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

Optimal. Leaf size=212 \[ -\frac{2 (f+g x)^{5/2} \left (e g (-a e g-2 b d g+3 b e f)-c \left (d^2 g^2-6 d e f g+6 e^2 f^2\right )\right )}{5 g^5}+\frac{2 \sqrt{f+g x} (e f-d g)^2 \left (a g^2-b f g+c f^2\right )}{g^5}-\frac{2 (f+g x)^{3/2} (e f-d g) (2 c f (2 e f-d g)-g (-2 a e g-b d g+3 b e f))}{3 g^5}-\frac{2 e (f+g x)^{7/2} (-b e g-2 c d g+4 c e f)}{7 g^5}+\frac{2 c e^2 (f+g x)^{9/2}}{9 g^5} \]

[Out]

(2*(e*f - d*g)^2*(c*f^2 - b*f*g + a*g^2)*Sqrt[f + g*x])/g^5 - (2*(e*f - d*g)*(2*
c*f*(2*e*f - d*g) - g*(3*b*e*f - b*d*g - 2*a*e*g))*(f + g*x)^(3/2))/(3*g^5) - (2
*(e*g*(3*b*e*f - 2*b*d*g - a*e*g) - c*(6*e^2*f^2 - 6*d*e*f*g + d^2*g^2))*(f + g*
x)^(5/2))/(5*g^5) - (2*e*(4*c*e*f - 2*c*d*g - b*e*g)*(f + g*x)^(7/2))/(7*g^5) +
(2*c*e^2*(f + g*x)^(9/2))/(9*g^5)

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Rubi [A]  time = 0.692864, antiderivative size = 212, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.074 \[ -\frac{2 (f+g x)^{5/2} \left (e g (-a e g-2 b d g+3 b e f)-c \left (d^2 g^2-6 d e f g+6 e^2 f^2\right )\right )}{5 g^5}+\frac{2 \sqrt{f+g x} (e f-d g)^2 \left (a g^2-b f g+c f^2\right )}{g^5}-\frac{2 (f+g x)^{3/2} (e f-d g) (2 c f (2 e f-d g)-g (-2 a e g-b d g+3 b e f))}{3 g^5}-\frac{2 e (f+g x)^{7/2} (-b e g-2 c d g+4 c e f)}{7 g^5}+\frac{2 c e^2 (f+g x)^{9/2}}{9 g^5} \]

Antiderivative was successfully verified.

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

[Out]

(2*(e*f - d*g)^2*(c*f^2 - b*f*g + a*g^2)*Sqrt[f + g*x])/g^5 - (2*(e*f - d*g)*(2*
c*f*(2*e*f - d*g) - g*(3*b*e*f - b*d*g - 2*a*e*g))*(f + g*x)^(3/2))/(3*g^5) - (2
*(e*g*(3*b*e*f - 2*b*d*g - a*e*g) - c*(6*e^2*f^2 - 6*d*e*f*g + d^2*g^2))*(f + g*
x)^(5/2))/(5*g^5) - (2*e*(4*c*e*f - 2*c*d*g - b*e*g)*(f + g*x)^(7/2))/(7*g^5) +
(2*c*e^2*(f + g*x)^(9/2))/(9*g^5)

