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

Optimal. Leaf size=285 \[ -\frac{2 e (f+g x)^{5/2} \left (e g (-a e g-3 b d g+4 b e f)-c \left (3 d^2 g^2-12 d e f g+10 e^2 f^2\right )\right )}{5 g^6}+\frac{2 (f+g x)^{3/2} (e f-d g) \left (3 e g (-a e g-b d g+2 b e f)-c \left (d^2 g^2-8 d e f g+10 e^2 f^2\right )\right )}{3 g^6}+\frac{2 (e f-d g)^3 \left (a g^2-b f g+c f^2\right )}{g^6 \sqrt{f+g x}}+\frac{2 \sqrt{f+g x} (e f-d g)^2 (c f (5 e f-2 d g)-g (-3 a e g-b d g+4 b e f))}{g^6}-\frac{2 e^2 (f+g x)^{7/2} (-b e g-3 c d g+5 c e f)}{7 g^6}+\frac{2 c e^3 (f+g x)^{9/2}}{9 g^6} \]

[Out]

(2*(e*f - d*g)^3*(c*f^2 - b*f*g + a*g^2))/(g^6*Sqrt[f + g*x]) + (2*(e*f - d*g)^2
*(c*f*(5*e*f - 2*d*g) - g*(4*b*e*f - b*d*g - 3*a*e*g))*Sqrt[f + g*x])/g^6 + (2*(
e*f - d*g)*(3*e*g*(2*b*e*f - b*d*g - a*e*g) - c*(10*e^2*f^2 - 8*d*e*f*g + d^2*g^
2))*(f + g*x)^(3/2))/(3*g^6) - (2*e*(e*g*(4*b*e*f - 3*b*d*g - a*e*g) - c*(10*e^2
*f^2 - 12*d*e*f*g + 3*d^2*g^2))*(f + g*x)^(5/2))/(5*g^6) - (2*e^2*(5*c*e*f - 3*c
*d*g - b*e*g)*(f + g*x)^(7/2))/(7*g^6) + (2*c*e^3*(f + g*x)^(9/2))/(9*g^6)

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Rubi [A]  time = 1.00847, antiderivative size = 285, 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 \[ -\frac{2 e (f+g x)^{5/2} \left (e g (-a e g-3 b d g+4 b e f)-c \left (3 d^2 g^2-12 d e f g+10 e^2 f^2\right )\right )}{5 g^6}+\frac{2 (f+g x)^{3/2} (e f-d g) \left (3 e g (-a e g-b d g+2 b e f)-c \left (d^2 g^2-8 d e f g+10 e^2 f^2\right )\right )}{3 g^6}+\frac{2 (e f-d g)^3 \left (a g^2-b f g+c f^2\right )}{g^6 \sqrt{f+g x}}+\frac{2 \sqrt{f+g x} (e f-d g)^2 (c f (5 e f-2 d g)-g (-3 a e g-b d g+4 b e f))}{g^6}-\frac{2 e^2 (f+g x)^{7/2} (-b e g-3 c d g+5 c e f)}{7 g^6}+\frac{2 c e^3 (f+g x)^{9/2}}{9 g^6} \]

Antiderivative was successfully verified.

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

[Out]

(2*(e*f - d*g)^3*(c*f^2 - b*f*g + a*g^2))/(g^6*Sqrt[f + g*x]) + (2*(e*f - d*g)^2
*(c*f*(5*e*f - 2*d*g) - g*(4*b*e*f - b*d*g - 3*a*e*g))*Sqrt[f + g*x])/g^6 + (2*(
e*f - d*g)*(3*e*g*(2*b*e*f - b*d*g - a*e*g) - c*(10*e^2*f^2 - 8*d*e*f*g + d^2*g^
2))*(f + g*x)^(3/2))/(3*g^6) - (2*e*(e*g*(4*b*e*f - 3*b*d*g - a*e*g) - c*(10*e^2
*f^2 - 12*d*e*f*g + 3*d^2*g^2))*(f + g*x)^(5/2))/(5*g^6) - (2*e^2*(5*c*e*f - 3*c
*d*g - b*e*g)*(f + g*x)^(7/2))/(7*g^6) + (2*c*e^3*(f + g*x)^(9/2))/(9*g^6)

