d^4+640 a A b c f^4 d^3+320 a^2 B c f^4 d^3+80 A b^2 c e f^3 d^3+160 a b B c e f^3 d^3+80 a^2 c C e f^3 d^3-44 b^2 B c e^2 f^2 d^3-88 a b c C e^2 f^2 d^3+28 b^2 c C e^3 f d^3-200 A b^2 c^2 f^4 d^2-400 a b B c^2 f^4 d^2-200 a^2 c^2 C f^4 d^2-44 b^2 B c^2 e f^3 d^2-88 a b c^2 C e f^3 d^2+26 b^2 c^2 C e^2 f^2 d^2+140 b^2 B c^3 f^4 d+280 a b c^3 C f^4 d+28 b^2 c^3 C e f^3 d-105 b^2 c^4 C f^4\right ) x}{3840 d^4 f^4}+\frac{315 b^2 C e^5 d^5+1920 a^2 A e f^4 d^5-1920 a A b e^2 f^3 d^5-960 a^2 B e^2 f^3 d^5+600 A b^2 e^3 f^2 d^5+1200 a b B e^3 f^2 d^5+600 a^2 C e^3 f^2 d^5-420 b^2 B e^4 f d^5-840 a b C e^4 f d^5+1920 a^2 A c f^5 d^4+1280 a A b c e f^4 d^4+640 a^2 B c e f^4 d^4-280 A b^2 c e^2 f^3 d^4-560 a b B c e^2 f^3 d^4-280 a^2 c C e^2 f^3 d^4+160 b^2 B c e^3 f^2 d^4+320 a b c C e^3 f^2 d^4-105 b^2 c C e^4 f d^4-1920 a A b c^2 f^5 d^3-960 a^2 B c^2 f^5 d^3-280 A b^2 c^2 e f^4 d^3-560 a b B c^2 e f^4 d^3-280 a^2 c^2 C e f^4 d^3+136 b^2 B c^2 e^2 f^3 d^3+272 a b c^2 C e^2 f^3 d^3-82 b^2 c^2 C e^3 f^2 d^3+600 A b^2 c^3 f^5 d^2+1200 a b B c^3 f^5 d^2+600 a^2 c^3 C f^5 d^2+160 b^2 B c^3 e f^4 d^2+320 a b c^3 C e f^4 d^2-82 b^2 c^3 C e^2 f^3 d^2-420 b^2 B c^4 f^5 d-840 a b c^4 C f^5 d-105 b^2 c^4 C e f^4 d+315 b^2 c^5 C f^5}{7680 d^5 f^5}\right )-\frac{(d e-c f)^2 \left (21 b^2 C e^4 d^4+128 a^2 A f^4 d^4-128 a A b e f^3 d^4-64 a^2 B e f^3 d^4+40 A b^2 e^2 f^2 d^4+80 a b B e^2 f^2 d^4+40 a^2 C e^2 f^2 d^4-28 b^2 B e^3 f d^4-56 a b C e^3 f d^4-128 a A b c f^4 d^3-64 a^2 B c f^4 d^3+48 A b^2 c e f^3 d^3+96 a b B c e f^3 d^3+48 a^2 c C e f^3 d^3-36 b^2 B c e^2 f^2 d^3-72 a b c C e^2 f^2 d^3+28 b^2 c C e^3 f d^3+40 A b^2 c^2 f^4 d^2+80 a b B c^2 f^4 d^2+40 a^2 c^2 C f^4 d^2-36 b^2 B c^2 e f^3 d^2-72 a b c^2 C e f^3 d^2+30 b^2 c^2 C e^2 f^2 d^2-28 b^2 B c^3 f^4 d-56 a b c^3 C f^4 d+28 b^2 c^3 C e f^3 d+21 b^2 c^4 C f^4\right ) \log \left (d e+c f+2 d f x+2 \sqrt{d} \sqrt{f} \sqrt{c+d x} \sqrt{e+f x}\right )}{1024 d^{11/2} f^{11/2}} \]

Antiderivative was successfully verified.