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \frac{2 c e^{2} \left (f + g x\right )^{\frac{9}{2}}}{9 g^{5}} + \frac{2 e \left (f + g x\right )^{\frac{7}{2}} \left (b e g + 2 c d g - 4 c e f\right )}{7 g^{5}} + \frac{2 \left (d g - e f\right )^{2} \left (a g^{2} - b f g + c f^{2}\right ) \int ^{\sqrt{f + g x}} \frac{1}{g^{4}}\, dx}{g} + \frac{2 \left (f + g x\right )^{\frac{5}{2}} \left (a e^{2} g^{2} + 2 b d e g^{2} - 3 b e^{2} f g + c d^{2} g^{2} - 6 c d e f g + 6 c e^{2} f^{2}\right )}{5 g^{5}} + \frac{2 \left (f + g x\right )^{\frac{3}{2}} \left (d g - e f\right ) \left (2 a e g^{2} + b d g^{2} - 3 b e f g - 2 c d f g + 4 c e f^{2}\right )}{3 g^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*c*e**2*(f + g*x)**(9/2)/(9*g**5) + 2*e*(f + g*x)**(7/2)*(b*e*g + 2*c*d*g - 4*c
*e*f)/(7*g**5) + 2*(d*g - e*f)**2*(a*g**2 - b*f*g + c*f**2)*Integral(g**(-4), (x
, sqrt(f + g*x)))/g + 2*(f + g*x)**(5/2)*(a*e**2*g**2 + 2*b*d*e*g**2 - 3*b*e**2*
f*g + c*d**2*g**2 - 6*c*d*e*f*g + 6*c*e**2*f**2)/(5*g**5) + 2*(f + g*x)**(3/2)*(
d*g - e*f)*(2*a*e*g**2 + b*d*g**2 - 3*b*e*f*g - 2*c*d*f*g + 4*c*e*f**2)/(3*g**5)

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Mathematica [A]  time = 0.330655, size = 256, normalized size = 1.21 \[ \frac{2 \sqrt{f+g x} \left (3 g \left (7 a g \left (15 d^2 g^2+10 d e g (g x-2 f)+e^2 \left (8 f^2-4 f g x+3 g^2 x^2\right )\right )+b \left (35 d^2 g^2 (g x-2 f)+14 d e g \left (8 f^2-4 f g x+3 g^2 x^2\right )-3 e^2 \left (16 f^3-8 f^2 g x+6 f g^2 x^2-5 g^3 x^3\right )\right )\right )+c \left (21 d^2 g^2 \left (8 f^2-4 f g x+3 g^2 x^2\right )+18 d e g \left (-16 f^3+8 f^2 g x-6 f g^2 x^2+5 g^3 x^3\right )+e^2 \left (128 f^4-64 f^3 g x+48 f^2 g^2 x^2-40 f g^3 x^3+35 g^4 x^4\right )\right )\right )}{315 g^5} \]

Antiderivative was successfully verified.

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

[Out]

(2*Sqrt[f + g*x]*(c*(21*d^2*g^2*(8*f^2 - 4*f*g*x + 3*g^2*x^2) + 18*d*e*g*(-16*f^
3 + 8*f^2*g*x - 6*f*g^2*x^2 + 5*g^3*x^3) + e^2*(128*f^4 - 64*f^3*g*x + 48*f^2*g^
2*x^2 - 40*f*g^3*x^3 + 35*g^4*x^4)) + 3*g*(7*a*g*(15*d^2*g^2 + 10*d*e*g*(-2*f +
g*x) + e^2*(8*f^2 - 4*f*g*x + 3*g^2*x^2)) + b*(35*d^2*g^2*(-2*f + g*x) + 14*d*e*
g*(8*f^2 - 4*f*g*x + 3*g^2*x^2) - 3*e^2*(16*f^3 - 8*f^2*g*x + 6*f*g^2*x^2 - 5*g^
3*x^3)))))/(315*g^5)

