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

Optimal. Leaf size=412 \[ \frac{16 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2} (c d f-a e g)^2 \left (8 a e^2 g+c d (e f-9 d g)\right ) \left (2 a e^2 g-c d (3 e f-d g)\right )}{315 c^5 d^5 e g \sqrt{d+e x}}-\frac{16 \sqrt{d+e x} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2} (c d f-a e g)^2 \left (8 a e^2 g+c d (e f-9 d g)\right )}{315 c^4 d^4 e}-\frac{4 (f+g x)^2 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2} (c d f-a e g) \left (8 a e^2 g+c d (e f-9 d g)\right )}{105 c^3 d^3 g \sqrt{d+e x}}-\frac{2 (f+g x)^3 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2} \left (8 a e^2 g+c d (e f-9 d g)\right )}{63 c^2 d^2 g \sqrt{d+e x}}+\frac{2 e (f+g x)^4 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{9 c d g \sqrt{d+e x}} \]

[Out]

(16*(c*d*f - a*e*g)^2*(8*a*e^2*g + c*d*(e*f - 9*d*g))*(2*a*e^2*g - c*d*(3*e*f -
d*g))*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(315*c^5*d^5*e*g*Sqrt[d + e*x
]) - (16*(c*d*f - a*e*g)^2*(8*a*e^2*g + c*d*(e*f - 9*d*g))*Sqrt[d + e*x]*Sqrt[a*
d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(315*c^4*d^4*e) - (4*(c*d*f - a*e*g)*(8*a*
e^2*g + c*d*(e*f - 9*d*g))*(f + g*x)^2*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^
2])/(105*c^3*d^3*g*Sqrt[d + e*x]) - (2*(8*a*e^2*g + c*d*(e*f - 9*d*g))*(f + g*x)
^3*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(63*c^2*d^2*g*Sqrt[d + e*x]) + (
2*e*(f + g*x)^4*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(9*c*d*g*Sqrt[d + e
*x])

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Rubi [A]  time = 1.6817, antiderivative size = 412, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 46, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.087 \[ \frac{16 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2} (c d f-a e g)^2 \left (8 a e^2 g+c d (e f-9 d g)\right ) \left (2 a e^2 g-c d (3 e f-d g)\right )}{315 c^5 d^5 e g \sqrt{d+e x}}-\frac{16 \sqrt{d+e x} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2} (c d f-a e g)^2 \left (8 a e^2 g+c d (e f-9 d g)\right )}{315 c^4 d^4 e}-\frac{4 (f+g x)^2 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2} (c d f-a e g) \left (8 a e^2 g+c d (e f-9 d g)\right )}{105 c^3 d^3 g \sqrt{d+e x}}-\frac{2 (f+g x)^3 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2} \left (8 a e^2 g+c d (e f-9 d g)\right )}{63 c^2 d^2 g \sqrt{d+e x}}+\frac{2 e (f+g x)^4 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{9 c d g \sqrt{d+e x}} \]

Antiderivative was successfully verified.

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

[Out]

(16*(c*d*f - a*e*g)^2*(8*a*e^2*g + c*d*(e*f - 9*d*g))*(2*a*e^2*g - c*d*(3*e*f -
d*g))*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(315*c^5*d^5*e*g*Sqrt[d + e*x
]) - (16*(c*d*f - a*e*g)^2*(8*a*e^2*g + c*d*(e*f - 9*d*g))*Sqrt[d + e*x]*Sqrt[a*
d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(315*c^4*d^4*e) - (4*(c*d*f - a*e*g)*(8*a*
e^2*g + c*d*(e*f - 9*d*g))*(f + g*x)^2*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^
2])/(105*c^3*d^3*g*Sqrt[d + e*x]) - (2*(8*a*e^2*g + c*d*(e*f - 9*d*g))*(f + g*x)
^3*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(63*c^2*d^2*g*Sqrt[d + e*x]) + (
2*e*(f + g*x)^4*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(9*c*d*g*Sqrt[d + e
*x])

