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]
________________________________________________________________________________________
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]
[Out]
Rule 43
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]
[Out]
________________________________________________________________________________________
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]
[Out]
________________________________________________________________________________________
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]
[Out]
________________________________________________________________________________________
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]
[Out]
________________________________________________________________________________________
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]
[Out]
________________________________________________________________________________________
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]
[Out]