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Maple [A]  time = 0.01, size = 315, normalized size = 1.5 \[{\frac{70\,{e}^{2}c{x}^{4}{g}^{4}+90\,b{e}^{2}{g}^{4}{x}^{3}+180\,cde{g}^{4}{x}^{3}-80\,c{e}^{2}f{g}^{3}{x}^{3}+126\,a{e}^{2}{g}^{4}{x}^{2}+252\,bde{g}^{4}{x}^{2}-108\,b{e}^{2}f{g}^{3}{x}^{2}+126\,c{d}^{2}{g}^{4}{x}^{2}-216\,cdef{g}^{3}{x}^{2}+96\,c{e}^{2}{f}^{2}{g}^{2}{x}^{2}+420\,ade{g}^{4}x-168\,a{e}^{2}f{g}^{3}x+210\,b{d}^{2}{g}^{4}x-336\,bdef{g}^{3}x+144\,b{e}^{2}{f}^{2}{g}^{2}x-168\,c{d}^{2}f{g}^{3}x+288\,cde{f}^{2}{g}^{2}x-128\,c{e}^{2}{f}^{3}gx+630\,{d}^{2}a{g}^{4}-840\,adef{g}^{3}+336\,a{e}^{2}{f}^{2}{g}^{2}-420\,b{d}^{2}f{g}^{3}+672\,bde{f}^{2}{g}^{2}-288\,b{e}^{2}{f}^{3}g+336\,c{d}^{2}{f}^{2}{g}^{2}-576\,cde{f}^{3}g+256\,c{e}^{2}{f}^{4}}{315\,{g}^{5}}\sqrt{gx+f}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/315*(g*x+f)^(1/2)*(35*c*e^2*g^4*x^4+45*b*e^2*g^4*x^3+90*c*d*e*g^4*x^3-40*c*e^2
*f*g^3*x^3+63*a*e^2*g^4*x^2+126*b*d*e*g^4*x^2-54*b*e^2*f*g^3*x^2+63*c*d^2*g^4*x^
2-108*c*d*e*f*g^3*x^2+48*c*e^2*f^2*g^2*x^2+210*a*d*e*g^4*x-84*a*e^2*f*g^3*x+105*
b*d^2*g^4*x-168*b*d*e*f*g^3*x+72*b*e^2*f^2*g^2*x-84*c*d^2*f*g^3*x+144*c*d*e*f^2*
g^2*x-64*c*e^2*f^3*g*x+315*a*d^2*g^4-420*a*d*e*f*g^3+168*a*e^2*f^2*g^2-210*b*d^2
*f*g^3+336*b*d*e*f^2*g^2-144*b*e^2*f^3*g+168*c*d^2*f^2*g^2-288*c*d*e*f^3*g+128*c
*e^2*f^4)/g^5

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Maxima [A]  time = 0.696518, size = 352, normalized size = 1.66 \[ \frac{2 \,{\left (35 \,{\left (g x + f\right )}^{\frac{9}{2}} c e^{2} - 45 \,{\left (4 \, c e^{2} f -{\left (2 \, c d e + b e^{2}\right )} g\right )}{\left (g x + f\right )}^{\frac{7}{2}} + 63 \,{\left (6 \, c e^{2} f^{2} - 3 \,{\left (2 \, c d e + b e^{2}\right )} f g +{\left (c d^{2} + 2 \, b d e + a e^{2}\right )} g^{2}\right )}{\left (g x + f\right )}^{\frac{5}{2}} - 105 \,{\left (4 \, c e^{2} f^{3} - 3 \,{\left (2 \, c d e + b e^{2}\right )} f^{2} g + 2 \,{\left (c d^{2} + 2 \, b d e + a e^{2}\right )} f g^{2} -{\left (b d^{2} + 2 \, a d e\right )} g^{3}\right )}{\left (g x + f\right )}^{\frac{3}{2}} + 315 \,{\left (c e^{2} f^{4} + a d^{2} g^{4} -{\left (2 \, c d e + b e^{2}\right )} f^{3} g +{\left (c d^{2} + 2 \, b d e + a e^{2}\right )} f^{2} g^{2} -{\left (b d^{2} + 2 \, a d e\right )} f g^{3}\right )} \sqrt{g x + f}\right )}}{315 \, g^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/315*(35*(g*x + f)^(9/2)*c*e^2 - 45*(4*c*e^2*f - (2*c*d*e + b*e^2)*g)*(g*x + f)
^(7/2) + 63*(6*c*e^2*f^2 - 3*(2*c*d*e + b*e^2)*f*g + (c*d^2 + 2*b*d*e + a*e^2)*g
^2)*(g*x + f)^(5/2) - 105*(4*c*e^2*f^3 - 3*(2*c*d*e + b*e^2)*f^2*g + 2*(c*d^2 +
2*b*d*e + a*e^2)*f*g^2 - (b*d^2 + 2*a*d*e)*g^3)*(g*x + f)^(3/2) + 315*(c*e^2*f^4
+ a*d^2*g^4 - (2*c*d*e + b*e^2)*f^3*g + (c*d^2 + 2*b*d*e + a*e^2)*f^2*g^2 - (b*
d^2 + 2*a*d*e)*f*g^3)*sqrt(g*x + f))/g^5

