3.5 \(\int \frac{\left (c+d x+e x^2+f x^3\right )^2}{\sqrt{a+b x}} \, dx\)

Optimal. Leaf size=320 \[ -\frac{2 (a+b x)^{9/2} \left (-15 a^2 f^2+10 a b e f+b^2 \left (-\left (2 d f+e^2\right )\right )\right )}{9 b^7}+\frac{4 (a+b x)^{7/2} \left (-10 a^3 f^2+10 a^2 b e f-2 a b^2 \left (2 d f+e^2\right )+b^3 (c f+d e)\right )}{7 b^7}+\frac{4 (a+b x)^{3/2} \left (3 a^2 f-2 a b e+b^2 d\right ) \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{3 b^7}+\frac{2 \sqrt{a+b x} \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )^2}{b^7}+\frac{2 (a+b x)^{5/2} \left (15 a^4 f^2-20 a^3 b e f+6 a^2 b^2 \left (2 d f+e^2\right )-6 a b^3 (c f+d e)+b^4 \left (2 c e+d^2\right )\right )}{5 b^7}+\frac{4 f (a+b x)^{11/2} (b e-3 a f)}{11 b^7}+\frac{2 f^2 (a+b x)^{13/2}}{13 b^7} \]

[Out]

(2*(b^3*c - a*b^2*d + a^2*b*e - a^3*f)^2*Sqrt[a + b*x])/b^7 + (4*(b^2*d - 2*a*b*
e + 3*a^2*f)*(b^3*c - a*b^2*d + a^2*b*e - a^3*f)*(a + b*x)^(3/2))/(3*b^7) + (2*(
b^4*(d^2 + 2*c*e) - 20*a^3*b*e*f + 15*a^4*f^2 - 6*a*b^3*(d*e + c*f) + 6*a^2*b^2*
(e^2 + 2*d*f))*(a + b*x)^(5/2))/(5*b^7) + (4*(10*a^2*b*e*f - 10*a^3*f^2 + b^3*(d
*e + c*f) - 2*a*b^2*(e^2 + 2*d*f))*(a + b*x)^(7/2))/(7*b^7) - (2*(10*a*b*e*f - 1
5*a^2*f^2 - b^2*(e^2 + 2*d*f))*(a + b*x)^(9/2))/(9*b^7) + (4*f*(b*e - 3*a*f)*(a
+ b*x)^(11/2))/(11*b^7) + (2*f^2*(a + b*x)^(13/2))/(13*b^7)

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Rubi [A]  time = 0.511365, antiderivative size = 320, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.037 \[ -\frac{2 (a+b x)^{9/2} \left (-15 a^2 f^2+10 a b e f+b^2 \left (-\left (2 d f+e^2\right )\right )\right )}{9 b^7}+\frac{4 (a+b x)^{7/2} \left (-10 a^3 f^2+10 a^2 b e f-2 a b^2 \left (2 d f+e^2\right )+b^3 (c f+d e)\right )}{7 b^7}+\frac{4 (a+b x)^{3/2} \left (3 a^2 f-2 a b e+b^2 d\right ) \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{3 b^7}+\frac{2 \sqrt{a+b x} \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )^2}{b^7}+\frac{2 (a+b x)^{5/2} \left (15 a^4 f^2-20 a^3 b e f+6 a^2 b^2 \left (2 d f+e^2\right )-6 a b^3 (c f+d e)+b^4 \left (2 c e+d^2\right )\right )}{5 b^7}+\frac{4 f (a+b x)^{11/2} (b e-3 a f)}{11 b^7}+\frac{2 f^2 (a+b x)^{13/2}}{13 b^7} \]

Antiderivative was successfully verified.

