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 = 0.0211512, 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, Rules used = {43} \[ \frac{4 b (b c-a d)}{d^3 \sqrt{c+d x}}-\frac{2 (b c-a d)^2}{3 d^3 (c+d x)^{3/2}}+\frac{2 b^2 \sqrt{c+d x}}{d^3} \]

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

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

\begin{align*} \int \frac{(a+b x)^2}{(c+d x)^{5/2}} \, dx &=\int \left (\frac{(-b c+a d)^2}{d^2 (c+d x)^{5/2}}-\frac{2 b (b c-a d)}{d^2 (c+d x)^{3/2}}+\frac{b^2}{d^2 \sqrt{c+d x}}\right ) \, dx\\ &=-\frac{2 (b c-a d)^2}{3 d^3 (c+d x)^{3/2}}+\frac{4 b (b c-a d)}{d^3 \sqrt{c+d x}}+\frac{2 b^2 \sqrt{c+d x}}{d^3}\\ \end{align*}

Mathematica [A]  time = 0.0321098, size = 62, normalized size = 0.93 \[ \frac{-2 a^2 d^2-4 a b d (2 c+3 d x)+2 b^2 \left (8 c^2+12 c d x+3 d^2 x^2\right )}{3 d^3 (c+d x)^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.004, size = 62, normalized size = 0.9 \begin{align*} -{\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}}}} \end{align*}

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 = 0.956515, size = 97, normalized size = 1.45 \begin{align*} \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} \end{align*}

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 = 2.084, size = 174, normalized size = 2.6 \begin{align*} \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 )} \sqrt{d x + c}}{3 \,{\left (d^{5} x^{2} + 2 \, c d^{4} x + c^{2} d^{3}\right )}} \end{align*}

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)*sqrt(d*x + c)/(d^5*x^2 + 2*c
*d^4*x + c^2*d^3)

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Sympy [A]  time = 1.20079, size = 265, normalized size = 3.96 \begin{align*} \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} \end{align*}

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*sqrt(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 [A]  time = 1.08627, size = 97, normalized size = 1.45 \begin{align*} \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}} \end{align*}

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)