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Fricas [A]  time = 0.276862, size = 351, normalized size = 1.66 \[ \frac{2 \,{\left (35 \, c e^{2} g^{4} x^{4} + 128 \, c e^{2} f^{4} + 315 \, a d^{2} g^{4} - 144 \,{\left (2 \, c d e + b e^{2}\right )} f^{3} g + 168 \,{\left (c d^{2} + 2 \, b d e + a e^{2}\right )} f^{2} g^{2} - 210 \,{\left (b d^{2} + 2 \, a d e\right )} f g^{3} - 5 \,{\left (8 \, c e^{2} f g^{3} - 9 \,{\left (2 \, c d e + b e^{2}\right )} g^{4}\right )} x^{3} + 3 \,{\left (16 \, c e^{2} f^{2} g^{2} - 18 \,{\left (2 \, c d e + b e^{2}\right )} f g^{3} + 21 \,{\left (c d^{2} + 2 \, b d e + a e^{2}\right )} g^{4}\right )} x^{2} -{\left (64 \, c e^{2} f^{3} g - 72 \,{\left (2 \, c d e + b e^{2}\right )} f^{2} g^{2} + 84 \,{\left (c d^{2} + 2 \, b d e + a e^{2}\right )} f g^{3} - 105 \,{\left (b d^{2} + 2 \, a d e\right )} g^{4}\right )} x\right )} \sqrt{g x + f}}{315 \, g^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/315*(35*c*e^2*g^4*x^4 + 128*c*e^2*f^4 + 315*a*d^2*g^4 - 144*(2*c*d*e + b*e^2)*
f^3*g + 168*(c*d^2 + 2*b*d*e + a*e^2)*f^2*g^2 - 210*(b*d^2 + 2*a*d*e)*f*g^3 - 5*
(8*c*e^2*f*g^3 - 9*(2*c*d*e + b*e^2)*g^4)*x^3 + 3*(16*c*e^2*f^2*g^2 - 18*(2*c*d*
e + b*e^2)*f*g^3 + 21*(c*d^2 + 2*b*d*e + a*e^2)*g^4)*x^2 - (64*c*e^2*f^3*g - 72*
(2*c*d*e + b*e^2)*f^2*g^2 + 84*(c*d^2 + 2*b*d*e + a*e^2)*f*g^3 - 105*(b*d^2 + 2*
a*d*e)*g^4)*x)*sqrt(g*x + f)/g^5

