3.1426 \(\int \frac{(a+b x)^3}{(c+d x)^{3/2}} \, dx\)

Optimal. Leaf size=94 \[ -\frac{2 b^2 (c+d x)^{3/2} (b c-a d)}{d^4}+\frac{6 b \sqrt{c+d x} (b c-a d)^2}{d^4}+\frac{2 (b c-a d)^3}{d^4 \sqrt{c+d x}}+\frac{2 b^3 (c+d x)^{5/2}}{5 d^4} \]

[Out]

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

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Rubi [A]  time = 0.0955831, antiderivative size = 94, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.059 \[ -\frac{2 b^2 (c+d x)^{3/2} (b c-a d)}{d^4}+\frac{6 b \sqrt{c+d x} (b c-a d)^2}{d^4}+\frac{2 (b c-a d)^3}{d^4 \sqrt{c+d x}}+\frac{2 b^3 (c+d x)^{5/2}}{5 d^4} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 21.9889, size = 87, normalized size = 0.93 \[ \frac{2 b^{3} \left (c + d x\right )^{\frac{5}{2}}}{5 d^{4}} + \frac{2 b^{2} \left (c + d x\right )^{\frac{3}{2}} \left (a d - b c\right )}{d^{4}} + \frac{6 b \sqrt{c + d x} \left (a d - b c\right )^{2}}{d^{4}} - \frac{2 \left (a d - b c\right )^{3}}{d^{4} \sqrt{c + d x}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*b**3*(c + d*x)**(5/2)/(5*d**4) + 2*b**2*(c + d*x)**(3/2)*(a*d - b*c)/d**4 + 6*
b*sqrt(c + d*x)*(a*d - b*c)**2/d**4 - 2*(a*d - b*c)**3/(d**4*sqrt(c + d*x))

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Mathematica [A]  time = 0.0852668, size = 99, normalized size = 1.05 \[ \frac{2 \left (-5 a^3 d^3+15 a^2 b d^2 (2 c+d x)+5 a b^2 d \left (-8 c^2-4 c d x+d^2 x^2\right )+b^3 \left (16 c^3+8 c^2 d x-2 c d^2 x^2+d^3 x^3\right )\right )}{5 d^4 \sqrt{c+d x}} \]

Antiderivative was successfully verified.

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

[Out]

(2*(-5*a^3*d^3 + 15*a^2*b*d^2*(2*c + d*x) + 5*a*b^2*d*(-8*c^2 - 4*c*d*x + d^2*x^
2) + b^3*(16*c^3 + 8*c^2*d*x - 2*c*d^2*x^2 + d^3*x^3)))/(5*d^4*Sqrt[c + d*x])

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Maple [A]  time = 0.009, size = 116, normalized size = 1.2 \[ -{\frac{-2\,{b}^{3}{x}^{3}{d}^{3}-10\,a{b}^{2}{d}^{3}{x}^{2}+4\,{b}^{3}c{d}^{2}{x}^{2}-30\,{a}^{2}b{d}^{3}x+40\,a{b}^{2}c{d}^{2}x-16\,{b}^{3}{c}^{2}dx+10\,{a}^{3}{d}^{3}-60\,{a}^{2}bc{d}^{2}+80\,a{b}^{2}{c}^{2}d-32\,{b}^{3}{c}^{3}}{5\,{d}^{4}}{\frac{1}{\sqrt{dx+c}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^3/(d*x+c)^(3/2),x)

[Out]

-2/5/(d*x+c)^(1/2)*(-b^3*d^3*x^3-5*a*b^2*d^3*x^2+2*b^3*c*d^2*x^2-15*a^2*b*d^3*x+
20*a*b^2*c*d^2*x-8*b^3*c^2*d*x+5*a^3*d^3-30*a^2*b*c*d^2+40*a*b^2*c^2*d-16*b^3*c^
3)/d^4

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Maxima [A]  time = 1.34382, size = 169, normalized size = 1.8 \[ \frac{2 \,{\left (\frac{{\left (d x + c\right )}^{\frac{5}{2}} b^{3} - 5 \,{\left (b^{3} c - a b^{2} d\right )}{\left (d x + c\right )}^{\frac{3}{2}} + 15 \,{\left (b^{3} c^{2} - 2 \, a b^{2} c d + a^{2} b d^{2}\right )} \sqrt{d x + c}}{d^{3}} + \frac{5 \,{\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )}}{\sqrt{d x + c} d^{3}}\right )}}{5 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/5*(((d*x + c)^(5/2)*b^3 - 5*(b^3*c - a*b^2*d)*(d*x + c)^(3/2) + 15*(b^3*c^2 -
2*a*b^2*c*d + a^2*b*d^2)*sqrt(d*x + c))/d^3 + 5*(b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2
*b*c*d^2 - a^3*d^3)/(sqrt(d*x + c)*d^3))/d

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Fricas [A]  time = 0.208917, size = 154, normalized size = 1.64 \[ \frac{2 \,{\left (b^{3} d^{3} x^{3} + 16 \, b^{3} c^{3} - 40 \, a b^{2} c^{2} d + 30 \, a^{2} b c d^{2} - 5 \, a^{3} d^{3} -{\left (2 \, b^{3} c d^{2} - 5 \, a b^{2} d^{3}\right )} x^{2} +{\left (8 \, b^{3} c^{2} d - 20 \, a b^{2} c d^{2} + 15 \, a^{2} b d^{3}\right )} x\right )}}{5 \, \sqrt{d x + c} d^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/5*(b^3*d^3*x^3 + 16*b^3*c^3 - 40*a*b^2*c^2*d + 30*a^2*b*c*d^2 - 5*a^3*d^3 - (2
*b^3*c*d^2 - 5*a*b^2*d^3)*x^2 + (8*b^3*c^2*d - 20*a*b^2*c*d^2 + 15*a^2*b*d^3)*x)
/(sqrt(d*x + c)*d^4)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.217979, size = 205, normalized size = 2.18 \[ \frac{2 \,{\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )}}{\sqrt{d x + c} d^{4}} + \frac{2 \,{\left ({\left (d x + c\right )}^{\frac{5}{2}} b^{3} d^{16} - 5 \,{\left (d x + c\right )}^{\frac{3}{2}} b^{3} c d^{16} + 15 \, \sqrt{d x + c} b^{3} c^{2} d^{16} + 5 \,{\left (d x + c\right )}^{\frac{3}{2}} a b^{2} d^{17} - 30 \, \sqrt{d x + c} a b^{2} c d^{17} + 15 \, \sqrt{d x + c} a^{2} b d^{18}\right )}}{5 \, d^{20}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*(b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3)/(sqrt(d*x + c)*d^4) + 2/5*
((d*x + c)^(5/2)*b^3*d^16 - 5*(d*x + c)^(3/2)*b^3*c*d^16 + 15*sqrt(d*x + c)*b^3*
c^2*d^16 + 5*(d*x + c)^(3/2)*a*b^2*d^17 - 30*sqrt(d*x + c)*a*b^2*c*d^17 + 15*sqr
t(d*x + c)*a^2*b*d^18)/d^20