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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((e*x+d)**3*(c*x**2+b*x+a)/(g*x+f)**(3/2),x)

[Out]

Timed out

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Mathematica [A]  time = 0.752229, size = 406, normalized size = 1.42 \[ \frac{2 \left (9 g \left (7 a g \left (-5 d^3 g^3+15 d^2 e g^2 (2 f+g x)+5 d e^2 g \left (-8 f^2-4 f g x+g^2 x^2\right )+e^3 \left (16 f^3+8 f^2 g x-2 f g^2 x^2+g^3 x^3\right )\right )+b \left (35 d^3 g^3 (2 f+g x)+35 d^2 e g^2 \left (-8 f^2-4 f g x+g^2 x^2\right )+21 d e^2 g \left (16 f^3+8 f^2 g x-2 f g^2 x^2+g^3 x^3\right )+e^3 \left (-128 f^4-64 f^3 g x+16 f^2 g^2 x^2-8 f g^3 x^3+5 g^4 x^4\right )\right )\right )+c \left (105 d^3 g^3 \left (-8 f^2-4 f g x+g^2 x^2\right )+189 d^2 e g^2 \left (16 f^3+8 f^2 g x-2 f g^2 x^2+g^3 x^3\right )+27 d e^2 g \left (-128 f^4-64 f^3 g x+16 f^2 g^2 x^2-8 f g^3 x^3+5 g^4 x^4\right )+5 e^3 \left (256 f^5+128 f^4 g x-32 f^3 g^2 x^2+16 f^2 g^3 x^3-10 f g^4 x^4+7 g^5 x^5\right )\right )\right )}{315 g^6 \sqrt{f+g x}} \]

Antiderivative was successfully verified.

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

[Out]

(2*(c*(105*d^3*g^3*(-8*f^2 - 4*f*g*x + g^2*x^2) + 189*d^2*e*g^2*(16*f^3 + 8*f^2*
g*x - 2*f*g^2*x^2 + g^3*x^3) + 27*d*e^2*g*(-128*f^4 - 64*f^3*g*x + 16*f^2*g^2*x^
2 - 8*f*g^3*x^3 + 5*g^4*x^4) + 5*e^3*(256*f^5 + 128*f^4*g*x - 32*f^3*g^2*x^2 + 1
6*f^2*g^3*x^3 - 10*f*g^4*x^4 + 7*g^5*x^5)) + 9*g*(7*a*g*(-5*d^3*g^3 + 15*d^2*e*g
^2*(2*f + g*x) + 5*d*e^2*g*(-8*f^2 - 4*f*g*x + g^2*x^2) + e^3*(16*f^3 + 8*f^2*g*
x - 2*f*g^2*x^2 + g^3*x^3)) + b*(35*d^3*g^3*(2*f + g*x) + 35*d^2*e*g^2*(-8*f^2 -
4*f*g*x + g^2*x^2) + 21*d*e^2*g*(16*f^3 + 8*f^2*g*x - 2*f*g^2*x^2 + g^3*x^3) +
e^3*(-128*f^4 - 64*f^3*g*x + 16*f^2*g^2*x^2 - 8*f*g^3*x^3 + 5*g^4*x^4)))))/(315*
g^6*Sqrt[f + g*x])

