3.1427 \(\int \frac{(a+b x)^2}{(c+d x)^{3/2}} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.0206038, 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} \[ -\frac{4 b \sqrt{c+d x} (b c-a d)}{d^3}-\frac{2 (b c-a d)^2}{d^3 \sqrt{c+d x}}+\frac{2 b^2 (c+d x)^{3/2}}{3 d^3} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x)^2/(c + d*x)^(3/2),x]

[Out]

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

Mathematica [A]  time = 0.0340529, size = 59, normalized size = 0.88 \[ \frac{2 \left (-3 a^2 d^2+6 a b d (2 c+d x)+b^2 \left (-8 c^2-4 c d x+d^2 x^2\right )\right )}{3 d^3 \sqrt{c+d x}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.005, size = 63, normalized size = 0.9 \begin{align*} -{\frac{-2\,{b}^{2}{x}^{2}{d}^{2}-12\,ab{d}^{2}x+8\,{b}^{2}cdx+6\,{a}^{2}{d}^{2}-24\,abcd+16\,{b}^{2}{c}^{2}}{3\,{d}^{3}}{\frac{1}{\sqrt{dx+c}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/3/(d*x+c)^(1/2)*(-b^2*d^2*x^2-6*a*b*d^2*x+4*b^2*c*d*x+3*a^2*d^2-12*a*b*c*d+8*b^2*c^2)/d^3

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Maxima [A]  time = 0.946276, size = 101, normalized size = 1.51 \begin{align*} \frac{2 \,{\left (\frac{{\left (d x + c\right )}^{\frac{3}{2}} b^{2} - 6 \,{\left (b^{2} c - a b d\right )} \sqrt{d x + c}}{d^{2}} - \frac{3 \,{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )}}{\sqrt{d x + c} 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)^(3/2),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 2.02985, size = 157, normalized size = 2.34 \begin{align*} \frac{2 \,{\left (b^{2} d^{2} x^{2} - 8 \, b^{2} c^{2} + 12 \, a b c d - 3 \, a^{2} d^{2} - 2 \,{\left (2 \, b^{2} c d - 3 \, a b d^{2}\right )} x\right )} \sqrt{d x + c}}{3 \,{\left (d^{4} x + c d^{3}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 7.99504, size = 65, normalized size = 0.97 \begin{align*} \frac{2 b^{2} \left (c + d x\right )^{\frac{3}{2}}}{3 d^{3}} + \frac{\sqrt{c + d x} \left (4 a b d - 4 b^{2} c\right )}{d^{3}} - \frac{2 \left (a d - b c\right )^{2}}{d^{3} \sqrt{c + d x}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.05313, size = 113, normalized size = 1.69 \begin{align*} -\frac{2 \,{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )}}{\sqrt{d x + c} d^{3}} + \frac{2 \,{\left ({\left (d x + c\right )}^{\frac{3}{2}} b^{2} d^{6} - 6 \, \sqrt{d x + c} b^{2} c d^{6} + 6 \, \sqrt{d x + c} a b d^{7}\right )}}{3 \, d^{9}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*(b^2*c^2 - 2*a*b*c*d + a^2*d^2)/(sqrt(d*x + c)*d^3) + 2/3*((d*x + c)^(3/2)*b^2*d^6 - 6*sqrt(d*x + c)*b^2*c*
d^6 + 6*sqrt(d*x + c)*a*b*d^7)/d^9