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

Optimal. Leaf size=175 \[ \frac{2 (f+g x)^{5/2} \left (a e^2 g^2+c \left (d^2 g^2-6 d e f g+6 e^2 f^2\right )\right )}{5 g^5}+\frac{2 \sqrt{f+g x} \left (a g^2+c f^2\right ) (e f-d g)^2}{g^5}-\frac{4 (f+g x)^{3/2} (e f-d g) \left (a e g^2+c f (2 e f-d g)\right )}{3 g^5}-\frac{4 c e (f+g x)^{7/2} (2 e f-d g)}{7 g^5}+\frac{2 c e^2 (f+g x)^{9/2}}{9 g^5} \]

[Out]

(2*(e*f - d*g)^2*(c*f^2 + a*g^2)*Sqrt[f + g*x])/g^5 - (4*(e*f - d*g)*(a*e*g^2 +
c*f*(2*e*f - d*g))*(f + g*x)^(3/2))/(3*g^5) + (2*(a*e^2*g^2 + c*(6*e^2*f^2 - 6*d
*e*f*g + d^2*g^2))*(f + g*x)^(5/2))/(5*g^5) - (4*c*e*(2*e*f - d*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 = 0.49055, antiderivative size = 175, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083 \[ \frac{2 (f+g x)^{5/2} \left (a e^2 g^2+c \left (d^2 g^2-6 d e f g+6 e^2 f^2\right )\right )}{5 g^5}+\frac{2 \sqrt{f+g x} \left (a g^2+c f^2\right ) (e f-d g)^2}{g^5}-\frac{4 (f+g x)^{3/2} (e f-d g) \left (a e g^2+c f (2 e f-d g)\right )}{3 g^5}-\frac{4 c e (f+g x)^{7/2} (2 e f-d g)}{7 g^5}+\frac{2 c e^2 (f+g x)^{9/2}}{9 g^5} \]

Antiderivative was successfully verified.

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

[Out]

(2*(e*f - d*g)^2*(c*f^2 + a*g^2)*Sqrt[f + g*x])/g^5 - (4*(e*f - d*g)*(a*e*g^2 +
c*f*(2*e*f - d*g))*(f + g*x)^(3/2))/(3*g^5) + (2*(a*e^2*g^2 + c*(6*e^2*f^2 - 6*d
*e*f*g + d^2*g^2))*(f + g*x)^(5/2))/(5*g^5) - (4*c*e*(2*e*f - d*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. \[ \frac{2 c e^{2} \left (f + g x\right )^{\frac{9}{2}}}{9 g^{5}} + \frac{4 c e \left (f + g x\right )^{\frac{7}{2}} \left (d g - 2 e f\right )}{7 g^{5}} + \frac{2 \left (a g^{2} + c f^{2}\right ) \left (d g - e f\right )^{2} \int ^{\sqrt{f + g x}} \frac{1}{g^{4}}\, dx}{g} + \frac{2 \left (f + g x\right )^{\frac{5}{2}} \left (a e^{2} g^{2} + c d^{2} g^{2} - 6 c d e f g + 6 c e^{2} f^{2}\right )}{5 g^{5}} + \frac{4 \left (f + g x\right )^{\frac{3}{2}} \left (d g - e f\right ) \left (a e g^{2} - c d f g + 2 c e f^{2}\right )}{3 g^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.186395, size = 177, normalized size = 1.01 \[ \frac{2 \sqrt{f+g x} \left (21 a g^2 \left (15 d^2 g^2+10 d e g (g x-2 f)+e^2 \left (8 f^2-4 f g x+3 g^2 x^2\right )\right )+c \left (21 d^2 g^2 \left (8 f^2-4 f g x+3 g^2 x^2\right )+18 d e g \left (-16 f^3+8 f^2 g x-6 f g^2 x^2+5 g^3 x^3\right )+e^2 \left (128 f^4-64 f^3 g x+48 f^2 g^2 x^2-40 f g^3 x^3+35 g^4 x^4\right )\right )\right )}{315 g^5} \]

Antiderivative was successfully verified.

