3.1436 \(\int \frac{(a+b x)^3}{(c+d x)^{5/2}} \, dx\)

Optimal. Leaf size=96 \[ -\frac{6 b^2 \sqrt{c+d x} (b c-a d)}{d^4}-\frac{6 b (b c-a d)^2}{d^4 \sqrt{c+d x}}+\frac{2 (b c-a d)^3}{3 d^4 (c+d x)^{3/2}}+\frac{2 b^3 (c+d x)^{3/2}}{3 d^4} \]

[Out]

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

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Rubi [A]  time = 0.0916707, antiderivative size = 96, 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 \[ -\frac{6 b^2 \sqrt{c+d x} (b c-a d)}{d^4}-\frac{6 b (b c-a d)^2}{d^4 \sqrt{c+d x}}+\frac{2 (b c-a d)^3}{3 d^4 (c+d x)^{3/2}}+\frac{2 b^3 (c+d x)^{3/2}}{3 d^4} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x)^3/(c + d*x)^(5/2),x]

[Out]

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

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Rubi in Sympy [A]  time = 20.6128, size = 88, normalized size = 0.92 \[ \frac{2 b^{3} \left (c + d x\right )^{\frac{3}{2}}}{3 d^{4}} + \frac{6 b^{2} \sqrt{c + d x} \left (a d - b c\right )}{d^{4}} - \frac{6 b \left (a d - b c\right )^{2}}{d^{4} \sqrt{c + d x}} - \frac{2 \left (a d - b c\right )^{3}}{3 d^{4} \left (c + d x\right )^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)**3/(d*x+c)**(5/2),x)

[Out]

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

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Mathematica [A]  time = 0.130574, size = 74, normalized size = 0.77 \[ \frac{2 \sqrt{c+d x} \left (b^2 (9 a d-8 b c)-\frac{9 b (b c-a d)^2}{c+d x}+\frac{(b c-a d)^3}{(c+d x)^2}+b^3 d x\right )}{3 d^4} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x)^3/(c + d*x)^(5/2),x]

[Out]

(2*Sqrt[c + d*x]*(b^2*(-8*b*c + 9*a*d) + b^3*d*x + (b*c - a*d)^3/(c + d*x)^2 - (
9*b*(b*c - a*d)^2)/(c + d*x)))/(3*d^4)

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Maple [A]  time = 0.007, size = 115, normalized size = 1.2 \[ -{\frac{-2\,{b}^{3}{x}^{3}{d}^{3}-18\,a{b}^{2}{d}^{3}{x}^{2}+12\,{b}^{3}c{d}^{2}{x}^{2}+18\,{a}^{2}b{d}^{3}x-72\,a{b}^{2}c{d}^{2}x+48\,{b}^{3}{c}^{2}dx+2\,{a}^{3}{d}^{3}+12\,{a}^{2}bc{d}^{2}-48\,a{b}^{2}{c}^{2}d+32\,{b}^{3}{c}^{3}}{3\,{d}^{4}} \left ( dx+c \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^3/(d*x+c)^(5/2),x)

[Out]

-2/3/(d*x+c)^(3/2)*(-b^3*d^3*x^3-9*a*b^2*d^3*x^2+6*b^3*c*d^2*x^2+9*a^2*b*d^3*x-3
6*a*b^2*c*d^2*x+24*b^3*c^2*d*x+a^3*d^3+6*a^2*b*c*d^2-24*a*b^2*c^2*d+16*b^3*c^3)/
d^4

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Maxima [A]  time = 1.3422, size = 165, normalized size = 1.72 \[ \frac{2 \,{\left (\frac{{\left (d x + c\right )}^{\frac{3}{2}} b^{3} - 9 \,{\left (b^{3} c - a b^{2} d\right )} \sqrt{d x + c}}{d^{3}} + \frac{b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3} - 9 \,{\left (b^{3} c^{2} - 2 \, a b^{2} c d + a^{2} b d^{2}\right )}{\left (d x + c\right )}}{{\left (d x + c\right )}^{\frac{3}{2}} d^{3}}\right )}}{3 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^3/(d*x + c)^(5/2),x, algorithm="maxima")

[Out]

2/3*(((d*x + c)^(3/2)*b^3 - 9*(b^3*c - a*b^2*d)*sqrt(d*x + c))/d^3 + (b^3*c^3 -
3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3 - 9*(b^3*c^2 - 2*a*b^2*c*d + a^2*b*d^2)*
(d*x + c))/((d*x + c)^(3/2)*d^3))/d

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Fricas [A]  time = 0.206703, size = 169, normalized size = 1.76 \[ \frac{2 \,{\left (b^{3} d^{3} x^{3} - 16 \, b^{3} c^{3} + 24 \, a b^{2} c^{2} d - 6 \, a^{2} b c d^{2} - a^{3} d^{3} - 3 \,{\left (2 \, b^{3} c d^{2} - 3 \, a b^{2} d^{3}\right )} x^{2} - 3 \,{\left (8 \, b^{3} c^{2} d - 12 \, a b^{2} c d^{2} + 3 \, a^{2} b d^{3}\right )} x\right )}}{3 \,{\left (d^{5} x + c d^{4}\right )} \sqrt{d x + c}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^3/(d*x + c)^(5/2),x, algorithm="fricas")

[Out]