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

[Out]

Sqrt[c + d*x]*Sqrt[e + f*x]*((315*b^2*C*d^5*e^5 - 105*b^2*c*C*d^4*e^4*f - 420*b^
2*B*d^5*e^4*f - 840*a*b*C*d^5*e^4*f - 82*b^2*c^2*C*d^3*e^3*f^2 + 160*b^2*B*c*d^4
*e^3*f^2 + 320*a*b*c*C*d^4*e^3*f^2 + 600*A*b^2*d^5*e^3*f^2 + 1200*a*b*B*d^5*e^3*
f^2 + 600*a^2*C*d^5*e^3*f^2 - 82*b^2*c^3*C*d^2*e^2*f^3 + 136*b^2*B*c^2*d^3*e^2*f
^3 + 272*a*b*c^2*C*d^3*e^2*f^3 - 280*A*b^2*c*d^4*e^2*f^3 - 560*a*b*B*c*d^4*e^2*f
^3 - 280*a^2*c*C*d^4*e^2*f^3 - 1920*a*A*b*d^5*e^2*f^3 - 960*a^2*B*d^5*e^2*f^3 -
105*b^2*c^4*C*d*e*f^4 + 160*b^2*B*c^3*d^2*e*f^4 + 320*a*b*c^3*C*d^2*e*f^4 - 280*
A*b^2*c^2*d^3*e*f^4 - 560*a*b*B*c^2*d^3*e*f^4 - 280*a^2*c^2*C*d^3*e*f^4 + 1280*a
*A*b*c*d^4*e*f^4 + 640*a^2*B*c*d^4*e*f^4 + 1920*a^2*A*d^5*e*f^4 + 315*b^2*c^5*C*
f^5 - 420*b^2*B*c^4*d*f^5 - 840*a*b*c^4*C*d*f^5 + 600*A*b^2*c^3*d^2*f^5 + 1200*a
*b*B*c^3*d^2*f^5 + 600*a^2*c^3*C*d^2*f^5 - 1920*a*A*b*c^2*d^3*f^5 - 960*a^2*B*c^
2*d^3*f^5 + 1920*a^2*A*c*d^4*f^5)/(7680*d^5*f^5) + ((-105*b^2*C*d^4*e^4 + 28*b^2
*c*C*d^3*e^3*f + 140*b^2*B*d^4*e^3*f + 280*a*b*C*d^4*e^3*f + 26*b^2*c^2*C*d^2*e^
2*f^2 - 44*b^2*B*c*d^3*e^2*f^2 - 88*a*b*c*C*d^3*e^2*f^2 - 200*A*b^2*d^4*e^2*f^2
- 400*a*b*B*d^4*e^2*f^2 - 200*a^2*C*d^4*e^2*f^2 + 28*b^2*c^3*C*d*e*f^3 - 44*b^2*
B*c^2*d^2*e*f^3 - 88*a*b*c^2*C*d^2*e*f^3 + 80*A*b^2*c*d^3*e*f^3 + 160*a*b*B*c*d^
3*e*f^3 + 80*a^2*c*C*d^3*e*f^3 + 640*a*A*b*d^4*e*f^3 + 320*a^2*B*d^4*e*f^3 - 105
*b^2*c^4*C*f^4 + 140*b^2*B*c^3*d*f^4 + 280*a*b*c^3*C*d*f^4 - 200*A*b^2*c^2*d^2*f
