Optimal. Leaf size=87 \[ \frac {16 d (c d-b e) (x (2 c d-b e)+b d)}{3 b^4 \sqrt {b x+c x^2}}-\frac {2 (d+e x)^2 (x (2 c d-b e)+b d)}{3 b^2 \left (b x+c x^2\right )^{3/2}} \]
________________________________________________________________________________________
Rubi [A] time = 0.04, antiderivative size = 87, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {722, 636} \begin {gather*} \frac {16 d (c d-b e) (x (2 c d-b e)+b d)}{3 b^4 \sqrt {b x+c x^2}}-\frac {2 (d+e x)^2 (x (2 c d-b e)+b d)}{3 b^2 \left (b x+c x^2\right )^{3/2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 636
Rule 722
Rubi steps
\begin {align*} \int \frac {(d+e x)^3}{\left (b x+c x^2\right )^{5/2}} \, dx &=-\frac {2 (d+e x)^2 (b d+(2 c d-b e) x)}{3 b^2 \left (b x+c x^2\right )^{3/2}}-\frac {(8 d (c d-b e)) \int \frac {d+e x}{\left (b x+c x^2\right )^{3/2}} \, dx}{3 b^2}\\ &=-\frac {2 (d+e x)^2 (b d+(2 c d-b e) x)}{3 b^2 \left (b x+c x^2\right )^{3/2}}+\frac {16 d (c d-b e) (b d+(2 c d-b e) x)}{3 b^4 \sqrt {b x+c x^2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.04, size = 105, normalized size = 1.21 \begin {gather*} \frac {2 \left (b^3 \left (-d^3-9 d^2 e x+9 d e^2 x^2+e^3 x^3\right )+6 b^2 c d x \left (d^2-6 d e x+e^2 x^2\right )+24 b c^2 d^2 x^2 (d-e x)+16 c^3 d^3 x^3\right )}{3 b^4 (x (b+c x))^{3/2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [A] time = 0.45, size = 143, normalized size = 1.64 \begin {gather*} \frac {2 \sqrt {b x+c x^2} \left (b^3 \left (-d^3\right )-9 b^3 d^2 e x+9 b^3 d e^2 x^2+b^3 e^3 x^3+6 b^2 c d^3 x-36 b^2 c d^2 e x^2+6 b^2 c d e^2 x^3+24 b c^2 d^3 x^2-24 b c^2 d^2 e x^3+16 c^3 d^3 x^3\right )}{3 b^4 x^2 (b+c x)^2} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.42, size = 147, normalized size = 1.69 \begin {gather*} -\frac {2 \, {\left (b^{3} d^{3} - {\left (16 \, c^{3} d^{3} - 24 \, b c^{2} d^{2} e + 6 \, b^{2} c d e^{2} + b^{3} e^{3}\right )} x^{3} - 3 \, {\left (8 \, b c^{2} d^{3} - 12 \, b^{2} c d^{2} e + 3 \, b^{3} d e^{2}\right )} x^{2} - 3 \, {\left (2 \, b^{2} c d^{3} - 3 \, b^{3} d^{2} e\right )} x\right )} \sqrt {c x^{2} + b x}}{3 \, {\left (b^{4} c^{2} x^{4} + 2 \, b^{5} c x^{3} + b^{6} x^{2}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.25, size = 127, normalized size = 1.46 \begin {gather*} -\frac {2 \, {\left (\frac {d^{3}}{b} - {\left (x {\left (\frac {{\left (16 \, c^{3} d^{3} - 24 \, b c^{2} d^{2} e + 6 \, b^{2} c d e^{2} + b^{3} e^{3}\right )} x}{b^{4}} + \frac {3 \, {\left (8 \, b c^{2} d^{3} - 12 \, b^{2} c d^{2} e + 3 \, b^{3} d e^{2}\right )}}{b^{4}}\right )} + \frac {3 \, {\left (2 \, b^{2} c d^{3} - 3 \, b^{3} d^{2} e\right )}}{b^{4}}\right )} x\right )}}{3 \, {\left (c x^{2} + b x\right )}^{\frac {3}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.05, size = 136, normalized size = 1.56 \begin {gather*} -\frac {2 \left (c x +b \right ) \left (-b^{3} e^{3} x^{3}-6 b^{2} c d \,e^{2} x^{3}+24 b \,c^{2} d^{2} e \,x^{3}-16 c^{3} d^{3} x^{3}-9 b^{3} d \,e^{2} x^{2}+36 b^{2} c \,d^{2} e \,x^{2}-24 b \,c^{2} d^{3} x^{2}+9 b^{3} d^{2} e x -6 b^{2} c \,d^{3} x +d^{3} b^{3}\right ) x}{3 \left (c \,x^{2}+b x \right )^{\frac {5}{2}} b^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [B] time = 1.39, size = 297, normalized size = 3.41 \begin {gather*} -\frac {e^{3} x^{2}}{{\left (c x^{2} + b x\right )}^{\frac {3}{2}} c} - \frac {4 \, c d^{3} x}{3 \, {\left (c x^{2} + b x\right )}^{\frac {3}{2}} b^{2}} + \frac {32 \, c^{2} d^{3} x}{3 \, \sqrt {c x^{2} + b x} b^{4}} + \frac {2 \, d^{2} e x}{{\left (c x^{2} + b x\right )}^{\frac {3}{2}} b} - \frac {16 \, c d^{2} e x}{\sqrt {c x^{2} + b x} b^{3}} + \frac {4 \, d e^{2} x}{\sqrt {c x^{2} + b x} b^{2}} - \frac {2 \, d e^{2} x}{{\left (c x^{2} + b x\right )}^{\frac {3}{2}} c} - \frac {b e^{3} x}{3 \, {\left (c x^{2} + b x\right )}^{\frac {3}{2}} c^{2}} + \frac {2 \, e^{3} x}{3 \, \sqrt {c x^{2} + b x} b c} - \frac {2 \, d^{3}}{3 \, {\left (c x^{2} + b x\right )}^{\frac {3}{2}} b} + \frac {16 \, c d^{3}}{3 \, \sqrt {c x^{2} + b x} b^{3}} - \frac {8 \, d^{2} e}{\sqrt {c x^{2} + b x} b^{2}} + \frac {2 \, d e^{2}}{\sqrt {c x^{2} + b x} b c} + \frac {e^{3}}{3 \, \sqrt {c x^{2} + b x} c^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.48, size = 129, normalized size = 1.48 \begin {gather*} \frac {2\,\left (-b^3\,d^3-9\,b^3\,d^2\,e\,x+9\,b^3\,d\,e^2\,x^2+b^3\,e^3\,x^3+6\,b^2\,c\,d^3\,x-36\,b^2\,c\,d^2\,e\,x^2+6\,b^2\,c\,d\,e^2\,x^3+24\,b\,c^2\,d^3\,x^2-24\,b\,c^2\,d^2\,e\,x^3+16\,c^3\,d^3\,x^3\right )}{3\,b^4\,{\left (c\,x^2+b\,x\right )}^{3/2}} \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 (d + e x\right )^{3}}{\left (x \left (b + c x\right )\right )^{\frac {5}{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________