Optimal. Leaf size=171 \[ -\frac {5 c^3 d x}{e^6}+\frac {c^3 x^2}{2 e^5}-\frac {\left (c d^2+a e^2\right )^3}{4 e^7 (d+e x)^4}+\frac {2 c d \left (c d^2+a e^2\right )^2}{e^7 (d+e x)^3}-\frac {3 c \left (c d^2+a e^2\right ) \left (5 c d^2+a e^2\right )}{2 e^7 (d+e x)^2}+\frac {4 c^2 d \left (5 c d^2+3 a e^2\right )}{e^7 (d+e x)}+\frac {3 c^2 \left (5 c d^2+a e^2\right ) \log (d+e x)}{e^7} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.11, antiderivative size = 171, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 1, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.059, Rules used = {711}
\begin {gather*} \frac {4 c^2 d \left (3 a e^2+5 c d^2\right )}{e^7 (d+e x)}+\frac {3 c^2 \left (a e^2+5 c d^2\right ) \log (d+e x)}{e^7}-\frac {3 c \left (a e^2+c d^2\right ) \left (a e^2+5 c d^2\right )}{2 e^7 (d+e x)^2}+\frac {2 c d \left (a e^2+c d^2\right )^2}{e^7 (d+e x)^3}-\frac {\left (a e^2+c d^2\right )^3}{4 e^7 (d+e x)^4}-\frac {5 c^3 d x}{e^6}+\frac {c^3 x^2}{2 e^5} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 711
Rubi steps
\begin {align*} \int \frac {\left (a+c x^2\right )^3}{(d+e x)^5} \, dx &=\int \left (-\frac {5 c^3 d}{e^6}+\frac {c^3 x}{e^5}+\frac {\left (c d^2+a e^2\right )^3}{e^6 (d+e x)^5}-\frac {6 c d \left (c d^2+a e^2\right )^2}{e^6 (d+e x)^4}+\frac {3 c \left (c d^2+a e^2\right ) \left (5 c d^2+a e^2\right )}{e^6 (d+e x)^3}-\frac {4 c^2 d \left (5 c d^2+3 a e^2\right )}{e^6 (d+e x)^2}+\frac {3 c^2 \left (5 c d^2+a e^2\right )}{e^6 (d+e x)}\right ) \, dx\\ &=-\frac {5 c^3 d x}{e^6}+\frac {c^3 x^2}{2 e^5}-\frac {\left (c d^2+a e^2\right )^3}{4 e^7 (d+e x)^4}+\frac {2 c d \left (c d^2+a e^2\right )^2}{e^7 (d+e x)^3}-\frac {3 c \left (c d^2+a e^2\right ) \left (5 c d^2+a e^2\right )}{2 e^7 (d+e x)^2}+\frac {4 c^2 d \left (5 c d^2+3 a e^2\right )}{e^7 (d+e x)}+\frac {3 c^2 \left (5 c d^2+a e^2\right ) \log (d+e x)}{e^7}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.04, size = 185, normalized size = 1.08 \begin {gather*} \frac {-a^3 e^6-a^2 c e^4 \left (d^2+4 d e x+6 e^2 x^2\right )+a c^2 d e^2 \left (25 d^3+88 d^2 e x+108 d e^2 x^2+48 e^3 x^3\right )+c^3 \left (57 d^6+168 d^5 e x+132 d^4 e^2 x^2-32 d^3 e^3 x^3-68 d^2 e^4 x^4-12 d e^5 x^5+2 e^6 x^6\right )+12 c^2 \left (5 c d^2+a e^2\right ) (d+e x)^4 \log (d+e x)}{4 e^7 (d+e x)^4} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.42, size = 205, normalized size = 1.20
method | result | size |
