Optimal. Leaf size=23 \[ \frac {a b^3 x}{d^3}+\frac {b^4 x^2}{2 d^3} \]
________________________________________________________________________________________
Rubi [A] time = 0.01, antiderivative size = 23, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {21} \begin {gather*} \frac {a b^3 x}{d^3}+\frac {b^4 x^2}{2 d^3} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 21
Rubi steps
\begin {align*} \int \frac {(a+b x)^4}{\left (\frac {a d}{b}+d x\right )^3} \, dx &=\frac {b^3 \int (a+b x) \, dx}{d^3}\\ &=\frac {a b^3 x}{d^3}+\frac {b^4 x^2}{2 d^3}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.00, size = 19, normalized size = 0.83 \begin {gather*} \frac {b^3 \left (a x+\frac {b x^2}{2}\right )}{d^3} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {(a+b x)^4}{\left (\frac {a d}{b}+d x\right )^3} \, dx \end {gather*}
Verification is not applicable to the result.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 1.25, size = 20, normalized size = 0.87 \begin {gather*} \frac {b^{4} x^{2} + 2 \, a b^{3} x}{2 \, d^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 1.19, size = 20, normalized size = 0.87 \begin {gather*} \frac {b^{4} x^{2} + 2 \, a b^{3} x}{2 \, d^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.00, size = 18, normalized size = 0.78 \begin {gather*} \frac {\left (\frac {1}{2} b \,x^{2}+a x \right ) b^{3}}{d^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 1.24, size = 20, normalized size = 0.87 \begin {gather*} \frac {b^{4} x^{2} + 2 \, a b^{3} x}{2 \, d^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.03, size = 16, normalized size = 0.70 \begin {gather*} \frac {b^3\,x\,\left (2\,a+b\,x\right )}{2\,d^3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 0.12, size = 20, normalized size = 0.87 \begin {gather*} \frac {a b^{3} x}{d^{3}} + \frac {b^{4} x^{2}}{2 d^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________