3.1416 \(\int \frac{(a+b x)^2}{\sqrt{c+d x}} \, dx\)

Optimal. Leaf size=69 \[ -\frac{4 b (c+d x)^{3/2} (b c-a d)}{3 d^3}+\frac{2 \sqrt{c+d x} (b c-a d)^2}{d^3}+\frac{2 b^2 (c+d x)^{5/2}}{5 d^3} \]

[Out]

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

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Rubi [A]  time = 0.021249, antiderivative size = 69, 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} \[ -\frac{4 b (c+d x)^{3/2} (b c-a d)}{3 d^3}+\frac{2 \sqrt{c+d x} (b c-a d)^2}{d^3}+\frac{2 b^2 (c+d x)^{5/2}}{5 d^3} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x)^2/Sqrt[c + d*x],x]

[Out]

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

Mathematica [A]  time = 0.0352896, size = 60, normalized size = 0.87 \[ \frac{2 \sqrt{c+d x} \left (15 a^2 d^2+10 a b d (d x-2 c)+b^2 \left (8 c^2-4 c d x+3 d^2 x^2\right )\right )}{15 d^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*Sqrt[c + d*x]*(15*a^2*d^2 + 10*a*b*d*(-2*c + d*x) + b^2*(8*c^2 - 4*c*d*x + 3*d^2*x^2)))/(15*d^3)

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Maple [A]  time = 0.004, size = 63, normalized size = 0.9 \begin{align*}{\frac{6\,{b}^{2}{x}^{2}{d}^{2}+20\,ab{d}^{2}x-8\,{b}^{2}cdx+30\,{a}^{2}{d}^{2}-40\,abcd+16\,{b}^{2}{c}^{2}}{15\,{d}^{3}}\sqrt{dx+c}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)^2/(d*x+c)^(1/2),x)

[Out]

2/15*(d*x+c)^(1/2)*(3*b^2*d^2*x^2+10*a*b*d^2*x-4*b^2*c*d*x+15*a^2*d^2-20*a*b*c*d+8*b^2*c^2)/d^3

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Maxima [A]  time = 0.964189, size = 111, normalized size = 1.61 \begin{align*} \frac{2 \,{\left (15 \, \sqrt{d x + c} a^{2} + \frac{10 \,{\left ({\left (d x + c\right )}^{\frac{3}{2}} - 3 \, \sqrt{d x + c} c\right )} a b}{d} + \frac{{\left (3 \,{\left (d x + c\right )}^{\frac{5}{2}} - 10 \,{\left (d x + c\right )}^{\frac{3}{2}} c + 15 \, \sqrt{d x + c} c^{2}\right )} b^{2}}{d^{2}}\right )}}{15 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^2/(d*x+c)^(1/2),x, algorithm="maxima")

[Out]

2/15*(15*sqrt(d*x + c)*a^2 + 10*((d*x + c)^(3/2) - 3*sqrt(d*x + c)*c)*a*b/d + (3*(d*x + c)^(5/2) - 10*(d*x + c
)^(3/2)*c + 15*sqrt(d*x + c)*c^2)*b^2/d^2)/d

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Fricas [A]  time = 1.76244, size = 146, normalized size = 2.12 \begin{align*} \frac{2 \,{\left (3 \, b^{2} d^{2} x^{2} + 8 \, b^{2} c^{2} - 20 \, a b c d + 15 \, a^{2} d^{2} - 2 \,{\left (2 \, b^{2} c d - 5 \, a b d^{2}\right )} x\right )} \sqrt{d x + c}}{15 \, d^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^2/(d*x+c)^(1/2),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 14.4817, size = 231, normalized size = 3.35 \begin{align*} \begin{cases} - \frac{\frac{2 a^{2} c}{\sqrt{c + d x}} + 2 a^{2} \left (- \frac{c}{\sqrt{c + d x}} - \sqrt{c + d x}\right ) + \frac{4 a b c \left (- \frac{c}{\sqrt{c + d x}} - \sqrt{c + d x}\right )}{d} + \frac{4 a b \left (\frac{c^{2}}{\sqrt{c + d x}} + 2 c \sqrt{c + d x} - \frac{\left (c + d x\right )^{\frac{3}{2}}}{3}\right )}{d} + \frac{2 b^{2} c \left (\frac{c^{2}}{\sqrt{c + d x}} + 2 c \sqrt{c + d x} - \frac{\left (c + d x\right )^{\frac{3}{2}}}{3}\right )}{d^{2}} + \frac{2 b^{2} \left (- \frac{c^{3}}{\sqrt{c + d x}} - 3 c^{2} \sqrt{c + d x} + c \left (c + d x\right )^{\frac{3}{2}} - \frac{\left (c + d x\right )^{\frac{5}{2}}}{5}\right )}{d^{2}}}{d} & \text{for}\: d \neq 0 \\\frac{\begin{cases} a^{2} x & \text{for}\: b = 0 \\\frac{\left (a + b x\right )^{3}}{3 b} & \text{otherwise} \end{cases}}{\sqrt{c}} & \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)**(1/2),x)

[Out]

Piecewise((-(2*a**2*c/sqrt(c + d*x) + 2*a**2*(-c/sqrt(c + d*x) - sqrt(c + d*x)) + 4*a*b*c*(-c/sqrt(c + d*x) -
sqrt(c + d*x))/d + 4*a*b*(c**2/sqrt(c + d*x) + 2*c*sqrt(c + d*x) - (c + d*x)**(3/2)/3)/d + 2*b**2*c*(c**2/sqrt
(c + d*x) + 2*c*sqrt(c + d*x) - (c + d*x)**(3/2)/3)/d**2 + 2*b**2*(-c**3/sqrt(c + d*x) - 3*c**2*sqrt(c + d*x)
+ c*(c + d*x)**(3/2) - (c + d*x)**(5/2)/5)/d**2)/d, Ne(d, 0)), (Piecewise((a**2*x, Eq(b, 0)), ((a + b*x)**3/(3
*b), True))/sqrt(c), True))

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Giac [A]  time = 1.10357, size = 111, normalized size = 1.61 \begin{align*} \frac{2 \,{\left (15 \, \sqrt{d x + c} a^{2} + \frac{10 \,{\left ({\left (d x + c\right )}^{\frac{3}{2}} - 3 \, \sqrt{d x + c} c\right )} a b}{d} + \frac{{\left (3 \,{\left (d x + c\right )}^{\frac{5}{2}} - 10 \,{\left (d x + c\right )}^{\frac{3}{2}} c + 15 \, \sqrt{d x + c} c^{2}\right )} b^{2}}{d^{2}}\right )}}{15 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^2/(d*x+c)^(1/2),x, algorithm="giac")

[Out]

2/15*(15*sqrt(d*x + c)*a^2 + 10*((d*x + c)^(3/2) - 3*sqrt(d*x + c)*c)*a*b/d + (3*(d*x + c)^(5/2) - 10*(d*x + c
)^(3/2)*c + 15*sqrt(d*x + c)*c^2)*b^2/d^2)/d