risch | \(\frac {c^{3} x^{2}}{2 e^{5}}-\frac {5 c^{3} d x}{e^{6}}+\frac {\left (12 d \,e^{4} c^{2} a +20 d^{3} e^{2} c^{3}\right ) x^{3}-\frac {3 e c \left (a^{2} e^{4}-18 a c \,d^{2} e^{2}-35 c^{2} d^{4}\right ) x^{2}}{2}+\left (-d \,e^{4} a^{2} c +22 d^{3} e^{2} c^{2} a +47 d^{5} c^{3}\right ) x -\frac {e^{6} a^{3}+e^{4} d^{2} a^{2} c -25 d^{4} e^{2} c^{2} a -57 d^{6} c^{3}}{4 e}}{e^{6} \left (e x +d \right )^{4}}+\frac {3 c^{2} \ln \left (e x +d \right ) a}{e^{5}}+\frac {15 c^{3} \ln \left (e x +d \right ) d^{2}}{e^{7}}\) | \(203\) |
norman | \(\frac {-\frac {e^{6} a^{3}+e^{4} d^{2} a^{2} c -25 d^{4} e^{2} c^{2} a -125 d^{6} c^{3}}{4 e^{7}}+\frac {c^{3} x^{6}}{2 e}-\frac {3 \left (e^{4} a^{2} c -18 d^{2} e^{2} c^{2} a -90 d^{4} c^{3}\right ) x^{2}}{2 e^{5}}-\frac {3 c^{3} d \,x^{5}}{e^{2}}+\frac {4 d \left (3 e^{2} c^{2} a +15 c^{3} d^{2}\right ) x^{3}}{e^{4}}-\frac {d \left (e^{4} a^{2} c -22 d^{2} e^{2} c^{2} a -110 d^{4} c^{3}\right ) x}{e^{6}}}{\left (e x +d \right )^{4}}+\frac {3 c^{2} \left (e^{2} a +5 c \,d^{2}\right ) \ln \left (e x +d \right )}{e^{7}}\) | \(204\) |
default | \(-\frac {c^{3} \left (-\frac {1}{2} e \,x^{2}+5 d x \right )}{e^{6}}+\frac {2 c d \left (a^{2} e^{4}+2 a c \,d^{2} e^{2}+c^{2} d^{4}\right )}{e^{7} \left (e x +d \right )^{3}}+\frac {4 c^{2} d \left (3 e^{2} a +5 c \,d^{2}\right )}{e^{7} \left (e x +d \right )}+\frac {3 c^{2} \left (e^{2} a +5 c \,d^{2}\right ) \ln \left (e x +d \right )}{e^{7}}-\frac {3 c \left (a^{2} e^{4}+6 a c \,d^{2} e^{2}+5 c^{2} d^{4}\right )}{2 e^{7} \left (e x +d \right )^{2}}-\frac {e^{6} a^{3}+3 e^{4} d^{2} a^{2} c +3 d^{4} e^{2} c^{2} a +d^{6} c^{3}}{4 e^{7} \left (e x +d \right )^{4}}\) | \(205\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.30, size = 224, normalized size = 1.31 \begin {gather*} 3 \, {\left (5 \, c^{3} d^{2} + a c^{2} e^{2}\right )} e^{\left (-7\right )} \log \left (x e + d\right ) + \frac {1}{2} \, {\left (c^{3} x^{2} e - 10 \, c^{3} d x\right )} e^{\left (-6\right )} + \frac {57 \, c^{3} d^{6} + 25 \, a c^{2} d^{4} e^{2} - a^{2} c d^{2} e^{4} + 16 \, {\left (5 \, c^{3} d^{3} e^{3} + 3 \, a c^{2} d e^{5}\right )} x^{3} - a^{3} e^{6} + 6 \, {\left (35 \, c^{3} d^{4} e^{2} + 18 \, a c^{2} d^{2} e^{4} - a^{2} c e^{6}\right )} x^{2} + 4 \, {\left (47 \, c^{3} d^{5} e + 22 \, a c^{2} d^{3} e^{3} - a^{2} c d e^{5}\right )} x}{4 \, {\left (x^{4} e^{11} + 4 \, d x^{3} e^{10} + 6 \, d^{2} x^{2} e^{9} + 4 \, d^{3} x e^{8} + d^{4} e^{7}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 332 vs.
\(2 (157) = 314\).