^4 - 400*a*b*B*c^2*d^2*f^4 - 200*a^2*c^2*C*d^2*f^4 + 640*a*A*b*c*d^3*f^4 + 320*a
^2*B*c*d^3*f^4 + 1920*a^2*A*d^4*f^4)*x)/(3840*d^4*f^4) + ((21*b^2*C*d^3*e^3 - 5*
b^2*c*C*d^2*e^2*f - 28*b^2*B*d^3*e^2*f - 56*a*b*C*d^3*e^2*f - 5*b^2*c^2*C*d*e*f^
2 + 8*b^2*B*c*d^2*e*f^2 + 16*a*b*c*C*d^2*e*f^2 + 40*A*b^2*d^3*e*f^2 + 80*a*b*B*d
^3*e*f^2 + 40*a^2*C*d^3*e*f^2 + 21*b^2*c^3*C*f^3 - 28*b^2*B*c^2*d*f^3 - 56*a*b*c
^2*C*d*f^3 + 40*A*b^2*c*d^2*f^3 + 80*a*b*B*c*d^2*f^3 + 40*a^2*c*C*d^2*f^3 + 640*
a*A*b*d^3*f^3 + 320*a^2*B*d^3*f^3)*x^2)/(960*d^3*f^3) + ((-9*b^2*C*d^2*e^2 + 2*b
^2*c*C*d*e*f + 12*b^2*B*d^2*e*f + 24*a*b*C*d^2*e*f - 9*b^2*c^2*C*f^2 + 12*b^2*B*
c*d*f^2 + 24*a*b*c*C*d*f^2 + 120*A*b^2*d^2*f^2 + 240*a*b*B*d^2*f^2 + 120*a^2*C*d
^2*f^2)*x^3)/(480*d^2*f^2) + (b*(b*C*d*e + b*c*C*f + 12*b*B*d*f + 24*a*C*d*f)*x^
4)/(60*d*f) + (b^2*C*x^5)/6) - ((d*e - c*f)^2*(21*b^2*C*d^4*e^4 + 28*b^2*c*C*d^3
*e^3*f - 28*b^2*B*d^4*e^3*f - 56*a*b*C*d^4*e^3*f + 30*b^2*c^2*C*d^2*e^2*f^2 - 36
*b^2*B*c*d^3*e^2*f^2 - 72*a*b*c*C*d^3*e^2*f^2 + 40*A*b^2*d^4*e^2*f^2 + 80*a*b*B*
d^4*e^2*f^2 + 40*a^2*C*d^4*e^2*f^2 + 28*b^2*c^3*C*d*e*f^3 - 36*b^2*B*c^2*d^2*e*f
^3 - 72*a*b*c^2*C*d^2*e*f^3 + 48*A*b^2*c*d^3*e*f^3 + 96*a*b*B*c*d^3*e*f^3 + 48*a
^2*c*C*d^3*e*f^3 - 128*a*A*b*d^4*e*f^3 - 64*a^2*B*d^4*e*f^3 + 21*b^2*c^4*C*f^4 -
28*b^2*B*c^3*d*f^4 - 56*a*b*c^3*C*d*f^4 + 40*A*b^2*c^2*d^2*f^4 + 80*a*b*B*c^2*d
^2*f^4 + 40*a^2*c^2*C*d^2*f^4 - 128*a*A*b*c*d^3*f^4 - 64*a^2*B*c*d^3*f^4 + 128*a
^2*A*d^4*f^4)*Log[d*e + c*f + 2*d*f*x + 2*Sqrt[d]*Sqrt[f]*Sqrt[c + d*x]*Sqrt[e +
f*x]])/(1024*d^(11/2)*f^(11/2))

