Optimal. Leaf size=61 \[ \frac{a c \sqrt{c x^2} (d x)^{m+4}}{d^4 (m+4) x}+\frac{b c \sqrt{c x^2} (d x)^{m+5}}{d^5 (m+5) x} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0294383, antiderivative size = 61, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.15, Rules used = {15, 16, 43} \[ \frac{a c \sqrt{c x^2} (d x)^{m+4}}{d^4 (m+4) x}+\frac{b c \sqrt{c x^2} (d x)^{m+5}}{d^5 (m+5) x} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 15
Rule 16
Rule 43
Rubi steps
\begin{align*} \int (d x)^m \left (c x^2\right )^{3/2} (a+b x) \, dx &=\frac{\left (c \sqrt{c x^2}\right ) \int x^3 (d x)^m (a+b x) \, dx}{x}\\ &=\frac{\left (c \sqrt{c x^2}\right ) \int (d x)^{3+m} (a+b x) \, dx}{d^3 x}\\ &=\frac{\left (c \sqrt{c x^2}\right ) \int \left (a (d x)^{3+m}+\frac{b (d x)^{4+m}}{d}\right ) \, dx}{d^3 x}\\ &=\frac{a c (d x)^{4+m} \sqrt{c x^2}}{d^4 (4+m) x}+\frac{b c (d x)^{5+m} \sqrt{c x^2}}{d^5 (5+m) x}\\ \end{align*}
Mathematica [A] time = 0.0304621, size = 38, normalized size = 0.62 \[ \frac{x \left (c x^2\right )^{3/2} (d x)^m (a (m+5)+b (m+4) x)}{(m+4) (m+5)} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.002, size = 40, normalized size = 0.7 \begin{align*}{\frac{ \left ( bmx+am+4\,bx+5\,a \right ) x \left ( dx \right ) ^{m}}{ \left ( 5+m \right ) \left ( 4+m \right ) } \left ( c{x}^{2} \right ) ^{{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.05781, size = 53, normalized size = 0.87 \begin{align*} \frac{b c^{\frac{3}{2}} d^{m} x^{5} x^{m}}{m + 5} + \frac{a c^{\frac{3}{2}} d^{m} x^{4} x^{m}}{m + 4} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.37498, size = 111, normalized size = 1.82 \begin{align*} \frac{{\left ({\left (b c m + 4 \, b c\right )} x^{4} +{\left (a c m + 5 \, a c\right )} x^{3}\right )} \sqrt{c x^{2}} \left (d x\right )^{m}}{m^{2} + 9 \, m + 20} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]