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Maple [B]  time = 0.01, size = 540, normalized size = 1.9 \[ -{\frac{-70\,{e}^{3}c{x}^{5}{g}^{5}-90\,b{e}^{3}{g}^{5}{x}^{4}-270\,cd{e}^{2}{g}^{5}{x}^{4}+100\,c{e}^{3}f{g}^{4}{x}^{4}-126\,a{e}^{3}{g}^{5}{x}^{3}-378\,bd{e}^{2}{g}^{5}{x}^{3}+144\,b{e}^{3}f{g}^{4}{x}^{3}-378\,c{d}^{2}e{g}^{5}{x}^{3}+432\,cd{e}^{2}f{g}^{4}{x}^{3}-160\,c{e}^{3}{f}^{2}{g}^{3}{x}^{3}-630\,ad{e}^{2}{g}^{5}{x}^{2}+252\,a{e}^{3}f{g}^{4}{x}^{2}-630\,b{d}^{2}e{g}^{5}{x}^{2}+756\,bd{e}^{2}f{g}^{4}{x}^{2}-288\,b{e}^{3}{f}^{2}{g}^{3}{x}^{2}-210\,c{d}^{3}{g}^{5}{x}^{2}+756\,c{d}^{2}ef{g}^{4}{x}^{2}-864\,cd{e}^{2}{f}^{2}{g}^{3}{x}^{2}+320\,c{e}^{3}{f}^{3}{g}^{2}{x}^{2}-1890\,a{d}^{2}e{g}^{5}x+2520\,ad{e}^{2}f{g}^{4}x-1008\,a{e}^{3}{f}^{2}{g}^{3}x-630\,b{d}^{3}{g}^{5}x+2520\,b{d}^{2}ef{g}^{4}x-3024\,bd{e}^{2}{f}^{2}{g}^{3}x+1152\,b{e}^{3}{f}^{3}{g}^{2}x+840\,c{d}^{3}f{g}^{4}x-3024\,c{d}^{2}e{f}^{2}{g}^{3}x+3456\,cd{e}^{2}{f}^{3}{g}^{2}x-1280\,c{e}^{3}{f}^{4}gx+630\,{d}^{3}a{g}^{5}-3780\,a{d}^{2}ef{g}^{4}+5040\,ad{e}^{2}{f}^{2}{g}^{3}-2016\,a{e}^{3}{f}^{3}{g}^{2}-1260\,b{d}^{3}f{g}^{4}+5040\,b{d}^{2}e{f}^{2}{g}^{3}-6048\,bd{e}^{2}{f}^{3}{g}^{2}+2304\,b{e}^{3}{f}^{4}g+1680\,c{d}^{3}{f}^{2}{g}^{3}-6048\,c{d}^{2}e{f}^{3}{g}^{2}+6912\,cd{e}^{2}{f}^{4}g-2560\,c{e}^{3}{f}^{5}}{315\,{g}^{6}}{\frac{1}{\sqrt{gx+f}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/315/(g*x+f)^(1/2)*(-35*c*e^3*g^5*x^5-45*b*e^3*g^5*x^4-135*c*d*e^2*g^5*x^4+50*
c*e^3*f*g^4*x^4-63*a*e^3*g^5*x^3-189*b*d*e^2*g^5*x^3+72*b*e^3*f*g^4*x^3-189*c*d^
2*e*g^5*x^3+216*c*d*e^2*f*g^4*x^3-80*c*e^3*f^2*g^3*x^3-315*a*d*e^2*g^5*x^2+126*a
*e^3*f*g^4*x^2-315*b*d^2*e*g^5*x^2+378*b*d*e^2*f*g^4*x^2-144*b*e^3*f^2*g^3*x^2-1
05*c*d^3*g^5*x^2+378*c*d^2*e*f*g^4*x^2-432*c*d*e^2*f^2*g^3*x^2+160*c*e^3*f^3*g^2
*x^2-945*a*d^2*e*g^5*x+1260*a*d*e^2*f*g^4*x-504*a*e^3*f^2*g^3*x-315*b*d^3*g^5*x+
1260*b*d^2*e*f*g^4*x-1512*b*d*e^2*f^2*g^3*x+576*b*e^3*f^3*g^2*x+420*c*d^3*f*g^4*
x-1512*c*d^2*e*f^2*g^3*x+1728*c*d*e^2*f^3*g^2*x-640*c*e^3*f^4*g*x+315*a*d^3*g^5-
1890*a*d^2*e*f*g^4+2520*a*d*e^2*f^2*g^3-1008*a*e^3*f^3*g^2-630*b*d^3*f*g^4+2520*
b*d^2*e*f^2*g^3-3024*b*d*e^2*f^3*g^2+1152*b*e^3*f^4*g+840*c*d^3*f^2*g^3-3024*c*d
^2*e*f^3*g^2+3456*c*d*e^2*f^4*g-1280*c*e^3*f^5)/g^6