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Sympy [A]  time = 64.9989, size = 1001, normalized size = 4.72 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Piecewise((-(2*a*d**2*f/sqrt(f + g*x) + 2*a*d**2*(-f/sqrt(f + g*x) - sqrt(f + g*
x)) + 4*a*d*e*f*(-f/sqrt(f + g*x) - sqrt(f + g*x))/g + 4*a*d*e*(f**2/sqrt(f + g*
x) + 2*f*sqrt(f + g*x) - (f + g*x)**(3/2)/3)/g + 2*a*e**2*f*(f**2/sqrt(f + g*x)
+ 2*f*sqrt(f + g*x) - (f + g*x)**(3/2)/3)/g**2 + 2*a*e**2*(-f**3/sqrt(f + g*x) -
3*f**2*sqrt(f + g*x) + f*(f + g*x)**(3/2) - (f + g*x)**(5/2)/5)/g**2 + 2*b*d**2
*f*(-f/sqrt(f + g*x) - sqrt(f + g*x))/g + 2*b*d**2*(f**2/sqrt(f + g*x) + 2*f*sqr
t(f + g*x) - (f + g*x)**(3/2)/3)/g + 4*b*d*e*f*(f**2/sqrt(f + g*x) + 2*f*sqrt(f
+ g*x) - (f + g*x)**(3/2)/3)/g**2 + 4*b*d*e*(-f**3/sqrt(f + g*x) - 3*f**2*sqrt(f
+ g*x) + f*(f + g*x)**(3/2) - (f + g*x)**(5/2)/5)/g**2 + 2*b*e**2*f*(-f**3/sqrt
(f + g*x) - 3*f**2*sqrt(f + g*x) + f*(f + g*x)**(3/2) - (f + g*x)**(5/2)/5)/g**3
+ 2*b*e**2*(f**4/sqrt(f + g*x) + 4*f**3*sqrt(f + g*x) - 2*f**2*(f + g*x)**(3/2)
+ 4*f*(f + g*x)**(5/2)/5 - (f + g*x)**(7/2)/7)/g**3 + 2*c*d**2*f*(f**2/sqrt(f +
g*x) + 2*f*sqrt(f + g*x) - (f + g*x)**(3/2)/3)/g**2 + 2*c*d**2*(-f**3/sqrt(f +
g*x) - 3*f**2*sqrt(f + g*x) + f*(f + g*x)**(3/2) - (f + g*x)**(5/2)/5)/g**2 + 4*
c*d*e*f*(-f**3/sqrt(f + g*x) - 3*f**2*sqrt(f + g*x) + f*(f + g*x)**(3/2) - (f +
g*x)**(5/2)/5)/g**3 + 4*c*d*e*(f**4/sqrt(f + g*x) + 4*f**3*sqrt(f + g*x) - 2*f**
2*(f + g*x)**(3/2) + 4*f*(f + g*x)**(5/2)/5 - (f + g*x)**(7/2)/7)/g**3 + 2*c*e**
2*f*(f**4/sqrt(f + g*x) + 4*f**3*sqrt(f + g*x) - 2*f**2*(f + g*x)**(3/2) + 4*f*(
f + g*x)**(5/2)/5 - (f + g*x)**(7/2)/7)/g**4 + 2*c*e**2*(-f**5/sqrt(f + g*x) - 5
*f**4*sqrt(f + g*x) + 10*f**3*(f + g*x)**(3/2)/3 - 2*f**2*(f + g*x)**(5/2) + 5*f
*(f + g*x)**(7/2)/7 - (f + g*x)**(9/2)/9)/g**4)/g, Ne(g, 0)), ((a*d**2*x + c*e**
2*x**5/5 + x**4*(b*e**2 + 2*c*d*e)/4 + x**3*(a*e**2 + 2*b*d*e + c*d**2)/3 + x**2
*(2*a*d*e + b*d**2)/2)/sqrt(f), True))