[In]  Int[(c + d*x + e*x^2 + f*x^3)^2/Sqrt[a + b*x],x]

[Out]

(2*(b^3*c - a*b^2*d + a^2*b*e - a^3*f)^2*Sqrt[a + b*x])/b^7 + (4*(b^2*d - 2*a*b*
e + 3*a^2*f)*(b^3*c - a*b^2*d + a^2*b*e - a^3*f)*(a + b*x)^(3/2))/(3*b^7) + (2*(
b^4*(d^2 + 2*c*e) - 20*a^3*b*e*f + 15*a^4*f^2 - 6*a*b^3*(d*e + c*f) + 6*a^2*b^2*
(e^2 + 2*d*f))*(a + b*x)^(5/2))/(5*b^7) + (4*(10*a^2*b*e*f - 10*a^3*f^2 + b^3*(d
*e + c*f) - 2*a*b^2*(e^2 + 2*d*f))*(a + b*x)^(7/2))/(7*b^7) - (2*(10*a*b*e*f - 1
5*a^2*f^2 - b^2*(e^2 + 2*d*f))*(a + b*x)^(9/2))/(9*b^7) + (4*f*(b*e - 3*a*f)*(a
+ b*x)^(11/2))/(11*b^7) + (2*f^2*(a + b*x)^(13/2))/(13*b^7)

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Rubi in Sympy [A]  time = 120.574, size = 348, normalized size = 1.09 \[ \frac{2 f^{2} \left (a + b x\right )^{\frac{13}{2}}}{13 b^{7}} - \frac{4 f \left (a + b x\right )^{\frac{11}{2}} \left (3 a f - b e\right )}{11 b^{7}} + \frac{2 \left (a + b x\right )^{\frac{9}{2}} \left (15 a^{2} f^{2} - 10 a b e f + 2 b^{2} d f + b^{2} e^{2}\right )}{9 b^{7}} - \frac{4 \left (a + b x\right )^{\frac{7}{2}} \left (10 a^{3} f^{2} - 10 a^{2} b e f + 4 a b^{2} d f + 2 a b^{2} e^{2} - b^{3} c f - b^{3} d e\right )}{7 b^{7}} + \frac{2 \left (a + b x\right )^{\frac{5}{2}} \left (15 a^{4} f^{2} - 20 a^{3} b e f + 12 a^{2} b^{2} d f + 6 a^{2} b^{2} e^{2} - 6 a b^{3} c f - 6 a b^{3} d e + 2 b^{4} c e + b^{4} d^{2}\right )}{5 b^{7}} - \frac{4 \left (a + b x\right )^{\frac{3}{2}} \left (3 a^{2} f - 2 a b e + b^{2} d\right ) \left (a^{3} f - a^{2} b e + a b^{2} d - b^{3} c\right )}{3 b^{7}} + \frac{2 \sqrt{a + b x} \left (a^{3} f - a^{2} b e + a b^{2} d - b^{3} c\right )^{2}}{b^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((f*x**3+e*x**2+d*x+c)**2/(b*x+a)**(1/2),x)

[Out]

2*f**2*(a + b*x)**(13/2)/(13*b**7) - 4*f*(a + b*x)**(11/2)*(3*a*f - b*e)/(11*b**
7) + 2*(a + b*x)**(9/2)*(15*a**2*f**2 - 10*a*b*e*f + 2*b**2*d*f + b**2*e**2)/(9*
b**7) - 4*(a + b*x)**(7/2)*(10*a**3*f**2 - 10*a**2*b*e*f + 4*a*b**2*d*f + 2*a*b*
*2*e**2 - b**3*c*f - b**3*d*e)/(7*b**7) + 2*(a + b*x)**(5/2)*(15*a**4*f**2 - 20*
a**3*b*e*f + 12*a**2*b**2*d*f + 6*a**2*b**2*e**2 - 6*a*b**3*c*f - 6*a*b**3*d*e +
2*b**4*c*e + b**4*d**2)/(5*b**7) - 4*(a + b*x)**(3/2)*(3*a**2*f - 2*a*b*e + b**
2*d)*(a**3*f - a**2*b*e + a*b**2*d - b**3*c)/(3*b**7) + 2*sqrt(a + b*x)*(a**3*f
- a**2*b*e + a*b**2*d - b**3*c)**2/b**7

