Optimal. Leaf size=154 \[ \frac {(d+e x)^{1+m} \left (a+c x^2\right )^{3/2} F_1\left (1+m;-\frac {3}{2},-\frac {3}{2};2+m;\frac {d+e x}{d-\frac {\sqrt {-a} e}{\sqrt {c}}},\frac {d+e x}{d+\frac {\sqrt {-a} e}{\sqrt {c}}}\right )}{e (1+m) \left (1-\frac {d+e x}{d-\frac {\sqrt {-a} e}{\sqrt {c}}}\right )^{3/2} \left (1-\frac {d+e x}{d+\frac {\sqrt {-a} e}{\sqrt {c}}}\right )^{3/2}} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.10, antiderivative size = 154, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {774, 138}
\begin {gather*} \frac {\left (a+c x^2\right )^{3/2} (d+e x)^{m+1} F_1\left (m+1;-\frac {3}{2},-\frac {3}{2};m+2;\frac {d+e x}{d-\frac {\sqrt {-a} e}{\sqrt {c}}},\frac {d+e x}{d+\frac {\sqrt {-a} e}{\sqrt {c}}}\right )}{e (m+1) \left (1-\frac {d+e x}{d-\frac {\sqrt {-a} e}{\sqrt {c}}}\right )^{3/2} \left (1-\frac {d+e x}{\frac {\sqrt {-a} e}{\sqrt {c}}+d}\right )^{3/2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 138
Rule 774
Rubi steps
\begin {align*} \int (d+e x)^m \left (a+c x^2\right )^{3/2} \, dx &=\frac {\left (a+c x^2\right )^{3/2} \text {Subst}\left (\int x^m \left (1-\frac {x}{d-\frac {\sqrt {-a} e}{\sqrt {c}}}\right )^{3/2} \left (1-\frac {x}{d+\frac {\sqrt {-a} e}{\sqrt {c}}}\right )^{3/2} \, dx,x,d+e x\right )}{e \left (1-\frac {d+e x}{d-\frac {\sqrt {-a} e}{\sqrt {c}}}\right )^{3/2} \left (1-\frac {d+e x}{d+\frac {\sqrt {-a} e}{\sqrt {c}}}\right )^{3/2}}\\ &=\frac {(d+e x)^{1+m} \left (a+c x^2\right )^{3/2} F_1\left (1+m;-\frac {3}{2},-\frac {3}{2};2+m;\frac {d+e x}{d-\frac {\sqrt {-a} e}{\sqrt {c}}},\frac {d+e x}{d+\frac {\sqrt {-a} e}{\sqrt {c}}}\right )}{e (1+m) \left (1-\frac {d+e x}{d-\frac {\sqrt {-a} e}{\sqrt {c}}}\right )^{3/2} \left (1-\frac {d+e x}{d+\frac {\sqrt {-a} e}{\sqrt {c}}}\right )^{3/2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [F]
time = 0.27, size = 0, normalized size = 0.00 \begin {gather*} \int (d+e x)^m \left (a+c x^2\right )^{3/2} \, dx \end {gather*}
Verification is not applicable to the result.
[In]
[Out]
________________________________________________________________________________________
Maple [F]
time = 0.11, size = 0, normalized size = 0.00 \[\int \left (e x +d \right )^{m} \left (c \,x^{2}+a \right )^{\frac {3}{2}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \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 \left (a + c x^{2}\right )^{\frac {3}{2}} \left (d + e x\right )^{m}\, dx \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*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int {\left (c\,x^2+a\right )}^{3/2}\,{\left (d+e\,x\right )}^m \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________