Optimal. Leaf size=67 \[ -\frac {2 (b B-A c) x^2}{3 b c \left (b x+c x^2\right )^{3/2}}+\frac {2 (b B+2 A c) x}{3 b^2 c \sqrt {b x+c x^2}} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.03, antiderivative size = 67, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {802, 650}
\begin {gather*} \frac {2 x (2 A c+b B)}{3 b^2 c \sqrt {b x+c x^2}}-\frac {2 x^2 (b B-A c)}{3 b c \left (b x+c x^2\right )^{3/2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 650
Rule 802
Rubi steps
\begin {align*} \int \frac {x^2 (A+B x)}{\left (b x+c x^2\right )^{5/2}} \, dx &=-\frac {2 (b B-A c) x^2}{3 b c \left (b x+c x^2\right )^{3/2}}+\frac {(b B+2 A c) \int \frac {x}{\left (b x+c x^2\right )^{3/2}} \, dx}{3 b c}\\ &=-\frac {2 (b B-A c) x^2}{3 b c \left (b x+c x^2\right )^{3/2}}+\frac {2 (b B+2 A c) x}{3 b^2 c \sqrt {b x+c x^2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.10, size = 35, normalized size = 0.52 \begin {gather*} \frac {2 x^2 (3 A b+b B x+2 A c x)}{3 b^2 (x (b+c x))^{3/2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(217\) vs.
\(2(59)=118\).
time = 0.56, size = 218, normalized size = 3.25
method | result | size |
trager | \(\frac {2 \left (2 A c x +b B x +3 A b \right ) \sqrt {c \,x^{2}+b x}}{3 b^{2} \left (c x +b \right )^{2}}\) | \(38\) |
gosper | \(\frac {2 x^{3} \left (c x +b \right ) \left (2 A c x +b B x +3 A b \right )}{3 b^{2} \left (c \,x^{2}+b x \right )^{\frac {5}{2}}}\) | \(39\) |
default | \(B \left (-\frac {x^{2}}{c \left (c \,x^{2}+b x \right )^{\frac {3}{2}}}+\frac {b \left (-\frac {x}{2 c \left (c \,x^{2}+b x \right )^{\frac {3}{2}}}-\frac {b \left (-\frac {1}{3 c \left (c \,x^{2}+b x \right )^{\frac {3}{2}}}-\frac {b \left (-\frac {2 \left (2 c x +b \right )}{3 b^{2} \left (c \,x^{2}+b x \right )^{\frac {3}{2}}}+\frac {16 c \left (2 c x +b \right )}{3 b^{4} \sqrt {c \,x^{2}+b x}}\right )}{2 c}\right )}{4 c}\right )}{2 c}\right )+A \left (-\frac {x}{2 c \left (c \,x^{2}+b x \right )^{\frac {3}{2}}}-\frac {b \left (-\frac {1}{3 c \left (c \,x^{2}+b x \right )^{\frac {3}{2}}}-\frac {b \left (-\frac {2 \left (2 c x +b \right )}{3 b^{2} \left (c \,x^{2}+b x \right )^{\frac {3}{2}}}+\frac {16 c \left (2 c x +b \right )}{3 b^{4} \sqrt {c \,x^{2}+b x}}\right )}{2 c}\right )}{4 c}\right )\) | \(218\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 134 vs.
\(2 (59) = 118\).
time = 0.27, size = 134, normalized size = 2.00 \begin {gather*} -\frac {B x^{2}}{{\left (c x^{2} + b x\right )}^{\frac {3}{2}} c} + \frac {4 \, A x}{3 \, \sqrt {c x^{2} + b x} b^{2}} - \frac {B b x}{3 \, {\left (c x^{2} + b x\right )}^{\frac {3}{2}} c^{2}} - \frac {2 \, A x}{3 \, {\left (c x^{2} + b x\right )}^{\frac {3}{2}} c} + \frac {2 \, B x}{3 \, \sqrt {c x^{2} + b x} b c} + \frac {B}{3 \, \sqrt {c x^{2} + b x} c^{2}} + \frac {2 \, A}{3 \, \sqrt {c x^{2} + b x} b c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 2.22, size = 51, normalized size = 0.76 \begin {gather*} \frac {2 \, \sqrt {c x^{2} + b x} {\left (3 \, A b + {\left (B b + 2 \, A c\right )} x\right )}}{3 \, {\left (b^{2} c^{2} x^{2} + 2 \, b^{3} c x + b^{4}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{2} \left (A + B x\right )}{\left (x \left (b + c x\right )\right )^{\frac {5}{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 119 vs.
\(2 (59) = 118\).
time = 0.83, size = 119, normalized size = 1.78 \begin {gather*} \frac {2 \, {\left (3 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{2} B c + 3 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )} B b \sqrt {c} + 3 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )} A c^{\frac {3}{2}} + B b^{2} + 2 \, A b c\right )}}{3 \, {\left ({\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )} \sqrt {c} + b\right )}^{3} c^{\frac {3}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 1.16, size = 37, normalized size = 0.55 \begin {gather*} \frac {2\,\sqrt {c\,x^2+b\,x}\,\left (3\,A\,b+2\,A\,c\,x+B\,b\,x\right )}{3\,b^2\,{\left (b+c\,x\right )}^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________