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Mathematica [A]  time = 0.565278, size = 320, normalized size = 1. \[ \frac{2 \sqrt{a+b x} \left (15360 a^6 f^2-2560 a^5 b f (13 e+3 f x)+128 a^4 b^2 \left (f \left (286 d+45 f x^2\right )+143 e^2+130 e f x\right )-32 a^3 b^3 \left (1287 c f+143 d (9 e+4 f x)+2 x \left (143 e^2+195 e f x+75 f^2 x^2\right )\right )+8 a^2 b^4 \left (858 c (7 e+3 f x)+3003 d^2+858 d x (3 e+2 f x)+x^2 \left (858 e^2+1300 e f x+525 f^2 x^2\right )\right )-4 a b^5 \left (429 c (35 d+x (14 e+9 f x))+x \left (3003 d^2+143 d x (27 e+20 f x)+5 x^2 \left (286 e^2+455 e f x+189 f^2 x^2\right )\right )\right )+b^6 \left (45045 c^2+858 c x (35 d+3 x (7 e+5 f x))+x^2 \left (9009 d^2+1430 d x (9 e+7 f x)+35 x^2 \left (143 e^2+234 e f x+99 f^2 x^2\right )\right )\right )\right )}{45045 b^7} \]

Antiderivative was successfully verified.

[In]  Integrate[(c + d*x + e*x^2 + f*x^3)^2/Sqrt[a + b*x],x]

[Out]

(2*Sqrt[a + b*x]*(15360*a^6*f^2 - 2560*a^5*b*f*(13*e + 3*f*x) + 128*a^4*b^2*(143
*e^2 + 130*e*f*x + f*(286*d + 45*f*x^2)) - 32*a^3*b^3*(1287*c*f + 143*d*(9*e + 4
*f*x) + 2*x*(143*e^2 + 195*e*f*x + 75*f^2*x^2)) + 8*a^2*b^4*(3003*d^2 + 858*d*x*
(3*e + 2*f*x) + 858*c*(7*e + 3*f*x) + x^2*(858*e^2 + 1300*e*f*x + 525*f^2*x^2))
+ b^6*(45045*c^2 + 858*c*x*(35*d + 3*x*(7*e + 5*f*x)) + x^2*(9009*d^2 + 1430*d*x
*(9*e + 7*f*x) + 35*x^2*(143*e^2 + 234*e*f*x + 99*f^2*x^2))) - 4*a*b^5*(429*c*(3
5*d + x*(14*e + 9*f*x)) + x*(3003*d^2 + 143*d*x*(27*e + 20*f*x) + 5*x^2*(286*e^2
+ 455*e*f*x + 189*f^2*x^2)))))/(45045*b^7)

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Maple [A]  time = 0.013, size = 447, normalized size = 1.4 \[{\frac{6930\,{f}^{2}{x}^{6}{b}^{6}-7560\,a{b}^{5}{f}^{2}{x}^{5}+16380\,{b}^{6}ef{x}^{5}+8400\,{a}^{2}{b}^{4}{f}^{2}{x}^{4}-18200\,a{b}^{5}ef{x}^{4}+20020\,{b}^{6}df{x}^{4}+10010\,{b}^{6}{e}^{2}{x}^{4}-9600\,{a}^{3}{b}^{3}{f}^{2}{x}^{3}+20800\,{a}^{2}{b}^{4}ef{x}^{3}-22880\,a{b}^{5}df{x}^{3}-11440\,a{b}^{5}{e}^{2}{x}^{3}+25740\,{b}^{6}cf{x}^{3}+25740\,{b}^{6}de{x}^{3}+11520\,{a}^{4}{b}^{2}{f}^{2}{x}^{2}-24960\,{a}^{3}{b}^{3}ef{x}^{2}+27456\,{a}^{2}{b}^{4}df{x}^{2}+13728\,{a}^{2}{b}^{4}{e}^{2}{x}^{2}-30888\,a{b}^{5}cf{x}^{2}-30888\,a{b}^{5}de{x}^{2}+36036\,{b}^{6}ce{x}^{2}+18018\,{b}^{6}{d}^{2}{x}^{2}-15360\,{a}^{5}b{f}^{2}x+33280\,{a}^{4}{b}^{2}efx-36608\,{a}^{3}{b}^{3}dfx-18304\,{a}^{3}{b}^{3}{e}^{2}x+41184\,{a}^{2}{b}^{4}cfx+41184\,{a}^{2}{b}^{4}dex-48048\,a{b}^{5}cex-24024\,a{b}^{5}{d}^{2}x+60060\,{b}^{6}cdx+30720\,{a}^{6}{f}^{2}-66560\,{a}^{5}bef+73216\,{a}^{4}{b}^{2}df+36608\,{a}^{4}{b}^{2}{e}^{2}-82368\,{a}^{3}{b}^{3}cf-82368\,{a}^{3}{b}^{3}de+96096\,{a}^{2}{b}^{4}ce+48048\,{a}^{2}{b}^{4}{d}^{2}-120120\,a{b}^{5}cd+90090\,{c}^{2}{b}^{6}}{45045\,{b}^{7}}\sqrt{bx+a}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((f*x^3+e*x^2+d*x+c)^2/(b*x+a)^(1/2),x)

