Optimal. Leaf size=43 \[ \frac {(a+b x) (d+e x)^{m+1}}{e (m+1) \sqrt {a^2+2 a b x+b^2 x^2}} \]
________________________________________________________________________________________
Rubi [A] time = 0.03, antiderivative size = 43, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {770, 21, 32} \begin {gather*} \frac {(a+b x) (d+e x)^{m+1}}{e (m+1) \sqrt {a^2+2 a b x+b^2 x^2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 21
Rule 32
Rule 770
Rubi steps
\begin {align*} \int \frac {(a+b x) (d+e x)^m}{\sqrt {a^2+2 a b x+b^2 x^2}} \, dx &=\frac {\left (a b+b^2 x\right ) \int \frac {(a+b x) (d+e x)^m}{a b+b^2 x} \, dx}{\sqrt {a^2+2 a b x+b^2 x^2}}\\ &=\frac {\left (a b+b^2 x\right ) \int (d+e x)^m \, dx}{b \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=\frac {(a+b x) (d+e x)^{1+m}}{e (1+m) \sqrt {a^2+2 a b x+b^2 x^2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.02, size = 34, normalized size = 0.79 \begin {gather*} \frac {(a+b x) (d+e x)^{m+1}}{e (m+1) \sqrt {(a+b x)^2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [F] time = 0.59, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {(a+b x) (d+e x)^m}{\sqrt {a^2+2 a b x+b^2 x^2}} \, dx \end {gather*}
Verification is not applicable to the result.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.44, size = 20, normalized size = 0.47 \begin {gather*} \frac {{\left (e x + d\right )} {\left (e x + d\right )}^{m}}{e m + e} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (b x + a\right )} {\left (e x + d\right )}^{m}}{\sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.04, size = 33, normalized size = 0.77 \begin {gather*} \frac {\left (b x +a \right ) \left (e x +d \right )^{m +1}}{\left (m +1\right ) \sqrt {\left (b x +a \right )^{2}}\, e} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.75, size = 21, normalized size = 0.49 \begin {gather*} \frac {{\left (e x + d\right )} {\left (e x + d\right )}^{m}}{e {\left (m + 1\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {\left (a+b\,x\right )\,{\left (d+e\,x\right )}^m}{\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}} \,d x \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 {\left (a + b x\right ) \left (d + e x\right )^{m}}{\sqrt {\left (a + b x\right )^{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________