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 = 0.02, antiderivative size = 69, normalized size of antiderivative = 1.00, 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 = 0.04, 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]
________________________________________________________________________________________
fricas [A] time = 0.46, size = 64, normalized size = 0.93 \[ \frac {2 \, {\left (3 \, b^{2} d^{2} x^{2} + 8 \, b^{2} c^{2} - 20 \, a b c d + 15 \, a^{2} d^{2} - 2 \, {\left (2 \, b^{2} c d - 5 \, a b d^{2}\right )} x\right )} \sqrt {d x + c}}{15 \, d^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 1.10, size = 82, normalized size = 1.19 \[ \frac {2 \, {\left (15 \, \sqrt {d x + c} a^{2} + \frac {10 \, {\left ({\left (d x + c\right )}^{\frac {3}{2}} - 3 \, \sqrt {d x + c} c\right )} a b}{d} + \frac {{\left (3 \, {\left (d x + c\right )}^{\frac {5}{2}} - 10 \, {\left (d x + c\right )}^{\frac {3}{2}} c + 15 \, \sqrt {d x + c} c^{2}\right )} b^{2}}{d^{2}}\right )}}{15 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.00, size = 63, normalized size = 0.91 \[ \frac {2 \sqrt {d x +c}\, \left (3 b^{2} x^{2} d^{2}+10 a b \,d^{2} x -4 b^{2} c d x +15 a^{2} d^{2}-20 a b c d +8 b^{2} c^{2}\right )}{15 d^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 1.37, size = 82, normalized size = 1.19 \[ \frac {2 \, {\left (15 \, \sqrt {d x + c} a^{2} + \frac {10 \, {\left ({\left (d x + c\right )}^{\frac {3}{2}} - 3 \, \sqrt {d x + c} c\right )} a b}{d} + \frac {{\left (3 \, {\left (d x + c\right )}^{\frac {5}{2}} - 10 \, {\left (d x + c\right )}^{\frac {3}{2}} c + 15 \, \sqrt {d x + c} c^{2}\right )} b^{2}}{d^{2}}\right )}}{15 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.07, size = 68, normalized size = 0.99 \[ \frac {2\,\sqrt {c+d\,x}\,\left (3\,b^2\,{\left (c+d\,x\right )}^2+15\,a^2\,d^2+15\,b^2\,c^2-10\,b^2\,c\,\left (c+d\,x\right )+10\,a\,b\,d\,\left (c+d\,x\right )-30\,a\,b\,c\,d\right )}{15\,d^3} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 20.94, size = 231, normalized size = 3.35 \[ \begin {cases} \frac {- \frac {2 a^{2} c}{\sqrt {c + d x}} - 2 a^{2} \left (- \frac {c}{\sqrt {c + d x}} - \sqrt {c + d x}\right ) - \frac {4 a b c \left (- \frac {c}{\sqrt {c + d x}} - \sqrt {c + d x}\right )}{d} - \frac {4 a b \left (\frac {c^{2}}{\sqrt {c + d x}} + 2 c \sqrt {c + d x} - \frac {\left (c + d x\right )^{\frac {3}{2}}}{3}\right )}{d} - \frac {2 b^{2} c \left (\frac {c^{2}}{\sqrt {c + d x}} + 2 c \sqrt {c + d x} - \frac {\left (c + d x\right )^{\frac {3}{2}}}{3}\right )}{d^{2}} - \frac {2 b^{2} \left (- \frac {c^{3}}{\sqrt {c + d x}} - 3 c^{2} \sqrt {c + d x} + c \left (c + d x\right )^{\frac {3}{2}} - \frac {\left (c + d x\right )^{\frac {5}{2}}}{5}\right )}{d^{2}}}{d} & \text {for}\: d \neq 0 \\\frac {\begin {cases} a^{2} x & \text {for}\: b = 0 \\\frac {\left (a + b x\right )^{3}}{3 b} & \text {otherwise} \end {cases}}{\sqrt {c}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________