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Maxima [A]  time = 0.69667, size = 590, normalized size = 2.07 \[ \frac{2 \,{\left (\frac{35 \,{\left (g x + f\right )}^{\frac{9}{2}} c e^{3} - 45 \,{\left (5 \, c e^{3} f -{\left (3 \, c d e^{2} + b e^{3}\right )} g\right )}{\left (g x + f\right )}^{\frac{7}{2}} + 63 \,{\left (10 \, c e^{3} f^{2} - 4 \,{\left (3 \, c d e^{2} + b e^{3}\right )} f g +{\left (3 \, c d^{2} e + 3 \, b d e^{2} + a e^{3}\right )} g^{2}\right )}{\left (g x + f\right )}^{\frac{5}{2}} - 105 \,{\left (10 \, c e^{3} f^{3} - 6 \,{\left (3 \, c d e^{2} + b e^{3}\right )} f^{2} g + 3 \,{\left (3 \, c d^{2} e + 3 \, b d e^{2} + a e^{3}\right )} f g^{2} -{\left (c d^{3} + 3 \, b d^{2} e + 3 \, a d e^{2}\right )} g^{3}\right )}{\left (g x + f\right )}^{\frac{3}{2}} + 315 \,{\left (5 \, c e^{3} f^{4} - 4 \,{\left (3 \, c d e^{2} + b e^{3}\right )} f^{3} g + 3 \,{\left (3 \, c d^{2} e + 3 \, b d e^{2} + a e^{3}\right )} f^{2} g^{2} - 2 \,{\left (c d^{3} + 3 \, b d^{2} e + 3 \, a d e^{2}\right )} f g^{3} +{\left (b d^{3} + 3 \, a d^{2} e\right )} g^{4}\right )} \sqrt{g x + f}}{g^{5}} + \frac{315 \,{\left (c e^{3} f^{5} - a d^{3} g^{5} -{\left (3 \, c d e^{2} + b e^{3}\right )} f^{4} g +{\left (3 \, c d^{2} e + 3 \, b d e^{2} + a e^{3}\right )} f^{3} g^{2} -{\left (c d^{3} + 3 \, b d^{2} e + 3 \, a d e^{2}\right )} f^{2} g^{3} +{\left (b d^{3} + 3 \, a d^{2} e\right )} f g^{4}\right )}}{\sqrt{g x + f} g^{5}}\right )}}{315 \, g} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/315*((35*(g*x + f)^(9/2)*c*e^3 - 45*(5*c*e^3*f - (3*c*d*e^2 + b*e^3)*g)*(g*x +
f)^(7/2) + 63*(10*c*e^3*f^2 - 4*(3*c*d*e^2 + b*e^3)*f*g + (3*c*d^2*e + 3*b*d*e^
2 + a*e^3)*g^2)*(g*x + f)^(5/2) - 105*(10*c*e^3*f^3 - 6*(3*c*d*e^2 + b*e^3)*f^2*
g + 3*(3*c*d^2*e + 3*b*d*e^2 + a*e^3)*f*g^2 - (c*d^3 + 3*b*d^2*e + 3*a*d*e^2)*g^
3)*(g*x + f)^(3/2) + 315*(5*c*e^3*f^4 - 4*(3*c*d*e^2 + b*e^3)*f^3*g + 3*(3*c*d^2
*e + 3*b*d*e^2 + a*e^3)*f^2*g^2 - 2*(c*d^3 + 3*b*d^2*e + 3*a*d*e^2)*f*g^3 + (b*d
^3 + 3*a*d^2*e)*g^4)*sqrt(g*x + f))/g^5 + 315*(c*e^3*f^5 - a*d^3*g^5 - (3*c*d*e^
2 + b*e^3)*f^4*g + (3*c*d^2*e + 3*b*d*e^2 + a*e^3)*f^3*g^2 - (c*d^3 + 3*b*d^2*e
+ 3*a*d*e^2)*f^2*g^3 + (b*d^3 + 3*a*d^2*e)*f*g^4)/(sqrt(g*x + f)*g^5))/g