[Out]

2/45045*(b*x+a)^(1/2)*(3465*b^6*f^2*x^6-3780*a*b^5*f^2*x^5+8190*b^6*e*f*x^5+4200
*a^2*b^4*f^2*x^4-9100*a*b^5*e*f*x^4+10010*b^6*d*f*x^4+5005*b^6*e^2*x^4-4800*a^3*
b^3*f^2*x^3+10400*a^2*b^4*e*f*x^3-11440*a*b^5*d*f*x^3-5720*a*b^5*e^2*x^3+12870*b
^6*c*f*x^3+12870*b^6*d*e*x^3+5760*a^4*b^2*f^2*x^2-12480*a^3*b^3*e*f*x^2+13728*a^
2*b^4*d*f*x^2+6864*a^2*b^4*e^2*x^2-15444*a*b^5*c*f*x^2-15444*a*b^5*d*e*x^2+18018
*b^6*c*e*x^2+9009*b^6*d^2*x^2-7680*a^5*b*f^2*x+16640*a^4*b^2*e*f*x-18304*a^3*b^3
*d*f*x-9152*a^3*b^3*e^2*x+20592*a^2*b^4*c*f*x+20592*a^2*b^4*d*e*x-24024*a*b^5*c*
e*x-12012*a*b^5*d^2*x+30030*b^6*c*d*x+15360*a^6*f^2-33280*a^5*b*e*f+36608*a^4*b^
2*d*f+18304*a^4*b^2*e^2-41184*a^3*b^3*c*f-41184*a^3*b^3*d*e+48048*a^2*b^4*c*e+24
024*a^2*b^4*d^2-60060*a*b^5*c*d+45045*b^6*c^2)/b^7

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Maxima [A]  time = 1.42126, size = 675, normalized size = 2.11 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((f*x^3 + e*x^2 + d*x + c)^2/sqrt(b*x + a),x, algorithm="maxima")

[Out]

2/45045*(45045*sqrt(b*x + a)*c^2 + 858*c*(35*((b*x + a)^(3/2) - 3*sqrt(b*x + a)*
a)*d/b + 7*(3*(b*x + a)^(5/2) - 10*(b*x + a)^(3/2)*a + 15*sqrt(b*x + a)*a^2)*e/b
^2 + 3*(5*(b*x + a)^(7/2) - 21*(b*x + a)^(5/2)*a + 35*(b*x + a)^(3/2)*a^2 - 35*s
qrt(b*x + a)*a^3)*f/b^3) + 3003*(3*(b*x + a)^(5/2) - 10*(b*x + a)^(3/2)*a + 15*s
qrt(b*x + a)*a^2)*d^2/b^2 + 143*(35*(b*x + a)^(9/2) - 180*(b*x + a)^(7/2)*a + 37
8*(b*x + a)^(5/2)*a^2 - 420*(b*x + a)^(3/2)*a^3 + 315*sqrt(b*x + a)*a^4)*e^2/b^4
+ 286*(35*(b*x + a)^(9/2)*f + 45*(b*e - 4*a*f)*(b*x + a)^(7/2) - 189*(a*b*e - 2
*a^2*f)*(b*x + a)^(5/2) + 105*(3*a^2*b*e - 4*a^3*f)*(b*x + a)^(3/2) - 315*(a^3*b
*e - a^4*f)*sqrt(b*x + a))*d/b^4 + 130*(63*(b*x + a)^(11/2) - 385*(b*x + a)^(9/2
)*a + 990*(b*x + a)^(7/2)*a^2 - 1386*(b*x + a)^(5/2)*a^3 + 1155*(b*x + a)^(3/2)*
a^4 - 693*sqrt(b*x + a)*a^5)*e*f/b^5 + 15*(231*(b*x + a)^(13/2) - 1638*(b*x + a)
^(11/2)*a + 5005*(b*x + a)^(9/2)*a^2 - 8580*(b*x + a)^(7/2)*a^3 + 9009*(b*x + a)
^(5/2)*a^4 - 6006*(b*x + a)^(3/2)*a^5 + 3003*sqrt(b*x + a)*a^6)*f^2/b^6)/b