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Rubi in Sympy [A]  time = 139.16, size = 418, normalized size = 1.01 \[ \frac{2 e \left (f + g x\right )^{4} \sqrt{a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )}}{9 c d g \sqrt{d + e x}} - \frac{2 \left (f + g x\right )^{3} \sqrt{a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )} \left (8 a e^{2} g - 9 c d^{2} g + c d e f\right )}{63 c^{2} d^{2} g \sqrt{d + e x}} + \frac{4 \left (f + g x\right )^{2} \left (a e g - c d f\right ) \sqrt{a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )} \left (8 a e^{2} g - 9 c d^{2} g + c d e f\right )}{105 c^{3} d^{3} g \sqrt{d + e x}} - \frac{16 \sqrt{d + e x} \left (a e g - c d f\right )^{2} \sqrt{a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )} \left (8 a e^{2} g - 9 c d^{2} g + c d e f\right )}{315 c^{4} d^{4} e} + \frac{16 \left (a e g - c d f\right )^{2} \sqrt{a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )} \left (2 a e^{2} g + c d^{2} g - 3 c d e f\right ) \left (8 a e^{2} g - 9 c d^{2} g + c d e f\right )}{315 c^{5} d^{5} e g \sqrt{d + e x}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*e*(f + g*x)**4*sqrt(a*d*e + c*d*e*x**2 + x*(a*e**2 + c*d**2))/(9*c*d*g*sqrt(d
+ e*x)) - 2*(f + g*x)**3*sqrt(a*d*e + c*d*e*x**2 + x*(a*e**2 + c*d**2))*(8*a*e**
2*g - 9*c*d**2*g + c*d*e*f)/(63*c**2*d**2*g*sqrt(d + e*x)) + 4*(f + g*x)**2*(a*e
*g - c*d*f)*sqrt(a*d*e + c*d*e*x**2 + x*(a*e**2 + c*d**2))*(8*a*e**2*g - 9*c*d**
2*g + c*d*e*f)/(105*c**3*d**3*g*sqrt(d + e*x)) - 16*sqrt(d + e*x)*(a*e*g - c*d*f
)**2*sqrt(a*d*e + c*d*e*x**2 + x*(a*e**2 + c*d**2))*(8*a*e**2*g - 9*c*d**2*g + c
*d*e*f)/(315*c**4*d**4*e) + 16*(a*e*g - c*d*f)**2*sqrt(a*d*e + c*d*e*x**2 + x*(a
*e**2 + c*d**2))*(2*a*e**2*g + c*d**2*g - 3*c*d*e*f)*(8*a*e**2*g - 9*c*d**2*g +
c*d*e*f)/(315*c**5*d**5*e*g*sqrt(d + e*x))

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Mathematica [A]  time = 0.429614, size = 264, normalized size = 0.64 \[ \frac{2 \sqrt{(d+e x) (a e+c d x)} \left (128 a^4 e^5 g^3-16 a^3 c d e^3 g^2 (9 d g+27 e f+4 e g x)+24 a^2 c^2 d^2 e^2 g \left (3 d g (7 f+g x)+e \left (21 f^2+9 f g x+2 g^2 x^2\right )\right )-2 a c^3 d^3 e \left (9 d g \left (35 f^2+14 f g x+3 g^2 x^2\right )+e \left (105 f^3+126 f^2 g x+81 f g^2 x^2+20 g^3 x^3\right )\right )+c^4 d^4 \left (9 d \left (35 f^3+35 f^2 g x+21 f g^2 x^2+5 g^3 x^3\right )+e x \left (105 f^3+189 f^2 g x+135 f g^2 x^2+35 g^3 x^3\right )\right )\right )}{315 c^5 d^5 \sqrt{d+e x}} \]

Antiderivative was successfully verified.