2/3*(b^3*d^3*x^3 - 16*b^3*c^3 + 24*a*b^2*c^2*d - 6*a^2*b*c*d^2 - a^3*d^3 - 3*(2*
b^3*c*d^2 - 3*a*b^2*d^3)*x^2 - 3*(8*b^3*c^2*d - 12*a*b^2*c*d^2 + 3*a^2*b*d^3)*x)
/((d^5*x + c*d^4)*sqrt(d*x + c))

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Sympy [A]  time = 2.11953, size = 461, normalized size = 4.8 \[ \begin{cases} - \frac{2 a^{3} d^{3}}{3 c d^{4} \sqrt{c + d x} + 3 d^{5} x \sqrt{c + d x}} - \frac{12 a^{2} b c d^{2}}{3 c d^{4} \sqrt{c + d x} + 3 d^{5} x \sqrt{c + d x}} - \frac{18 a^{2} b d^{3} x}{3 c d^{4} \sqrt{c + d x} + 3 d^{5} x \sqrt{c + d x}} + \frac{48 a b^{2} c^{2} d}{3 c d^{4} \sqrt{c + d x} + 3 d^{5} x \sqrt{c + d x}} + \frac{72 a b^{2} c d^{2} x}{3 c d^{4} \sqrt{c + d x} + 3 d^{5} x \sqrt{c + d x}} + \frac{18 a b^{2} d^{3} x^{2}}{3 c d^{4} \sqrt{c + d x} + 3 d^{5} x \sqrt{c + d x}} - \frac{32 b^{3} c^{3}}{3 c d^{4} \sqrt{c + d x} + 3 d^{5} x \sqrt{c + d x}} - \frac{48 b^{3} c^{2} d x}{3 c d^{4} \sqrt{c + d x} + 3 d^{5} x \sqrt{c + d x}} - \frac{12 b^{3} c d^{2} x^{2}}{3 c d^{4} \sqrt{c + d x} + 3 d^{5} x \sqrt{c + d x}} + \frac{2 b^{3} d^{3} x^{3}}{3 c d^{4} \sqrt{c + d x} + 3 d^{5} x \sqrt{c + d x}} & \text{for}\: d \neq 0 \\\frac{a^{3} x + \frac{3 a^{2} b x^{2}}{2} + a b^{2} x^{3} + \frac{b^{3} x^{4}}{4}}{c^{\frac{5}{2}}} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x+a)**3/(d*x+c)**(5/2),x)

[Out]

Piecewise((-2*a**3*d**3/(3*c*d**4*sqrt(c + d*x) + 3*d**5*x*sqrt(c + d*x)) - 12*a
**2*b*c*d**2/(3*c*d**4*sqrt(c + d*x) + 3*d**5*x*sqrt(c + d*x)) - 18*a**2*b*d**3*
x/(3*c*d**4*sqrt(c + d*x) + 3*d**5*x*sqrt(c + d*x)) + 48*a*b**2*c**2*d/(3*c*d**4
*sqrt(c + d*x) + 3*d**5*x*sqrt(c + d*x)) + 72*a*b**2*c*d**2*x/(3*c*d**4*sqrt(c +
d*x) + 3*d**5*x*sqrt(c + d*x)) + 18*a*b**2*d**3*x**2/(3*c*d**4*sqrt(c + d*x) +
3*d**5*x*sqrt(c + d*x)) - 32*b**3*c**3/(3*c*d**4*sqrt(c + d*x) + 3*d**5*x*sqrt(c
+ d*x)) - 48*b**3*c**2*d*x/(3*c*d**4*sqrt(c + d*x) + 3*d**5*x*sqrt(c + d*x)) -
12*b**3*c*d**2*x**2/(3*c*d**4*sqrt(c + d*x) + 3*d**5*x*sqrt(c + d*x)) + 2*b**3*d
**3*x**3/(3*c*d**4*sqrt(c + d*x) + 3*d**5*x*sqrt(c + d*x)), Ne(d, 0)), ((a**3*x
+ 3*a**2*b*x**2/2 + a*b**2*x**3 + b**3*x**4/4)/c**(5/2), True))

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GIAC/XCAS [A]  time = 0.222827, size = 190, normalized size = 1.98 \[ -\frac{2 \,{\left (9 \,{\left (d x + c\right )} b^{3} c^{2} - b^{3} c^{3} - 18 \,{\left (d x + c\right )} a b^{2} c d + 3 \, a b^{2} c^{2} d + 9 \,{\left (d x + c\right )} a^{2} b d^{2} - 3 \, a^{2} b c d^{2} + a^{3} d^{3}\right )}}{3 \,{\left (d x + c\right )}^{\frac{3}{2}} d^{4}} + \frac{2 \,{\left ({\left (d x + c\right )}^{\frac{3}{2}} b^{3} d^{8} - 9 \, \sqrt{d x + c} b^{3} c d^{8} + 9 \, \sqrt{d x + c} a b^{2} d^{9}\right )}}{3 \, d^{12}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^3/(d*x + c)^(5/2),x, algorithm="giac")

[Out]

-2/3*(9*(d*x + c)*b^3*c^2 - b^3*c^3 - 18*(d*x + c)*a*b^2*c*d + 3*a*b^2*c^2*d + 9
*(d*x + c)*a^2*b*d^2 - 3*a^2*b*c*d^2 + a^3*d^3)/((d*x + c)^(3/2)*d^4) + 2/3*((d*
x + c)^(3/2)*b^3*d^8 - 9*sqrt(d*x + c)*b^3*c*d^8 + 9*sqrt(d*x + c)*a*b^2*d^9)/d^
12