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Fricas [A]  time = 0.218249, size = 563, normalized size = 1.76 \[ \frac{2 \,{\left (3465 \, b^{6} f^{2} x^{6} + 45045 \, b^{6} c^{2} - 60060 \, a b^{5} c d + 24024 \, a^{2} b^{4} d^{2} + 18304 \, a^{4} b^{2} e^{2} + 15360 \, a^{6} f^{2} + 630 \,{\left (13 \, b^{6} e f - 6 \, a b^{5} f^{2}\right )} x^{5} + 35 \,{\left (143 \, b^{6} e^{2} + 120 \, a^{2} b^{4} f^{2} + 26 \,{\left (11 \, b^{6} d - 10 \, a b^{5} e\right )} f\right )} x^{4} + 10 \,{\left (1287 \, b^{6} d e - 572 \, a b^{5} e^{2} - 480 \, a^{3} b^{3} f^{2} + 13 \,{\left (99 \, b^{6} c - 88 \, a b^{5} d + 80 \, a^{2} b^{4} e\right )} f\right )} x^{3} + 3 \,{\left (3003 \, b^{6} d^{2} + 2288 \, a^{2} b^{4} e^{2} + 1920 \, a^{4} b^{2} f^{2} + 858 \,{\left (7 \, b^{6} c - 6 \, a b^{5} d\right )} e - 52 \,{\left (99 \, a b^{5} c - 88 \, a^{2} b^{4} d + 80 \, a^{3} b^{3} e\right )} f\right )} x^{2} + 6864 \,{\left (7 \, a^{2} b^{4} c - 6 \, a^{3} b^{3} d\right )} e - 416 \,{\left (99 \, a^{3} b^{3} c - 88 \, a^{4} b^{2} d + 80 \, a^{5} b e\right )} f + 2 \,{\left (15015 \, b^{6} c d - 6006 \, a b^{5} d^{2} - 4576 \, a^{3} b^{3} e^{2} - 3840 \, a^{5} b f^{2} - 1716 \,{\left (7 \, a b^{5} c - 6 \, a^{2} b^{4} d\right )} e + 104 \,{\left (99 \, a^{2} b^{4} c - 88 \, a^{3} b^{3} d + 80 \, a^{4} b^{2} e\right )} f\right )} x\right )} \sqrt{b x + a}}{45045 \, b^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((f*x^3 + e*x^2 + d*x + c)^2/sqrt(b*x + a),x, algorithm="fricas")

[Out]

2/45045*(3465*b^6*f^2*x^6 + 45045*b^6*c^2 - 60060*a*b^5*c*d + 24024*a^2*b^4*d^2
+ 18304*a^4*b^2*e^2 + 15360*a^6*f^2 + 630*(13*b^6*e*f - 6*a*b^5*f^2)*x^5 + 35*(1
43*b^6*e^2 + 120*a^2*b^4*f^2 + 26*(11*b^6*d - 10*a*b^5*e)*f)*x^4 + 10*(1287*b^6*
d*e - 572*a*b^5*e^2 - 480*a^3*b^3*f^2 + 13*(99*b^6*c - 88*a*b^5*d + 80*a^2*b^4*e
)*f)*x^3 + 3*(3003*b^6*d^2 + 2288*a^2*b^4*e^2 + 1920*a^4*b^2*f^2 + 858*(7*b^6*c
- 6*a*b^5*d)*e - 52*(99*a*b^5*c - 88*a^2*b^4*d + 80*a^3*b^3*e)*f)*x^2 + 6864*(7*
a^2*b^4*c - 6*a^3*b^3*d)*e - 416*(99*a^3*b^3*c - 88*a^4*b^2*d + 80*a^5*b*e)*f +
2*(15015*b^6*c*d - 6006*a*b^5*d^2 - 4576*a^3*b^3*e^2 - 3840*a^5*b*f^2 - 1716*(7*
a*b^5*c - 6*a^2*b^4*d)*e + 104*(99*a^2*b^4*c - 88*a^3*b^3*d + 80*a^4*b^2*e)*f)*x
)*sqrt(b*x + a)/b^7