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Fricas [A]  time = 0.27232, size = 578, normalized size = 2.03 \[ \frac{2 \,{\left (35 \, c e^{3} g^{5} x^{5} + 1280 \, c e^{3} f^{5} - 315 \, a d^{3} g^{5} - 1152 \,{\left (3 \, c d e^{2} + b e^{3}\right )} f^{4} g + 1008 \,{\left (3 \, c d^{2} e + 3 \, b d e^{2} + a e^{3}\right )} f^{3} g^{2} - 840 \,{\left (c d^{3} + 3 \, b d^{2} e + 3 \, a d e^{2}\right )} f^{2} g^{3} + 630 \,{\left (b d^{3} + 3 \, a d^{2} e\right )} f g^{4} - 5 \,{\left (10 \, c e^{3} f g^{4} - 9 \,{\left (3 \, c d e^{2} + b e^{3}\right )} g^{5}\right )} x^{4} +{\left (80 \, c e^{3} f^{2} g^{3} - 72 \,{\left (3 \, c d e^{2} + b e^{3}\right )} f g^{4} + 63 \,{\left (3 \, c d^{2} e + 3 \, b d e^{2} + a e^{3}\right )} g^{5}\right )} x^{3} -{\left (160 \, c e^{3} f^{3} g^{2} - 144 \,{\left (3 \, c d e^{2} + b e^{3}\right )} f^{2} g^{3} + 126 \,{\left (3 \, c d^{2} e + 3 \, b d e^{2} + a e^{3}\right )} f g^{4} - 105 \,{\left (c d^{3} + 3 \, b d^{2} e + 3 \, a d e^{2}\right )} g^{5}\right )} x^{2} +{\left (640 \, c e^{3} f^{4} g - 576 \,{\left (3 \, c d e^{2} + b e^{3}\right )} f^{3} g^{2} + 504 \,{\left (3 \, c d^{2} e + 3 \, b d e^{2} + a e^{3}\right )} f^{2} g^{3} - 420 \,{\left (c d^{3} + 3 \, b d^{2} e + 3 \, a d e^{2}\right )} f g^{4} + 315 \,{\left (b d^{3} + 3 \, a d^{2} e\right )} g^{5}\right )} x\right )}}{315 \, \sqrt{g x + f} g^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/315*(35*c*e^3*g^5*x^5 + 1280*c*e^3*f^5 - 315*a*d^3*g^5 - 1152*(3*c*d*e^2 + b*e
^3)*f^4*g + 1008*(3*c*d^2*e + 3*b*d*e^2 + a*e^3)*f^3*g^2 - 840*(c*d^3 + 3*b*d^2*
e + 3*a*d*e^2)*f^2*g^3 + 630*(b*d^3 + 3*a*d^2*e)*f*g^4 - 5*(10*c*e^3*f*g^4 - 9*(
3*c*d*e^2 + b*e^3)*g^5)*x^4 + (80*c*e^3*f^2*g^3 - 72*(3*c*d*e^2 + b*e^3)*f*g^4 +
63*(3*c*d^2*e + 3*b*d*e^2 + a*e^3)*g^5)*x^3 - (160*c*e^3*f^3*g^2 - 144*(3*c*d*e
^2 + b*e^3)*f^2*g^3 + 126*(3*c*d^2*e + 3*b*d*e^2 + a*e^3)*f*g^4 - 105*(c*d^3 + 3
*b*d^2*e + 3*a*d*e^2)*g^5)*x^2 + (640*c*e^3*f^4*g - 576*(3*c*d*e^2 + b*e^3)*f^3*
g^2 + 504*(3*c*d^2*e + 3*b*d*e^2 + a*e^3)*f^2*g^3 - 420*(c*d^3 + 3*b*d^2*e + 3*a
*d*e^2)*f*g^4 + 315*(b*d^3 + 3*a*d^2*e)*g^5)*x)/(sqrt(g*x + f)*g^6)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (d + e x\right )^{3} \left (a + b x + c x^{2}\right )}{\left (f + g x\right )^{\frac{3}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral((d + e*x)**3*(a + b*x + c*x**2)/(f + g*x)**(3/2), x)

