Optimal. Leaf size=105 \[ -\frac {a^2 d^4 x (d x)^{m-4}}{c^2 (4-m) \sqrt {c x^2}}-\frac {2 a b d^3 x (d x)^{m-3}}{c^2 (3-m) \sqrt {c x^2}}-\frac {b^2 d^2 x (d x)^{m-2}}{c^2 (2-m) \sqrt {c x^2}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.05, antiderivative size = 105, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {15, 16, 43} \[ -\frac {a^2 d^4 x (d x)^{m-4}}{c^2 (4-m) \sqrt {c x^2}}-\frac {2 a b d^3 x (d x)^{m-3}}{c^2 (3-m) \sqrt {c x^2}}-\frac {b^2 d^2 x (d x)^{m-2}}{c^2 (2-m) \sqrt {c x^2}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 15
Rule 16
Rule 43
Rubi steps
\begin {align*} \int \frac {(d x)^m (a+b x)^2}{\left (c x^2\right )^{5/2}} \, dx &=\frac {x \int \frac {(d x)^m (a+b x)^2}{x^5} \, dx}{c^2 \sqrt {c x^2}}\\ &=\frac {\left (d^5 x\right ) \int (d x)^{-5+m} (a+b x)^2 \, dx}{c^2 \sqrt {c x^2}}\\ &=\frac {\left (d^5 x\right ) \int \left (a^2 (d x)^{-5+m}+\frac {2 a b (d x)^{-4+m}}{d}+\frac {b^2 (d x)^{-3+m}}{d^2}\right ) \, dx}{c^2 \sqrt {c x^2}}\\ &=-\frac {a^2 d^4 x (d x)^{-4+m}}{c^2 (4-m) \sqrt {c x^2}}-\frac {2 a b d^3 x (d x)^{-3+m}}{c^2 (3-m) \sqrt {c x^2}}-\frac {b^2 d^2 x (d x)^{-2+m}}{c^2 (2-m) \sqrt {c x^2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.05, size = 72, normalized size = 0.69 \[ \frac {x (d x)^m \left (a^2 \left (m^2-5 m+6\right )+2 a b \left (m^2-6 m+8\right ) x+b^2 \left (m^2-7 m+12\right ) x^2\right )}{(m-4) (m-3) (m-2) \left (c x^2\right )^{5/2}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.48, size = 106, normalized size = 1.01 \[ \frac {{\left (a^{2} m^{2} - 5 \, a^{2} m + {\left (b^{2} m^{2} - 7 \, b^{2} m + 12 \, b^{2}\right )} x^{2} + 6 \, a^{2} + 2 \, {\left (a b m^{2} - 6 \, a b m + 8 \, a b\right )} x\right )} \sqrt {c x^{2}} \left (d x\right )^{m}}{{\left (c^{3} m^{3} - 9 \, c^{3} m^{2} + 26 \, c^{3} m - 24 \, c^{3}\right )} x^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (b x + a\right )}^{2} \left (d x\right )^{m}}{\left (c x^{2}\right )^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.01, size = 95, normalized size = 0.90 \[ \frac {\left (b^{2} m^{2} x^{2}+2 a b \,m^{2} x -7 b^{2} m \,x^{2}+a^{2} m^{2}-12 a b m x +12 b^{2} x^{2}-5 a^{2} m +16 a b x +6 a^{2}\right ) x \left (d x \right )^{m}}{\left (m -2\right ) \left (m -3\right ) \left (m -4\right ) \left (c \,x^{2}\right )^{\frac {5}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 1.57, size = 64, normalized size = 0.61 \[ \frac {b^{2} d^{m} x^{m}}{c^{\frac {5}{2}} {\left (m - 2\right )} x^{2}} + \frac {2 \, a b d^{m} x^{m}}{c^{\frac {5}{2}} {\left (m - 3\right )} x^{3}} + \frac {a^{2} d^{m} x^{m}}{c^{\frac {5}{2}} {\left (m - 4\right )} x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.34, size = 82, normalized size = 0.78 \[ \frac {a^2\,{\left (d\,x\right )}^m}{c^2\,x^3\,\sqrt {c\,x^2}\,\left (m-4\right )}+\frac {b^2\,{\left (d\,x\right )}^m}{c^2\,x\,\sqrt {c\,x^2}\,\left (m-2\right )}+\frac {2\,a\,b\,{\left (d\,x\right )}^m}{c^2\,x^2\,\sqrt {c\,x^2}\,\left (m-3\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________