Optimal. Leaf size=66 \[ -\frac {a^2}{7 c^2 x^6 \sqrt {c x^2}}-\frac {a b}{3 c^2 x^5 \sqrt {c x^2}}-\frac {b^2}{5 c^2 x^4 \sqrt {c x^2}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.01, antiderivative size = 66, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {15, 43} \[ -\frac {a^2}{7 c^2 x^6 \sqrt {c x^2}}-\frac {a b}{3 c^2 x^5 \sqrt {c x^2}}-\frac {b^2}{5 c^2 x^4 \sqrt {c x^2}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 15
Rule 43
Rubi steps
\begin {align*} \int \frac {(a+b x)^2}{x^3 \left (c x^2\right )^{5/2}} \, dx &=\frac {x \int \frac {(a+b x)^2}{x^8} \, dx}{c^2 \sqrt {c x^2}}\\ &=\frac {x \int \left (\frac {a^2}{x^8}+\frac {2 a b}{x^7}+\frac {b^2}{x^6}\right ) \, dx}{c^2 \sqrt {c x^2}}\\ &=-\frac {a^2}{7 c^2 x^6 \sqrt {c x^2}}-\frac {a b}{3 c^2 x^5 \sqrt {c x^2}}-\frac {b^2}{5 c^2 x^4 \sqrt {c x^2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.01, size = 33, normalized size = 0.50 \[ \frac {c \left (-15 a^2-35 a b x-21 b^2 x^2\right )}{105 \left (c x^2\right )^{7/2}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.43, size = 34, normalized size = 0.52 \[ -\frac {{\left (21 \, b^{2} x^{2} + 35 \, a b x + 15 \, a^{2}\right )} \sqrt {c x^{2}}}{105 \, c^{3} x^{8}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (b x + a\right )}^{2}}{\left (c x^{2}\right )^{\frac {5}{2}} x^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.00, size = 32, normalized size = 0.48 \[ -\frac {21 b^{2} x^{2}+35 a b x +15 a^{2}}{105 \left (c \,x^{2}\right )^{\frac {5}{2}} x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 1.37, size = 33, normalized size = 0.50 \[ -\frac {b^{2}}{5 \, c^{\frac {5}{2}} x^{5}} - \frac {a b}{3 \, c^{\frac {5}{2}} x^{6}} - \frac {a^{2}}{7 \, c^{\frac {5}{2}} x^{7}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.18, size = 42, normalized size = 0.64 \[ -\frac {15\,a^2\,\sqrt {x^2}+21\,b^2\,x^2\,\sqrt {x^2}+35\,a\,b\,x\,\sqrt {x^2}}{105\,c^{5/2}\,x^8} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 1.70, size = 56, normalized size = 0.85 \[ - \frac {a^{2}}{7 c^{\frac {5}{2}} x^{2} \left (x^{2}\right )^{\frac {5}{2}}} - \frac {a b}{3 c^{\frac {5}{2}} x \left (x^{2}\right )^{\frac {5}{2}}} - \frac {b^{2}}{5 c^{\frac {5}{2}} \left (x^{2}\right )^{\frac {5}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________