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GIAC/XCAS [A]  time = 0.266469, size = 579, normalized size = 2.73 \[ \frac{2 \,{\left (315 \, \sqrt{g x + f} a d^{2} + \frac{105 \,{\left ({\left (g x + f\right )}^{\frac{3}{2}} - 3 \, \sqrt{g x + f} f\right )} b d^{2}}{g} + \frac{210 \,{\left ({\left (g x + f\right )}^{\frac{3}{2}} - 3 \, \sqrt{g x + f} f\right )} a d e}{g} + \frac{21 \,{\left (3 \,{\left (g x + f\right )}^{\frac{5}{2}} g^{8} - 10 \,{\left (g x + f\right )}^{\frac{3}{2}} f g^{8} + 15 \, \sqrt{g x + f} f^{2} g^{8}\right )} c d^{2}}{g^{10}} + \frac{42 \,{\left (3 \,{\left (g x + f\right )}^{\frac{5}{2}} g^{8} - 10 \,{\left (g x + f\right )}^{\frac{3}{2}} f g^{8} + 15 \, \sqrt{g x + f} f^{2} g^{8}\right )} b d e}{g^{10}} + \frac{21 \,{\left (3 \,{\left (g x + f\right )}^{\frac{5}{2}} g^{8} - 10 \,{\left (g x + f\right )}^{\frac{3}{2}} f g^{8} + 15 \, \sqrt{g x + f} f^{2} g^{8}\right )} a e^{2}}{g^{10}} + \frac{18 \,{\left (5 \,{\left (g x + f\right )}^{\frac{7}{2}} g^{18} - 21 \,{\left (g x + f\right )}^{\frac{5}{2}} f g^{18} + 35 \,{\left (g x + f\right )}^{\frac{3}{2}} f^{2} g^{18} - 35 \, \sqrt{g x + f} f^{3} g^{18}\right )} c d e}{g^{21}} + \frac{9 \,{\left (5 \,{\left (g x + f\right )}^{\frac{7}{2}} g^{18} - 21 \,{\left (g x + f\right )}^{\frac{5}{2}} f g^{18} + 35 \,{\left (g x + f\right )}^{\frac{3}{2}} f^{2} g^{18} - 35 \, \sqrt{g x + f} f^{3} g^{18}\right )} b e^{2}}{g^{21}} + \frac{{\left (35 \,{\left (g x + f\right )}^{\frac{9}{2}} g^{32} - 180 \,{\left (g x + f\right )}^{\frac{7}{2}} f g^{32} + 378 \,{\left (g x + f\right )}^{\frac{5}{2}} f^{2} g^{32} - 420 \,{\left (g x + f\right )}^{\frac{3}{2}} f^{3} g^{32} + 315 \, \sqrt{g x + f} f^{4} g^{32}\right )} c e^{2}}{g^{36}}\right )}}{315 \, g} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/315*(315*sqrt(g*x + f)*a*d^2 + 105*((g*x + f)^(3/2) - 3*sqrt(g*x + f)*f)*b*d^2
/g + 210*((g*x + f)^(3/2) - 3*sqrt(g*x + f)*f)*a*d*e/g + 21*(3*(g*x + f)^(5/2)*g
^8 - 10*(g*x + f)^(3/2)*f*g^8 + 15*sqrt(g*x + f)*f^2*g^8)*c*d^2/g^10 + 42*(3*(g*
x + f)^(5/2)*g^8 - 10*(g*x + f)^(3/2)*f*g^8 + 15*sqrt(g*x + f)*f^2*g^8)*b*d*e/g^
10 + 21*(3*(g*x + f)^(5/2)*g^8 - 10*(g*x + f)^(3/2)*f*g^8 + 15*sqrt(g*x + f)*f^2
*g^8)*a*e^2/g^10 + 18*(5*(g*x + f)^(7/2)*g^18 - 21*(g*x + f)^(5/2)*f*g^18 + 35*(
g*x + f)^(3/2)*f^2*g^18 - 35*sqrt(g*x + f)*f^3*g^18)*c*d*e/g^21 + 9*(5*(g*x + f)
^(7/2)*g^18 - 21*(g*x + f)^(5/2)*f*g^18 + 35*(g*x + f)^(3/2)*f^2*g^18 - 35*sqrt(
g*x + f)*f^3*g^18)*b*e^2/g^21 + (35*(g*x + f)^(9/2)*g^32 - 180*(g*x + f)^(7/2)*f
*g^32 + 378*(g*x + f)^(5/2)*f^2*g^32 - 420*(g*x + f)^(3/2)*f^3*g^32 + 315*sqrt(g
*x + f)*f^4*g^32)*c*e^2/g^36)/g