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

[Out]

(2*Sqrt[f + g*x]*(21*a*g^2*(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)) + c*(21*d^2*g^2*(8*f^2 - 4*f*g*x + 3*g^2*x^2) + 18*d*e*g*(-1
6*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))))/(315*g^5)

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Maple [A]  time = 0.01, size = 215, normalized size = 1.2 \[{\frac{70\,{e}^{2}c{x}^{4}{g}^{4}+180\,cde{g}^{4}{x}^{3}-80\,c{e}^{2}f{g}^{3}{x}^{3}+126\,a{e}^{2}{g}^{4}{x}^{2}+126\,c{d}^{2}{g}^{4}{x}^{2}-216\,cdef{g}^{3}{x}^{2}+96\,c{e}^{2}{f}^{2}{g}^{2}{x}^{2}+420\,ade{g}^{4}x-168\,a{e}^{2}f{g}^{3}x-168\,c{d}^{2}f{g}^{3}x+288\,cde{f}^{2}{g}^{2}x-128\,c{e}^{2}{f}^{3}gx+630\,{d}^{2}a{g}^{4}-840\,adef{g}^{3}+336\,a{e}^{2}{f}^{2}{g}^{2}+336\,c{d}^{2}{f}^{2}{g}^{2}-576\,cde{f}^{3}g+256\,c{e}^{2}{f}^{4}}{315\,{g}^{5}}\sqrt{gx+f}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/315*(g*x+f)^(1/2)*(35*c*e^2*g^4*x^4+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+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-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+168*c*d^2*f^2*g^2-288*c*d*e*f^3*g+12
8*c*e^2*f^4)/g^5

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Maxima [A]  time = 0.704692, size = 266, normalized size = 1.52 \[ \frac{2 \,{\left (35 \,{\left (g x + f\right )}^{\frac{9}{2}} c e^{2} - 90 \,{\left (2 \, c e^{2} f - c d e g\right )}{\left (g x + f\right )}^{\frac{7}{2}} + 63 \,{\left (6 \, c e^{2} f^{2} - 6 \, c d e f g +{\left (c d^{2} + a e^{2}\right )} g^{2}\right )}{\left (g x + f\right )}^{\frac{5}{2}} - 210 \,{\left (2 \, c e^{2} f^{3} - 3 \, c d e f^{2} g - a d e g^{3} +{\left (c d^{2} + a e^{2}\right )} f g^{2}\right )}{\left (g x + f\right )}^{\frac{3}{2}} + 315 \,{\left (c e^{2} f^{4} - 2 \, c d e f^{3} g - 2 \, a d e f g^{3} + a d^{2} g^{4} +{\left (c d^{2} + a e^{2}\right )} f^{2} g^{2}\right )} \sqrt{g x + f}\right )}}{315 \, g^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/315*(35*(g*x + f)^(9/2)*c*e^2 - 90*(2*c*e^2*f - c*d*e*g)*(g*x + f)^(7/2) + 63*
(6*c*e^2*f^2 - 6*c*d*e*f*g + (c*d^2 + a*e^2)*g^2)*(g*x + f)^(5/2) - 210*(2*c*e^2
*f^3 - 3*c*d*e*f^2*g - a*d*e*g^3 + (c*d^2 + a*e^2)*f*g^2)*(g*x + f)^(3/2) + 315*
(c*e^2*f^4 - 2*c*d*e*f^3*g - 2*a*d*e*f*g^3 + a*d^2*g^4 + (c*d^2 + a*e^2)*f^2*g^2
)*sqrt(g*x + f))/g^5