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Maple [B]  time = 0.062, size = 6728, normalized size = 5. \[ \text{output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

result too large to display

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((C*x^2 + B*x + A)*(b*x + a)^2*sqrt(d*x + c)*sqrt(f*x + e),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 2.4495, size = 1, normalized size = 0. \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((C*x^2 + B*x + A)*(b*x + a)^2*sqrt(d*x + c)*sqrt(f*x + e),x, algorithm="fricas")

[Out]

[1/30720*(4*(1280*C*b^2*d^5*f^5*x^5 + 315*C*b^2*d^5*e^5 - 105*(C*b^2*c*d^4 + 4*(
2*C*a*b + B*b^2)*d^5)*e^4*f - 2*(41*C*b^2*c^2*d^3 - 80*(2*C*a*b + B*b^2)*c*d^4 -
300*(C*a^2 + 2*B*a*b + A*b^2)*d^5)*e^3*f^2 - 2*(41*C*b^2*c^3*d^2 - 68*(2*C*a*b
+ B*b^2)*c^2*d^3 + 140*(C*a^2 + 2*B*a*b + A*b^2)*c*d^4 + 480*(B*a^2 + 2*A*a*b)*d
^5)*e^2*f^3 - 5*(21*C*b^2*c^4*d - 384*A*a^2*d^5 - 32*(2*C*a*b + B*b^2)*c^3*d^2 +
56*(C*a^2 + 2*B*a*b + A*b^2)*c^2*d^3 - 128*(B*a^2 + 2*A*a*b)*c*d^4)*e*f^4 + 15*
(21*C*b^2*c^5 + 128*A*a^2*c*d^4 - 28*(2*C*a*b + B*b^2)*c^4*d + 40*(C*a^2 + 2*B*a
*b + A*b^2)*c^3*d^2 - 64*(B*a^2 + 2*A*a*b)*c^2*d^3)*f^5 + 128*(C*b^2*d^5*e*f^4 +
(C*b^2*c*d^4 + 12*(2*C*a*b + B*b^2)*d^5)*f^5)*x^4 - 16*(9*C*b^2*d^5*e^2*f^3 - 2
*(C*b^2*c*d^4 + 6*(2*C*a*b + B*b^2)*d^5)*e*f^4 + 3*(3*C*b^2*c^2*d^3 - 4*(2*C*a*b
+ B*b^2)*c*d^4 - 40*(C*a^2 + 2*B*a*b + A*b^2)*d^5)*f^5)*x^3 + 8*(21*C*b^2*d^5*e
^3*f^2 - (5*C*b^2*c*d^4 + 28*(2*C*a*b + B*b^2)*d^5)*e^2*f^3 - (5*C*b^2*c^2*d^3 -
8*(2*C*a*b + B*b^2)*c*d^4 - 40*(C*a^2 + 2*B*a*b + A*b^2)*d^5)*e*f^4 + (21*C*b^2
*c^3*d^2 - 28*(2*C*a*b + B*b^2)*c^2*d^3 + 40*(C*a^2 + 2*B*a*b + A*b^2)*c*d^4 + 3
20*(B*a^2 + 2*A*a*b)*d^5)*f^5)*x^2 - 2*(105*C*b^2*d^5*e^4*f - 28*(C*b^2*c*d^4 +
5*(2*C*a*b + B*b^2)*d^5)*e^3*f^2 - 2*(13*C*b^2*c^2*d^3 - 22*(2*C*a*b + B*b^2)*c*
d^4 - 100*(C*a^2 + 2*B*a*b + A*b^2)*d^5)*e^2*f^3 - 4*(7*C*b^2*c^3*d^2 - 11*(2*C*
a*b + B*b^2)*c^2*d^3 + 20*(C*a^2 + 2*B*a*b + A*b^2)*c*d^4 + 80*(B*a^2 + 2*A*a*b)