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Sympy [A]  time = 52.4947, size = 1365, normalized size = 4.27 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((f*x**3+e*x**2+d*x+c)**2/(b*x+a)**(1/2),x)

[Out]

Piecewise((-(2*a*c**2/sqrt(a + b*x) + 4*a*c*d*(-a/sqrt(a + b*x) - sqrt(a + b*x))
/b + 4*a*c*e*(a**2/sqrt(a + b*x) + 2*a*sqrt(a + b*x) - (a + b*x)**(3/2)/3)/b**2
+ 2*a*d**2*(a**2/sqrt(a + b*x) + 2*a*sqrt(a + b*x) - (a + b*x)**(3/2)/3)/b**2 +
4*a*c*f*(-a**3/sqrt(a + b*x) - 3*a**2*sqrt(a + b*x) + a*(a + b*x)**(3/2) - (a +
b*x)**(5/2)/5)/b**3 + 4*a*d*e*(-a**3/sqrt(a + b*x) - 3*a**2*sqrt(a + b*x) + a*(a
+ b*x)**(3/2) - (a + b*x)**(5/2)/5)/b**3 + 4*a*d*f*(a**4/sqrt(a + b*x) + 4*a**3
*sqrt(a + b*x) - 2*a**2*(a + b*x)**(3/2) + 4*a*(a + b*x)**(5/2)/5 - (a + b*x)**(
7/2)/7)/b**4 + 2*a*e**2*(a**4/sqrt(a + b*x) + 4*a**3*sqrt(a + b*x) - 2*a**2*(a +
b*x)**(3/2) + 4*a*(a + b*x)**(5/2)/5 - (a + b*x)**(7/2)/7)/b**4 + 4*a*e*f*(-a**
5/sqrt(a + b*x) - 5*a**4*sqrt(a + b*x) + 10*a**3*(a + b*x)**(3/2)/3 - 2*a**2*(a
+ b*x)**(5/2) + 5*a*(a + b*x)**(7/2)/7 - (a + b*x)**(9/2)/9)/b**5 + 2*a*f**2*(a*
*6/sqrt(a + b*x) + 6*a**5*sqrt(a + b*x) - 5*a**4*(a + b*x)**(3/2) + 4*a**3*(a +
b*x)**(5/2) - 15*a**2*(a + b*x)**(7/2)/7 + 2*a*(a + b*x)**(9/2)/3 - (a + b*x)**(
11/2)/11)/b**6 + 2*c**2*(-a/sqrt(a + b*x) - sqrt(a + b*x)) + 4*c*d*(a**2/sqrt(a
+ b*x) + 2*a*sqrt(a + b*x) - (a + b*x)**(3/2)/3)/b + 4*c*e*(-a**3/sqrt(a + b*x)
- 3*a**2*sqrt(a + b*x) + a*(a + b*x)**(3/2) - (a + b*x)**(5/2)/5)/b**2 + 2*d**2*
(-a**3/sqrt(a + b*x) - 3*a**2*sqrt(a + b*x) + a*(a + b*x)**(3/2) - (a + b*x)**(5
/2)/5)/b**2 + 4*c*f*(a**4/sqrt(a + b*x) + 4*a**3*sqrt(a + b*x) - 2*a**2*(a + b*x
)**(3/2) + 4*a*(a + b*x)**(5/2)/5 - (a + b*x)**(7/2)/7)/b**3 + 4*d*e*(a**4/sqrt(
a + b*x) + 4*a**3*sqrt(a + b*x) - 2*a**2*(a + b*x)**(3/2) + 4*a*(a + b*x)**(5/2)
/5 - (a + b*x)**(7/2)/7)/b**3 + 4*d*f*(-a**5/sqrt(a + b*x) - 5*a**4*sqrt(a + b*x
) + 10*a**3*(a + b*x)**(3/2)/3 - 2*a**2*(a + b*x)**(5/2) + 5*a*(a + b*x)**(7/2)/
7 - (a + b*x)**(9/2)/9)/b**4 + 2*e**2*(-a**5/sqrt(a + b*x) - 5*a**4*sqrt(a + b*x
) + 10*a**3*(a + b*x)**(3/2)/3 - 2*a**2*(a + b*x)**(5/2) + 5*a*(a + b*x)**(7/2)/
7 - (a + b*x)**(9/2)/9)/b**4 + 4*e*f*(a**6/sqrt(a + b*x) + 6*a**5*sqrt(a + b*x)
- 5*a**4*(a + b*x)**(3/2) + 4*a**3*(a + b*x)**(5/2) - 15*a**2*(a + b*x)**(7/2)/7
+ 2*a*(a + b*x)**(9/2)/3 - (a + b*x)**(11/2)/11)/b**5 + 2*f**2*(-a**7/sqrt(a +
b*x) - 7*a**6*sqrt(a + b*x) + 7*a**5*(a + b*x)**(3/2) - 7*a**4*(a + b*x)**(5/2)
+ 5*a**3*(a + b*x)**(7/2) - 7*a**2*(a + b*x)**(9/2)/3 + 7*a*(a + b*x)**(11/2)/11
- (a + b*x)**(13/2)/13)/b**6)/b, Ne(b, 0)), ((c**2*x + c*d*x**2 + e*f*x**6/3 +
f**2*x**7/7 + x**5*(2*d*f + e**2)/5 + x**4*(2*c*f + 2*d*e)/4 + x**3*(2*c*e + d**
2)/3)/sqrt(a), True))