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Fricas [A]  time = 0.271521, size = 266, normalized size = 1.52 \[ \frac{2 \,{\left (35 \, c e^{2} g^{4} x^{4} + 128 \, c e^{2} f^{4} - 288 \, c d e f^{3} g - 420 \, a d e f g^{3} + 315 \, a d^{2} g^{4} + 168 \,{\left (c d^{2} + a e^{2}\right )} f^{2} g^{2} - 10 \,{\left (4 \, c e^{2} f g^{3} - 9 \, c d e g^{4}\right )} x^{3} + 3 \,{\left (16 \, c e^{2} f^{2} g^{2} - 36 \, c d e f g^{3} + 21 \,{\left (c d^{2} + a e^{2}\right )} g^{4}\right )} x^{2} - 2 \,{\left (32 \, c e^{2} f^{3} g - 72 \, c d e f^{2} g^{2} - 105 \, a d e g^{4} + 42 \,{\left (c d^{2} + a e^{2}\right )} f g^{3}\right )} x\right )} \sqrt{g x + f}}{315 \, g^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + 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 - 288*c*d*e*f^3*g - 420*a*d*e*f*g^3 + 31
5*a*d^2*g^4 + 168*(c*d^2 + a*e^2)*f^2*g^2 - 10*(4*c*e^2*f*g^3 - 9*c*d*e*g^4)*x^3
+ 3*(16*c*e^2*f^2*g^2 - 36*c*d*e*f*g^3 + 21*(c*d^2 + a*e^2)*g^4)*x^2 - 2*(32*c*
e^2*f^3*g - 72*c*d*e*f^2*g^2 - 105*a*d*e*g^4 + 42*(c*d^2 + a*e^2)*f*g^3)*x)*sqrt
(g*x + f)/g^5

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Sympy [A]  time = 44.9736, size = 673, normalized size = 3.85 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x+d)**2*(c*x**2+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*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 + a*d*e*x**2 + c*d*e*x**4/2 + c*e**2*x**5/5 + x**3*(a*e**2 + c*d**2)/3
)/sqrt(f), True))

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GIAC/XCAS [A]  time = 0.270121, size = 389, normalized size = 2.22 \[ \frac{2 \,{\left (315 \, \sqrt{g x + f} a d^{2} + \frac{210 \,{\left ({\left (g x + f\right )}^{\frac{3}{2}} - 3 \, \sqrt{g x + f} f\right )} a d e}{g} + \frac{21 \,{\left (3 \,{\left (g x + f\right )}^{\frac{5}{2}} g^{8} - 10 \,{\left (g x + f\right )}^{\frac{3}{2}} f g^{8} + 15 \, \sqrt{g x + f} f^{2} g^{8}\right )} c d^{2}}{g^{10}} + \frac{21 \,{\left (3 \,{\left (g x + f\right )}^{\frac{5}{2}} g^{8} - 10 \,{\left (g x + f\right )}^{\frac{3}{2}} f g^{8} + 15 \, \sqrt{g x + f} f^{2} g^{8}\right )} a e^{2}}{g^{10}} + \frac{18 \,{\left (5 \,{\left (g x + f\right )}^{\frac{7}{2}} g^{18} - 21 \,{\left (g x + f\right )}^{\frac{5}{2}} f g^{18} + 35 \,{\left (g x + f\right )}^{\frac{3}{2}} f^{2} g^{18} - 35 \, \sqrt{g x + f} f^{3} g^{18}\right )} c d e}{g^{21}} + \frac{{\left (35 \,{\left (g x + f\right )}^{\frac{9}{2}} g^{32} - 180 \,{\left (g x + f\right )}^{\frac{7}{2}} f g^{32} + 378 \,{\left (g x + f\right )}^{\frac{5}{2}} f^{2} g^{32} - 420 \,{\left (g x + f\right )}^{\frac{3}{2}} f^{3} g^{32} + 315 \, \sqrt{g x + f} f^{4} g^{32}\right )} c e^{2}}{g^{36}}\right )}}{315 \, g} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/315*(315*sqrt(g*x + f)*a*d^2 + 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 + 21*(3*(g*x + f)^(5/2)*g^8 - 10*(g*x + f)^(3/2)*f*g^8 + 15*sqr
t(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 +
(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