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

[Out]

(2*Sqrt[(a*e + c*d*x)*(d + e*x)]*(128*a^4*e^5*g^3 - 16*a^3*c*d*e^3*g^2*(27*e*f +
9*d*g + 4*e*g*x) + 24*a^2*c^2*d^2*e^2*g*(3*d*g*(7*f + g*x) + e*(21*f^2 + 9*f*g*
x + 2*g^2*x^2)) - 2*a*c^3*d^3*e*(9*d*g*(35*f^2 + 14*f*g*x + 3*g^2*x^2) + e*(105*
f^3 + 126*f^2*g*x + 81*f*g^2*x^2 + 20*g^3*x^3)) + c^4*d^4*(9*d*(35*f^3 + 35*f^2*
g*x + 21*f*g^2*x^2 + 5*g^3*x^3) + e*x*(105*f^3 + 189*f^2*g*x + 135*f*g^2*x^2 + 3
5*g^3*x^3))))/(315*c^5*d^5*Sqrt[d + e*x])

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Maple [A]  time = 0.013, size = 425, normalized size = 1. \[{\frac{ \left ( 2\,cdx+2\,ae \right ) \left ( 35\,e{g}^{3}{x}^{4}{c}^{4}{d}^{4}-40\,a{c}^{3}{d}^{3}{e}^{2}{g}^{3}{x}^{3}+45\,{c}^{4}{d}^{5}{g}^{3}{x}^{3}+135\,{c}^{4}{d}^{4}ef{g}^{2}{x}^{3}+48\,{a}^{2}{c}^{2}{d}^{2}{e}^{3}{g}^{3}{x}^{2}-54\,a{c}^{3}{d}^{4}e{g}^{3}{x}^{2}-162\,a{c}^{3}{d}^{3}{e}^{2}f{g}^{2}{x}^{2}+189\,{c}^{4}{d}^{5}f{g}^{2}{x}^{2}+189\,{c}^{4}{d}^{4}e{f}^{2}g{x}^{2}-64\,{a}^{3}cd{e}^{4}{g}^{3}x+72\,{a}^{2}{c}^{2}{d}^{3}{e}^{2}{g}^{3}x+216\,{a}^{2}{c}^{2}{d}^{2}{e}^{3}f{g}^{2}x-252\,a{c}^{3}{d}^{4}ef{g}^{2}x-252\,a{c}^{3}{d}^{3}{e}^{2}{f}^{2}gx+315\,{c}^{4}{d}^{5}{f}^{2}gx+105\,{c}^{4}{d}^{4}e{f}^{3}x+128\,{a}^{4}{e}^{5}{g}^{3}-144\,{a}^{3}c{d}^{2}{e}^{3}{g}^{3}-432\,{a}^{3}cd{e}^{4}f{g}^{2}+504\,{a}^{2}{c}^{2}{d}^{3}{e}^{2}f{g}^{2}+504\,{a}^{2}{c}^{2}{d}^{2}{e}^{3}{f}^{2}g-630\,a{c}^{3}{d}^{4}e{f}^{2}g-210\,a{c}^{3}{d}^{3}{e}^{2}{f}^{3}+315\,{d}^{5}{f}^{3}{c}^{4} \right ) }{315\,{c}^{5}{d}^{5}}\sqrt{ex+d}{\frac{1}{\sqrt{cde{x}^{2}+a{e}^{2}x+c{d}^{2}x+ade}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/315*(c*d*x+a*e)*(35*c^4*d^4*e*g^3*x^4-40*a*c^3*d^3*e^2*g^3*x^3+45*c^4*d^5*g^3*
x^3+135*c^4*d^4*e*f*g^2*x^3+48*a^2*c^2*d^2*e^3*g^3*x^2-54*a*c^3*d^4*e*g^3*x^2-16
2*a*c^3*d^3*e^2*f*g^2*x^2+189*c^4*d^5*f*g^2*x^2+189*c^4*d^4*e*f^2*g*x^2-64*a^3*c
*d*e^4*g^3*x+72*a^2*c^2*d^3*e^2*g^3*x+216*a^2*c^2*d^2*e^3*f*g^2*x-252*a*c^3*d^4*
e*f*g^2*x-252*a*c^3*d^3*e^2*f^2*g*x+315*c^4*d^5*f^2*g*x+105*c^4*d^4*e*f^3*x+128*
a^4*e^5*g^3-144*a^3*c*d^2*e^3*g^3-432*a^3*c*d*e^4*f*g^2+504*a^2*c^2*d^3*e^2*f*g^
2+504*a^2*c^2*d^2*e^3*f^2*g-630*a*c^3*d^4*e*f^2*g-210*a*c^3*d^3*e^2*f^3+315*c^4*
d^5*f^3)*(e*x+d)^(1/2)/c^5/d^5/(c*d*e*x^2+a*e^2*x+c*d^2*x+a*d*e)^(1/2)