*d^5)*e*f^4 + 5*(21*C*b^2*c^4*d - 384*A*a^2*d^5 - 28*(2*C*a*b + B*b^2)*c^3*d^2 +
40*(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)*f^5)*x)*sqrt
(d*f)*sqrt(d*x + c)*sqrt(f*x + e) + 15*(21*C*b^2*d^6*e^6 - 14*(C*b^2*c*d^5 + 2*(
2*C*a*b + B*b^2)*d^6)*e^5*f - 5*(C*b^2*c^2*d^4 - 4*(2*C*a*b + B*b^2)*c*d^5 - 8*(
C*a^2 + 2*B*a*b + A*b^2)*d^6)*e^4*f^2 - 4*(C*b^2*c^3*d^3 - 2*(2*C*a*b + B*b^2)*c
^2*d^4 + 8*(C*a^2 + 2*B*a*b + A*b^2)*c*d^5 + 16*(B*a^2 + 2*A*a*b)*d^6)*e^3*f^3 -
(5*C*b^2*c^4*d^2 - 128*A*a^2*d^6 - 8*(2*C*a*b + B*b^2)*c^3*d^3 + 16*(C*a^2 + 2*
B*a*b + A*b^2)*c^2*d^4 - 64*(B*a^2 + 2*A*a*b)*c*d^5)*e^2*f^4 - 2*(7*C*b^2*c^5*d
+ 128*A*a^2*c*d^5 - 10*(2*C*a*b + B*b^2)*c^4*d^2 + 16*(C*a^2 + 2*B*a*b + A*b^2)*
c^3*d^3 - 32*(B*a^2 + 2*A*a*b)*c^2*d^4)*e*f^5 + (21*C*b^2*c^6 + 128*A*a^2*c^2*d^
4 - 28*(2*C*a*b + B*b^2)*c^5*d + 40*(C*a^2 + 2*B*a*b + A*b^2)*c^4*d^2 - 64*(B*a^
2 + 2*A*a*b)*c^3*d^3)*f^6)*log(-4*(2*d^2*f^2*x + d^2*e*f + c*d*f^2)*sqrt(d*x + c
)*sqrt(f*x + e) + (8*d^2*f^2*x^2 + d^2*e^2 + 6*c*d*e*f + c^2*f^2 + 8*(d^2*e*f +
c*d*f^2)*x)*sqrt(d*f)))/(sqrt(d*f)*d^5*f^5), 1/15360*(2*(1280*C*b^2*d^5*f^5*x^5
+ 315*C*b^2*d^5*e^5 - 105*(C*b^2*c*d^4 + 4*(2*C*a*b + B*b^2)*d^5)*e^4*f - 2*(41*
C*b^2*c^2*d^3 - 80*(2*C*a*b + B*b^2)*c*d^4 - 300*(C*a^2 + 2*B*a*b + A*b^2)*d^5)*
e^3*f^2 - 2*(41*C*b^2*c^3*d^2 - 68*(2*C*a*b + B*b^2)*c^2*d^3 + 140*(C*a^2 + 2*B*
a*b + A*b^2)*c*d^4 + 480*(B*a^2 + 2*A*a*b)*d^5)*e^2*f^3 - 5*(21*C*b^2*c^4*d - 38
4*A*a^2*d^5 - 32*(2*C*a*b + B*b^2)*c^3*d^2 + 56*(C*a^2 + 2*B*a*b + A*b^2)*c^2*d^
3 - 128*(B*a^2 + 2*A*a*b)*c*d^4)*e*f^4 + 15*(21*C*b^2*c^5 + 128*A*a^2*c*d^4 - 28
*(2*C*a*b + B*b^2)*c^4*d + 40*(C*a^2 + 2*B*a*b + A*b^2)*c^3*d^2 - 64*(B*a^2 + 2*
A*a*b)*c^2*d^3)*f^5 + 128*(C*b^2*d^5*e*f^4 + (C*b^2*c*d^4 + 12*(2*C*a*b + B*b^2)
*d^5)*f^5)*x^4 - 16*(9*C*b^2*d^5*e^2*f^3 - 2*(C*b^2*c*d^4 + 6*(2*C*a*b + B*b^2)*
d^5)*e*f^4 + 3*(3*C*b^2*c^2*d^3 - 4*(2*C*a*b + B*b^2)*c*d^4 - 40*(C*a^2 + 2*B*a*
