3.1437 \(\int \frac{(a+b x)^2}{(c+d x)^{5/2}} \, dx\)

Optimal. Leaf size=67 \[ \frac{4 b (b c-a d)}{d^3 \sqrt{c+d x}}-\frac{2 (b c-a d)^2}{3 d^3 (c+d x)^{3/2}}+\frac{2 b^2 \sqrt{c+d x}}{d^3} \]

[Out]

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

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Rubi [A]  time = 0.0668144, antiderivative size = 67, 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 \[ \frac{4 b (b c-a d)}{d^3 \sqrt{c+d x}}-\frac{2 (b c-a d)^2}{3 d^3 (c+d x)^{3/2}}+\frac{2 b^2 \sqrt{c+d x}}{d^3} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 14.1267, size = 61, normalized size = 0.91 \[ \frac{2 b^{2} \sqrt{c + d x}}{d^{3}} - \frac{4 b \left (a d - b c\right )}{d^{3} \sqrt{c + d x}} - \frac{2 \left (a d - b c\right )^{2}}{3 d^{3} \left (c + d x\right )^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0594631, size = 64, normalized size = 0.96 \[ \sqrt{c+d x} \left (\frac{4 b (b c-a d)}{d^3 (c+d x)}-\frac{2 (a d-b c)^2}{3 d^3 (c+d x)^2}+\frac{2 b^2}{d^3}\right ) \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.007, size = 62, normalized size = 0.9 \[ -{\frac{-6\,{b}^{2}{x}^{2}{d}^{2}+12\,ab{d}^{2}x-24\,{b}^{2}cdx+2\,{a}^{2}{d}^{2}+8\,abcd-16\,{b}^{2}{c}^{2}}{3\,{d}^{3}} \left ( dx+c \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^2/(d*x+c)^(5/2),x)

[Out]

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

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Maxima [A]  time = 1.33749, size = 97, normalized size = 1.45 \[ \frac{2 \,{\left (\frac{3 \, \sqrt{d x + c} b^{2}}{d^{2}} - \frac{b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2} - 6 \,{\left (b^{2} c - a b d\right )}{\left (d x + c\right )}}{{\left (d x + c\right )}^{\frac{3}{2}} d^{2}}\right )}}{3 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.205433, size = 100, normalized size = 1.49 \[ \frac{2 \,{\left (3 \, b^{2} d^{2} x^{2} + 8 \, b^{2} c^{2} - 4 \, a b c d - a^{2} d^{2} + 6 \,{\left (2 \, b^{2} c d - a b d^{2}\right )} x\right )}}{3 \,{\left (d^{4} x + c d^{3}\right )} \sqrt{d x + c}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 1.90835, size = 265, normalized size = 3.96 \[ \begin{cases} - \frac{2 a^{2} d^{2}}{3 c d^{3} \sqrt{c + d x} + 3 d^{4} x \sqrt{c + d x}} - \frac{8 a b c d}{3 c d^{3} \sqrt{c + d x} + 3 d^{4} x \sqrt{c + d x}} - \frac{12 a b d^{2} x}{3 c d^{3} \sqrt{c + d x} + 3 d^{4} x \sqrt{c + d x}} + \frac{16 b^{2} c^{2}}{3 c d^{3} \sqrt{c + d x} + 3 d^{4} x \sqrt{c + d x}} + \frac{24 b^{2} c d x}{3 c d^{3} \sqrt{c + d x} + 3 d^{4} x \sqrt{c + d x}} + \frac{6 b^{2} d^{2} x^{2}}{3 c d^{3} \sqrt{c + d x} + 3 d^{4} x \sqrt{c + d x}} & \text{for}\: d \neq 0 \\\frac{a^{2} x + a b x^{2} + \frac{b^{2} x^{3}}{3}}{c^{\frac{5}{2}}} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Piecewise((-2*a**2*d**2/(3*c*d**3*sqrt(c + d*x) + 3*d**4*x*sqrt(c + d*x)) - 8*a*
b*c*d/(3*c*d**3*sqrt(c + d*x) + 3*d**4*x*sqrt(c + d*x)) - 12*a*b*d**2*x/(3*c*d**
3*sqrt(c + d*x) + 3*d**4*x*sqrt(c + d*x)) + 16*b**2*c**2/(3*c*d**3*sqrt(c + d*x)
+ 3*d**4*x*sqrt(c + d*x)) + 24*b**2*c*d*x/(3*c*d**3*sqrt(c + d*x) + 3*d**4*x*sq
rt(c + d*x)) + 6*b**2*d**2*x**2/(3*c*d**3*sqrt(c + d*x) + 3*d**4*x*sqrt(c + d*x)
), Ne(d, 0)), ((a**2*x + a*b*x**2 + b**2*x**3/3)/c**(5/2), True))

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GIAC/XCAS [A]  time = 0.220985, size = 97, normalized size = 1.45 \[ \frac{2 \, \sqrt{d x + c} b^{2}}{d^{3}} + \frac{2 \,{\left (6 \,{\left (d x + c\right )} b^{2} c - b^{2} c^{2} - 6 \,{\left (d x + c\right )} a b d + 2 \, a b c d - a^{2} d^{2}\right )}}{3 \,{\left (d x + c\right )}^{\frac{3}{2}} d^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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