Optimal. Leaf size=52 \[ -\frac {16 c d^3}{3 \sqrt {a+b x+c x^2}}-\frac {2 d^3 (b+2 c x)^2}{3 \left (a+b x+c x^2\right )^{3/2}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.02, antiderivative size = 52, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {686, 629} \[ -\frac {16 c d^3}{3 \sqrt {a+b x+c x^2}}-\frac {2 d^3 (b+2 c x)^2}{3 \left (a+b x+c x^2\right )^{3/2}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 629
Rule 686
Rubi steps
\begin {align*} \int \frac {(b d+2 c d x)^3}{\left (a+b x+c x^2\right )^{5/2}} \, dx &=-\frac {2 d^3 (b+2 c x)^2}{3 \left (a+b x+c x^2\right )^{3/2}}+\frac {1}{3} \left (8 c d^2\right ) \int \frac {b d+2 c d x}{\left (a+b x+c x^2\right )^{3/2}} \, dx\\ &=-\frac {2 d^3 (b+2 c x)^2}{3 \left (a+b x+c x^2\right )^{3/2}}-\frac {16 c d^3}{3 \sqrt {a+b x+c x^2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.03, size = 42, normalized size = 0.81 \[ -\frac {2 d^3 \left (4 c \left (2 a+3 c x^2\right )+b^2+12 b c x\right )}{3 (a+x (b+c x))^{3/2}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 1.54, size = 83, normalized size = 1.60 \[ -\frac {2 \, {\left (12 \, c^{2} d^{3} x^{2} + 12 \, b c d^{3} x + {\left (b^{2} + 8 \, a c\right )} d^{3}\right )} \sqrt {c x^{2} + b x + a}}{3 \, {\left (c^{2} x^{4} + 2 \, b c x^{3} + 2 \, a b x + {\left (b^{2} + 2 \, a c\right )} x^{2} + a^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.22, size = 74, normalized size = 1.42 \[ -\frac {2 \, {\left (b^{2} d^{3} - 4 \, a c d^{3} + 12 \, {\left (c d x^{2} + b d x + a d\right )} c d^{2}\right )} d}{3 \, {\left (c d x^{2} + b d x + a d\right )} \sqrt {\frac {c d x^{2} + b d x + a d}{d}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.05, size = 39, normalized size = 0.75 \[ -\frac {2 \left (12 c^{2} x^{2}+12 b c x +8 a c +b^{2}\right ) d^{3}}{3 \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.70, size = 62, normalized size = 1.19 \[ -\frac {2\,b^2\,d^3+24\,c\,d^3\,\left (c\,x^2+b\,x+a\right )-8\,a\,c\,d^3}{\sqrt {c\,x^2+b\,x+a}\,\left (3\,c\,x^2+3\,b\,x+3\,a\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [B] time = 1.75, size = 264, normalized size = 5.08 \[ - \frac {16 a c d^{3}}{3 a \sqrt {a + b x + c x^{2}} + 3 b x \sqrt {a + b x + c x^{2}} + 3 c x^{2} \sqrt {a + b x + c x^{2}}} - \frac {2 b^{2} d^{3}}{3 a \sqrt {a + b x + c x^{2}} + 3 b x \sqrt {a + b x + c x^{2}} + 3 c x^{2} \sqrt {a + b x + c x^{2}}} - \frac {24 b c d^{3} x}{3 a \sqrt {a + b x + c x^{2}} + 3 b x \sqrt {a + b x + c x^{2}} + 3 c x^{2} \sqrt {a + b x + c x^{2}}} - \frac {24 c^{2} d^{3} x^{2}}{3 a \sqrt {a + b x + c x^{2}} + 3 b x \sqrt {a + b x + c x^{2}} + 3 c x^{2} \sqrt {a + b x + c x^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________