time = 1.85, size = 332, normalized size = 1.94 \begin {gather*} \frac {168 \, c^{3} d^{5} x e + 57 \, c^{3} d^{6} + {\left (2 \, c^{3} x^{6} - 6 \, a^{2} c x^{2} - a^{3}\right )} e^{6} - 4 \, {\left (3 \, c^{3} d x^{5} - 12 \, a c^{2} d x^{3} + a^{2} c d x\right )} e^{5} - {\left (68 \, c^{3} d^{2} x^{4} - 108 \, a c^{2} d^{2} x^{2} + a^{2} c d^{2}\right )} e^{4} - 8 \, {\left (4 \, c^{3} d^{3} x^{3} - 11 \, a c^{2} d^{3} x\right )} e^{3} + {\left (132 \, c^{3} d^{4} x^{2} + 25 \, a c^{2} d^{4}\right )} e^{2} + 12 \, {\left (20 \, c^{3} d^{5} x e + 5 \, c^{3} d^{6} + a c^{2} x^{4} e^{6} + 4 \, a c^{2} d x^{3} e^{5} + {\left (5 \, c^{3} d^{2} x^{4} + 6 \, a c^{2} d^{2} x^{2}\right )} e^{4} + 4 \, {\left (5 \, c^{3} d^{3} x^{3} + a c^{2} d^{3} x\right )} e^{3} + {\left (30 \, c^{3} d^{4} x^{2} + a c^{2} d^{4}\right )} e^{2}\right )} \log \left (x e + d\right )}{4 \, {\left (x^{4} e^{11} + 4 \, d x^{3} e^{10} + 6 \, d^{2} x^{2} e^{9} + 4 \, d^{3} x e^{8} + d^{4} e^{7}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A]
time = 1.93, size = 243, normalized size = 1.42 \begin {gather*} - \frac {5 c^{3} d x}{e^{6}} + \frac {c^{3} x^{2}}{2 e^{5}} + \frac {3 c^{2} \left (a e^{2} + 5 c d^{2}\right ) \log {\left (d + e x \right )}}{e^{7}} + \frac {- a^{3} e^{6} - a^{2} c d^{2} e^{4} + 25 a c^{2} d^{4} e^{2} + 57 c^{3} d^{6} + x^{3} \cdot \left (48 a c^{2} d e^{5} + 80 c^{3} d^{3} e^{3}\right ) + x^{2} \left (- 6 a^{2} c e^{6} + 108 a c^{2} d^{2} e^{4} + 210 c^{3} d^{4} e^{2}\right ) + x \left (- 4 a^{2} c d e^{5} + 88 a c^{2} d^{3} e^{3} + 188 c^{3} d^{5} e\right )}{4 d^{4} e^{7} + 16 d^{3} e^{8} x + 24 d^{2} e^{9} x^{2} + 16 d e^{10} x^{3} + 4 e^{11} x^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A]
time = 0.88, size = 288, normalized size = 1.68 \begin {gather*} \frac {1}{2} \, {\left (c^{3} - \frac {12 \, c^{3} d}{x e + d}\right )} {\left (x e + d\right )}^{2} e^{\left (-7\right )} - 3 \, {\left (5 \, c^{3} d^{2} + a c^{2} e^{2}\right )} e^{\left (-7\right )} \log \left (\frac {{\left | x e + d \right |} e^{\left (-1\right )}}{{\left (x e + d\right )}^{2}}\right ) + \frac {1}{4} \, {\left (\frac {80 \, c^{3} d^{3} e^{29}}{x e + d} - \frac {30 \, c^{3} d^{4} e^{29}}{{\left (x e + d\right )}^{2}} + \frac {8 \, c^{3} d^{5} e^{29}}{{\left (x e + d\right )}^{3}} - \frac {c^{3} d^{6} e^{29}}{{\left (x e + d\right )}^{4}} + \frac {48 \, a c^{2} d e^{31}}{x e + d} - \frac {36 \, a c^{2} d^{2} e^{31}}{{\left (x e + d\right )}^{2}} + \frac {16 \, a c^{2} d^{3} e^{31}}{{\left (x e + d\right )}^{3}} - \frac {3 \, a c^{2} d^{4} e^{31}}{{\left (x e + d\right )}^{4}} - \frac {6 \, a^{2} c e^{33}}{{\left (x e + d\right )}^{2}} + \frac {8 \, a^{2} c d e^{33}}{{\left (x e + d\right )}^{3}} - \frac {3 \, a^{2} c d^{2} e^{33}}{{\left (x e + d\right )}^{4}} - \frac {a^{3} e^{35}}{{\left (x e + d\right )}^{4}}\right )} e^{\left (-36\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 0.09, size = 236, normalized size = 1.38 \begin {gather*} \frac {x^2\,\left (-\frac {3\,a^2\,c\,e^5}{2}+27\,a\,c^2\,d^2\,e^3+\frac {105\,c^3\,d^4\,e}{2}\right )-\frac {a^3\,e^6+a^2\,c\,d^2\,e^4-25\,a\,c^2\,d^4\,e^2-57\,c^3\,d^6}{4\,e}+x\,\left (-a^2\,c\,d\,e^4+22\,a\,c^2\,d^3\,e^2+47\,c^3\,d^5\right )+x^3\,\left (20\,c^3\,d^3\,e^2+12\,a\,c^2\,d\,e^4\right )}{d^4\,e^6+4\,d^3\,e^7\,x+6\,d^2\,e^8\,x^2+4\,d\,e^9\,x^3+e^{10}\,x^4}+\frac {\ln \left (d+e\,x\right )\,\left (15\,c^3\,d^2+3\,a\,c^2\,e^2\right )}{e^7}+\frac {c^3\,x^2}{2\,e^5}-\frac {5\,c^3\,d\,x}{e^6} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________