b + A*b^2)*d^5)*f^5)*x^3 + 8*(21*C*b^2*d^5*e^3*f^2 - (5*C*b^2*c*d^4 + 28*(2*C*a*
b + B*b^2)*d^5)*e^2*f^3 - (5*C*b^2*c^2*d^3 - 8*(2*C*a*b + B*b^2)*c*d^4 - 40*(C*a
^2 + 2*B*a*b + A*b^2)*d^5)*e*f^4 + (21*C*b^2*c^3*d^2 - 28*(2*C*a*b + B*b^2)*c^2*
d^3 + 40*(C*a^2 + 2*B*a*b + A*b^2)*c*d^4 + 320*(B*a^2 + 2*A*a*b)*d^5)*f^5)*x^2 -
2*(105*C*b^2*d^5*e^4*f - 28*(C*b^2*c*d^4 + 5*(2*C*a*b + B*b^2)*d^5)*e^3*f^2 - 2
*(13*C*b^2*c^2*d^3 - 22*(2*C*a*b + B*b^2)*c*d^4 - 100*(C*a^2 + 2*B*a*b + A*b^2)*
d^5)*e^2*f^3 - 4*(7*C*b^2*c^3*d^2 - 11*(2*C*a*b + B*b^2)*c^2*d^3 + 20*(C*a^2 + 2
*B*a*b + A*b^2)*c*d^4 + 80*(B*a^2 + 2*A*a*b)*d^5)*e*f^4 + 5*(21*C*b^2*c^4*d - 38
4*A*a^2*d^5 - 28*(2*C*a*b + B*b^2)*c^3*d^2 + 40*(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)*f^5)*x)*sqrt(-d*f)*sqrt(d*x + c)*sqrt(f*x + e) -
15*(21*C*b^2*d^6*e^6 - 14*(C*b^2*c*d^5 + 2*(2*C*a*b + B*b^2)*d^6)*e^5*f - 5*(C*
b^2*c^2*d^4 - 4*(2*C*a*b + B*b^2)*c*d^5 - 8*(C*a^2 + 2*B*a*b + A*b^2)*d^6)*e^4*f
^2 - 4*(C*b^2*c^3*d^3 - 2*(2*C*a*b + B*b^2)*c^2*d^4 + 8*(C*a^2 + 2*B*a*b + A*b^2
)*c*d^5 + 16*(B*a^2 + 2*A*a*b)*d^6)*e^3*f^3 - (5*C*b^2*c^4*d^2 - 128*A*a^2*d^6 -
8*(2*C*a*b + B*b^2)*c^3*d^3 + 16*(C*a^2 + 2*B*a*b + A*b^2)*c^2*d^4 - 64*(B*a^2
+ 2*A*a*b)*c*d^5)*e^2*f^4 - 2*(7*C*b^2*c^5*d + 128*A*a^2*c*d^5 - 10*(2*C*a*b + B
*b^2)*c^4*d^2 + 16*(C*a^2 + 2*B*a*b + A*b^2)*c^3*d^3 - 32*(B*a^2 + 2*A*a*b)*c^2*
d^4)*e*f^5 + (21*C*b^2*c^6 + 128*A*a^2*c^2*d^4 - 28*(2*C*a*b + B*b^2)*c^5*d + 40
*(C*a^2 + 2*B*a*b + A*b^2)*c^4*d^2 - 64*(B*a^2 + 2*A*a*b)*c^3*d^3)*f^6)*arctan(1
/2*(2*d*f*x + d*e + c*f)*sqrt(-d*f)/(sqrt(d*x + c)*sqrt(f*x + e)*d*f)))/(sqrt(-d
*f)*d^5*f^5)]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \left (a + b x\right )^{2} \sqrt{c + d x} \sqrt{e + f x} \left (A + B x + C x^{2}\right )\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.480446, size = 1, normalized size = 0. \[ \mathit{Done} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((C*x^2 + B*x + A)*(b*x + a)^2*sqrt(d*x + c)*sqrt(f*x + e),x, algorithm="giac")

[Out]

Done