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GIAC/XCAS [A]  time = 0.218871, size = 846, normalized size = 2.64 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((f*x^3 + e*x^2 + d*x + c)^2/sqrt(b*x + a),x, algorithm="giac")

[Out]

2/45045*(45045*sqrt(b*x + a)*c^2 + 30030*((b*x + a)^(3/2) - 3*sqrt(b*x + a)*a)*c
*d/b + 3003*(3*(b*x + a)^(5/2)*b^8 - 10*(b*x + a)^(3/2)*a*b^8 + 15*sqrt(b*x + a)
*a^2*b^8)*d^2/b^10 + 6006*(3*(b*x + a)^(5/2)*b^8 - 10*(b*x + a)^(3/2)*a*b^8 + 15
*sqrt(b*x + a)*a^2*b^8)*c*e/b^10 + 2574*(5*(b*x + a)^(7/2)*b^18 - 21*(b*x + a)^(
5/2)*a*b^18 + 35*(b*x + a)^(3/2)*a^2*b^18 - 35*sqrt(b*x + a)*a^3*b^18)*c*f/b^21
+ 2574*(5*(b*x + a)^(7/2)*b^18 - 21*(b*x + a)^(5/2)*a*b^18 + 35*(b*x + a)^(3/2)*
a^2*b^18 - 35*sqrt(b*x + a)*a^3*b^18)*d*e/b^21 + 286*(35*(b*x + a)^(9/2)*b^32 -
180*(b*x + a)^(7/2)*a*b^32 + 378*(b*x + a)^(5/2)*a^2*b^32 - 420*(b*x + a)^(3/2)*
a^3*b^32 + 315*sqrt(b*x + a)*a^4*b^32)*d*f/b^36 + 143*(35*(b*x + a)^(9/2)*b^32 -
180*(b*x + a)^(7/2)*a*b^32 + 378*(b*x + a)^(5/2)*a^2*b^32 - 420*(b*x + a)^(3/2)
*a^3*b^32 + 315*sqrt(b*x + a)*a^4*b^32)*e^2/b^36 + 130*(63*(b*x + a)^(11/2)*b^50
- 385*(b*x + a)^(9/2)*a*b^50 + 990*(b*x + a)^(7/2)*a^2*b^50 - 1386*(b*x + a)^(5
/2)*a^3*b^50 + 1155*(b*x + a)^(3/2)*a^4*b^50 - 693*sqrt(b*x + a)*a^5*b^50)*f*e/b
^55 + 15*(231*(b*x + a)^(13/2)*b^72 - 1638*(b*x + a)^(11/2)*a*b^72 + 5005*(b*x +
a)^(9/2)*a^2*b^72 - 8580*(b*x + a)^(7/2)*a^3*b^72 + 9009*(b*x + a)^(5/2)*a^4*b^
72 - 6006*(b*x + a)^(3/2)*a^5*b^72 + 3003*sqrt(b*x + a)*a^6*b^72)*f^2/b^78)/b