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Maxima [A]  time = 0.755974, size = 653, normalized size = 1.58 \[ \frac{2 \,{\left (c^{2} d^{2} e x^{2} + 3 \, a c d^{2} e - 2 \, a^{2} e^{3} +{\left (3 \, c^{2} d^{3} - a c d e^{2}\right )} x\right )} f^{3}}{3 \, \sqrt{c d x + a e} c^{2} d^{2}} + \frac{2 \,{\left (3 \, c^{3} d^{3} e x^{3} - 10 \, a^{2} c d^{2} e^{2} + 8 \, a^{3} e^{4} +{\left (5 \, c^{3} d^{4} - a c^{2} d^{2} e^{2}\right )} x^{2} -{\left (5 \, a c^{2} d^{3} e - 4 \, a^{2} c d e^{3}\right )} x\right )} f^{2} g}{5 \, \sqrt{c d x + a e} c^{3} d^{3}} + \frac{2 \,{\left (15 \, c^{4} d^{4} e x^{4} + 56 \, a^{3} c d^{2} e^{3} - 48 \, a^{4} e^{5} + 3 \,{\left (7 \, c^{4} d^{5} - a c^{3} d^{3} e^{2}\right )} x^{3} -{\left (7 \, a c^{3} d^{4} e - 6 \, a^{2} c^{2} d^{2} e^{3}\right )} x^{2} + 4 \,{\left (7 \, a^{2} c^{2} d^{3} e^{2} - 6 \, a^{3} c d e^{4}\right )} x\right )} f g^{2}}{35 \, \sqrt{c d x + a e} c^{4} d^{4}} + \frac{2 \,{\left (35 \, c^{5} d^{5} e x^{5} - 144 \, a^{4} c d^{2} e^{4} + 128 \, a^{5} e^{6} + 5 \,{\left (9 \, c^{5} d^{6} - a c^{4} d^{4} e^{2}\right )} x^{4} -{\left (9 \, a c^{4} d^{5} e - 8 \, a^{2} c^{3} d^{3} e^{3}\right )} x^{3} + 2 \,{\left (9 \, a^{2} c^{3} d^{4} e^{2} - 8 \, a^{3} c^{2} d^{2} e^{4}\right )} x^{2} - 8 \,{\left (9 \, a^{3} c^{2} d^{3} e^{3} - 8 \, a^{4} c d e^{5}\right )} x\right )} g^{3}}{315 \, \sqrt{c d x + a e} c^{5} d^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/3*(c^2*d^2*e*x^2 + 3*a*c*d^2*e - 2*a^2*e^3 + (3*c^2*d^3 - a*c*d*e^2)*x)*f^3/(s
qrt(c*d*x + a*e)*c^2*d^2) + 2/5*(3*c^3*d^3*e*x^3 - 10*a^2*c*d^2*e^2 + 8*a^3*e^4
+ (5*c^3*d^4 - a*c^2*d^2*e^2)*x^2 - (5*a*c^2*d^3*e - 4*a^2*c*d*e^3)*x)*f^2*g/(sq
rt(c*d*x + a*e)*c^3*d^3) + 2/35*(15*c^4*d^4*e*x^4 + 56*a^3*c*d^2*e^3 - 48*a^4*e^
5 + 3*(7*c^4*d^5 - a*c^3*d^3*e^2)*x^3 - (7*a*c^3*d^4*e - 6*a^2*c^2*d^2*e^3)*x^2
+ 4*(7*a^2*c^2*d^3*e^2 - 6*a^3*c*d*e^4)*x)*f*g^2/(sqrt(c*d*x + a*e)*c^4*d^4) + 2
/315*(35*c^5*d^5*e*x^5 - 144*a^4*c*d^2*e^4 + 128*a^5*e^6 + 5*(9*c^5*d^6 - a*c^4*
d^4*e^2)*x^4 - (9*a*c^4*d^5*e - 8*a^2*c^3*d^3*e^3)*x^3 + 2*(9*a^2*c^3*d^4*e^2 -
8*a^3*c^2*d^2*e^4)*x^2 - 8*(9*a^3*c^2*d^3*e^3 - 8*a^4*c*d*e^5)*x)*g^3/(sqrt(c*d*
x + a*e)*c^5*d^5)