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GIAC/XCAS [A]  time = 0.276629, size = 903, normalized size = 3.17 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*(c*d^3*f^2*g^3 - b*d^3*f*g^4 + a*d^3*g^5 - 3*c*d^2*f^3*g^2*e + 3*b*d^2*f^2*g^
3*e - 3*a*d^2*f*g^4*e + 3*c*d*f^4*g*e^2 - 3*b*d*f^3*g^2*e^2 + 3*a*d*f^2*g^3*e^2
- c*f^5*e^3 + b*f^4*g*e^3 - a*f^3*g^2*e^3)/(sqrt(g*x + f)*g^6) + 2/315*(105*(g*x
+ f)^(3/2)*c*d^3*g^51 - 630*sqrt(g*x + f)*c*d^3*f*g^51 + 315*sqrt(g*x + f)*b*d^
3*g^52 + 189*(g*x + f)^(5/2)*c*d^2*g^50*e - 945*(g*x + f)^(3/2)*c*d^2*f*g^50*e +
2835*sqrt(g*x + f)*c*d^2*f^2*g^50*e + 315*(g*x + f)^(3/2)*b*d^2*g^51*e - 1890*s
qrt(g*x + f)*b*d^2*f*g^51*e + 945*sqrt(g*x + f)*a*d^2*g^52*e + 135*(g*x + f)^(7/
2)*c*d*g^49*e^2 - 756*(g*x + f)^(5/2)*c*d*f*g^49*e^2 + 1890*(g*x + f)^(3/2)*c*d*
f^2*g^49*e^2 - 3780*sqrt(g*x + f)*c*d*f^3*g^49*e^2 + 189*(g*x + f)^(5/2)*b*d*g^5
0*e^2 - 945*(g*x + f)^(3/2)*b*d*f*g^50*e^2 + 2835*sqrt(g*x + f)*b*d*f^2*g^50*e^2
+ 315*(g*x + f)^(3/2)*a*d*g^51*e^2 - 1890*sqrt(g*x + f)*a*d*f*g^51*e^2 + 35*(g*
x + f)^(9/2)*c*g^48*e^3 - 225*(g*x + f)^(7/2)*c*f*g^48*e^3 + 630*(g*x + f)^(5/2)
*c*f^2*g^48*e^3 - 1050*(g*x + f)^(3/2)*c*f^3*g^48*e^3 + 1575*sqrt(g*x + f)*c*f^4
*g^48*e^3 + 45*(g*x + f)^(7/2)*b*g^49*e^3 - 252*(g*x + f)^(5/2)*b*f*g^49*e^3 + 6
30*(g*x + f)^(3/2)*b*f^2*g^49*e^3 - 1260*sqrt(g*x + f)*b*f^3*g^49*e^3 + 63*(g*x
+ f)^(5/2)*a*g^50*e^3 - 315*(g*x + f)^(3/2)*a*f*g^50*e^3 + 945*sqrt(g*x + f)*a*f
^2*g^50*e^3)/g^54