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Fricas [A]  time = 0.27576, size = 1061, normalized size = 2.58 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/315*(35*c^5*d^5*e^2*g^3*x^6 + 5*(27*c^5*d^5*e^2*f*g^2 + (16*c^5*d^6*e - a*c^4*
d^4*e^3)*g^3)*x^5 + (189*c^5*d^5*e^2*f^2*g + 27*(12*c^5*d^6*e - a*c^4*d^4*e^3)*f
*g^2 + (45*c^5*d^7 - 14*a*c^4*d^5*e^2 + 8*a^2*c^3*d^3*e^4)*g^3)*x^4 + 105*(3*a*c
^4*d^6*e - 2*a^2*c^3*d^4*e^3)*f^3 - 126*(5*a^2*c^3*d^5*e^2 - 4*a^3*c^2*d^3*e^4)*
f^2*g + 72*(7*a^3*c^2*d^4*e^3 - 6*a^4*c*d^2*e^5)*f*g^2 - 16*(9*a^4*c*d^3*e^4 - 8
*a^5*d*e^6)*g^3 + (105*c^5*d^5*e^2*f^3 + 63*(8*c^5*d^6*e - a*c^4*d^4*e^3)*f^2*g
+ 9*(21*c^5*d^7 - 10*a*c^4*d^5*e^2 + 6*a^2*c^3*d^3*e^4)*f*g^2 - (9*a*c^4*d^6*e -
26*a^2*c^3*d^4*e^3 + 16*a^3*c^2*d^2*e^5)*g^3)*x^3 + (105*(4*c^5*d^6*e - a*c^4*d
^4*e^3)*f^3 + 63*(5*c^5*d^7 - 6*a*c^4*d^5*e^2 + 4*a^2*c^3*d^3*e^4)*f^2*g - 9*(7*
a*c^4*d^6*e - 34*a^2*c^3*d^4*e^3 + 24*a^3*c^2*d^2*e^5)*f*g^2 + 2*(9*a^2*c^3*d^5*
e^2 - 44*a^3*c^2*d^3*e^4 + 32*a^4*c*d*e^6)*g^3)*x^2 + (105*(3*c^5*d^7 + 2*a*c^4*
d^5*e^2 - 2*a^2*c^3*d^3*e^4)*f^3 - 63*(5*a*c^4*d^6*e + 6*a^2*c^3*d^4*e^3 - 8*a^3
*c^2*d^2*e^5)*f^2*g + 36*(7*a^2*c^3*d^5*e^2 + 8*a^3*c^2*d^3*e^4 - 12*a^4*c*d*e^6
)*f*g^2 - 8*(9*a^3*c^2*d^4*e^3 + 10*a^4*c*d^2*e^5 - 16*a^5*e^7)*g^3)*x)/(sqrt(c*
d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*sqrt(e*x + d)*c^5*d^5)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.39072, size = 1, normalized size = 0. \[ \mathit{Done} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Done