Optimal. Leaf size=1485 \[ \frac {3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{2/3}}{4 c d}+\frac {3 \left (c d^2-a e^2\right ) \sqrt {\left (c d^2+a e^2+2 c d e x\right )^2} \sqrt {\left (a e^2+c d (d+2 e x)\right )^2}}{2 \sqrt [3]{2} c^{5/3} d^{5/3} e^{2/3} \left (c d^2+a e^2+2 c d e x\right ) \left (\left (1+\sqrt {3}\right ) \left (c d^2-a e^2\right )^{2/3}+2^{2/3} \sqrt [3]{c} \sqrt [3]{d} \sqrt [3]{e} \sqrt [3]{(a e+c d x) (d+e x)}\right )}-\frac {3 \sqrt [4]{3} \sqrt {2-\sqrt {3}} \left (c d^2-a e^2\right )^{5/3} \sqrt {\left (c d^2+a e^2+2 c d e x\right )^2} \left (\left (c d^2-a e^2\right )^{2/3}+2^{2/3} \sqrt [3]{c} \sqrt [3]{d} \sqrt [3]{e} \sqrt [3]{(a e+c d x) (d+e x)}\right ) \sqrt {\frac {\left (c d^2-a e^2\right )^{4/3}-2^{2/3} \sqrt [3]{c} \sqrt [3]{d} \sqrt [3]{e} \left (c d^2-a e^2\right )^{2/3} \sqrt [3]{(a e+c d x) (d+e x)}+2 \sqrt [3]{2} c^{2/3} d^{2/3} e^{2/3} ((a e+c d x) (d+e x))^{2/3}}{\left (\left (1+\sqrt {3}\right ) \left (c d^2-a e^2\right )^{2/3}+2^{2/3} \sqrt [3]{c} \sqrt [3]{d} \sqrt [3]{e} \sqrt [3]{(a e+c d x) (d+e x)}\right )^2}} E\left (\sin ^{-1}\left (\frac {\left (1-\sqrt {3}\right ) \left (c d^2-a e^2\right )^{2/3}+2^{2/3} \sqrt [3]{c} \sqrt [3]{d} \sqrt [3]{e} \sqrt [3]{(a e+c d x) (d+e x)}}{\left (1+\sqrt {3}\right ) \left (c d^2-a e^2\right )^{2/3}+2^{2/3} \sqrt [3]{c} \sqrt [3]{d} \sqrt [3]{e} \sqrt [3]{(a e+c d x) (d+e x)}}\right )|-7-4 \sqrt {3}\right )}{4 \sqrt [3]{2} c^{5/3} d^{5/3} e^{2/3} \left (c d^2+a e^2+2 c d e x\right ) \sqrt {\frac {\left (c d^2-a e^2\right )^{2/3} \left (\left (c d^2-a e^2\right )^{2/3}+2^{2/3} \sqrt [3]{c} \sqrt [3]{d} \sqrt [3]{e} \sqrt [3]{(a e+c d x) (d+e x)}\right )}{\left (\left (1+\sqrt {3}\right ) \left (c d^2-a e^2\right )^{2/3}+2^{2/3} \sqrt [3]{c} \sqrt [3]{d} \sqrt [3]{e} \sqrt [3]{(a e+c d x) (d+e x)}\right )^2}} \sqrt {\left (a e^2+c d (d+2 e x)\right )^2}}+\frac {3^{3/4} \left (c d^2-a e^2\right )^{5/3} \sqrt {\left (c d^2+a e^2+2 c d e x\right )^2} \left (\left (c d^2-a e^2\right )^{2/3}+2^{2/3} \sqrt [3]{c} \sqrt [3]{d} \sqrt [3]{e} \sqrt [3]{(a e+c d x) (d+e x)}\right ) \sqrt {\frac {\left (c d^2-a e^2\right )^{4/3}-2^{2/3} \sqrt [3]{c} \sqrt [3]{d} \sqrt [3]{e} \left (c d^2-a e^2\right )^{2/3} \sqrt [3]{(a e+c d x) (d+e x)}+2 \sqrt [3]{2} c^{2/3} d^{2/3} e^{2/3} ((a e+c d x) (d+e x))^{2/3}}{\left (\left (1+\sqrt {3}\right ) \left (c d^2-a e^2\right )^{2/3}+2^{2/3} \sqrt [3]{c} \sqrt [3]{d} \sqrt [3]{e} \sqrt [3]{(a e+c d x) (d+e x)}\right )^2}} F\left (\sin ^{-1}\left (\frac {\left (1-\sqrt {3}\right ) \left (c d^2-a e^2\right )^{2/3}+2^{2/3} \sqrt [3]{c} \sqrt [3]{d} \sqrt [3]{e} \sqrt [3]{(a e+c d x) (d+e x)}}{\left (1+\sqrt {3}\right ) \left (c d^2-a e^2\right )^{2/3}+2^{2/3} \sqrt [3]{c} \sqrt [3]{d} \sqrt [3]{e} \sqrt [3]{(a e+c d x) (d+e x)}}\right )|-7-4 \sqrt {3}\right )}{2^{5/6} c^{5/3} d^{5/3} e^{2/3} \left (c d^2+a e^2+2 c d e x\right ) \sqrt {\frac {\left (c d^2-a e^2\right )^{2/3} \left (\left (c d^2-a e^2\right )^{2/3}+2^{2/3} \sqrt [3]{c} \sqrt [3]{d} \sqrt [3]{e} \sqrt [3]{(a e+c d x) (d+e x)}\right )}{\left (\left (1+\sqrt {3}\right ) \left (c d^2-a e^2\right )^{2/3}+2^{2/3} \sqrt [3]{c} \sqrt [3]{d} \sqrt [3]{e} \sqrt [3]{(a e+c d x) (d+e x)}\right )^2}} \sqrt {\left (a e^2+c d (d+2 e x)\right )^2}} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 1.52, antiderivative size = 1485, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 5, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {654, 637, 309,
224, 1891} \begin {gather*} -\frac {3 \sqrt [4]{3} \sqrt {2-\sqrt {3}} \sqrt {\left (c d^2+2 c e x d+a e^2\right )^2} \left (\left (c d^2-a e^2\right )^{2/3}+2^{2/3} \sqrt [3]{c} \sqrt [3]{d} \sqrt [3]{e} \sqrt [3]{(a e+c d x) (d+e x)}\right ) \sqrt {\frac {\left (c d^2-a e^2\right )^{4/3}-2^{2/3} \sqrt [3]{c} \sqrt [3]{d} \sqrt [3]{e} \sqrt [3]{(a e+c d x) (d+e x)} \left (c d^2-a e^2\right )^{2/3}+2 \sqrt [3]{2} c^{2/3} d^{2/3} e^{2/3} ((a e+c d x) (d+e x))^{2/3}}{\left (\left (1+\sqrt {3}\right ) \left (c d^2-a e^2\right )^{2/3}+2^{2/3} \sqrt [3]{c} \sqrt [3]{d} \sqrt [3]{e} \sqrt [3]{(a e+c d x) (d+e x)}\right )^2}} E\left (\text {ArcSin}\left (\frac {\left (1-\sqrt {3}\right ) \left (c d^2-a e^2\right )^{2/3}+2^{2/3} \sqrt [3]{c} \sqrt [3]{d} \sqrt [3]{e} \sqrt [3]{(a e+c d x) (d+e x)}}{\left (1+\sqrt {3}\right ) \left (c d^2-a e^2\right )^{2/3}+2^{2/3} \sqrt [3]{c} \sqrt [3]{d} \sqrt [3]{e} \sqrt [3]{(a e+c d x) (d+e x)}}\right )|-7-4 \sqrt {3}\right ) \left (c d^2-a e^2\right )^{5/3}}{4 \sqrt [3]{2} c^{5/3} d^{5/3} e^{2/3} \left (c d^2+2 c e x d+a e^2\right ) \sqrt {\frac {\left (c d^2-a e^2\right )^{2/3} \left (\left (c d^2-a e^2\right )^{2/3}+2^{2/3} \sqrt [3]{c} \sqrt [3]{d} \sqrt [3]{e} \sqrt [3]{(a e+c d x) (d+e x)}\right )}{\left (\left (1+\sqrt {3}\right ) \left (c d^2-a e^2\right )^{2/3}+2^{2/3} \sqrt [3]{c} \sqrt [3]{d} \sqrt [3]{e} \sqrt [3]{(a e+c d x) (d+e x)}\right )^2}} \sqrt {\left (a e^2+c d (d+2 e x)\right )^2}}+\frac {3^{3/4} \sqrt {\left (c d^2+2 c e x d+a e^2\right )^2} \left (\left (c d^2-a e^2\right )^{2/3}+2^{2/3} \sqrt [3]{c} \sqrt [3]{d} \sqrt [3]{e} \sqrt [3]{(a e+c d x) (d+e x)}\right ) \sqrt {\frac {\left (c d^2-a e^2\right )^{4/3}-2^{2/3} \sqrt [3]{c} \sqrt [3]{d} \sqrt [3]{e} \sqrt [3]{(a e+c d x) (d+e x)} \left (c d^2-a e^2\right )^{2/3}+2 \sqrt [3]{2} c^{2/3} d^{2/3} e^{2/3} ((a e+c d x) (d+e x))^{2/3}}{\left (\left (1+\sqrt {3}\right ) \left (c d^2-a e^2\right )^{2/3}+2^{2/3} \sqrt [3]{c} \sqrt [3]{d} \sqrt [3]{e} \sqrt [3]{(a e+c d x) (d+e x)}\right )^2}} F\left (\text {ArcSin}\left (\frac {\left (1-\sqrt {3}\right ) \left (c d^2-a e^2\right )^{2/3}+2^{2/3} \sqrt [3]{c} \sqrt [3]{d} \sqrt [3]{e} \sqrt [3]{(a e+c d x) (d+e x)}}{\left (1+\sqrt {3}\right ) \left (c d^2-a e^2\right )^{2/3}+2^{2/3} \sqrt [3]{c} \sqrt [3]{d} \sqrt [3]{e} \sqrt [3]{(a e+c d x) (d+e x)}}\right )|-7-4 \sqrt {3}\right ) \left (c d^2-a e^2\right )^{5/3}}{2^{5/6} c^{5/3} d^{5/3} e^{2/3} \left (c d^2+2 c e x d+a e^2\right ) \sqrt {\frac {\left (c d^2-a e^2\right )^{2/3} \left (\left (c d^2-a e^2\right )^{2/3}+2^{2/3} \sqrt [3]{c} \sqrt [3]{d} \sqrt [3]{e} \sqrt [3]{(a e+c d x) (d+e x)}\right )}{\left (\left (1+\sqrt {3}\right ) \left (c d^2-a e^2\right )^{2/3}+2^{2/3} \sqrt [3]{c} \sqrt [3]{d} \sqrt [3]{e} \sqrt [3]{(a e+c d x) (d+e x)}\right )^2}} \sqrt {\left (a e^2+c d (d+2 e x)\right )^2}}+\frac {3 \sqrt {\left (c d^2+2 c e x d+a e^2\right )^2} \sqrt {\left (a e^2+c d (d+2 e x)\right )^2} \left (c d^2-a e^2\right )}{2 \sqrt [3]{2} c^{5/3} d^{5/3} e^{2/3} \left (c d^2+2 c e x d+a e^2\right ) \left (\left (1+\sqrt {3}\right ) \left (c d^2-a e^2\right )^{2/3}+2^{2/3} \sqrt [3]{c} \sqrt [3]{d} \sqrt [3]{e} \sqrt [3]{(a e+c d x) (d+e x)}\right )}+\frac {3 \left (c d e x^2+\left (c d^2+a e^2\right ) x+a d e\right )^{2/3}}{4 c d} \end {gather*}
Warning: Unable to verify antiderivative.
[In]
[Out]
Rule 224
Rule 309
Rule 637
Rule 654
Rule 1891
Rubi steps
\begin {align*} \int \frac {d+e x}{\sqrt [3]{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx &=\frac {3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{2/3}}{4 c d}+\frac {\left (d^2-\frac {a e^2}{c}\right ) \int \frac {1}{\sqrt [3]{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx}{2 d}\\ &=\frac {3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{2/3}}{4 c d}+\frac {\left (3 \left (d^2-\frac {a e^2}{c}\right ) \sqrt {\left (c d^2+a e^2+2 c d e x\right )^2}\right ) \text {Subst}\left (\int \frac {x}{\sqrt {-4 a c d^2 e^2+\left (c d^2+a e^2\right )^2+4 c d e x^3}} \, dx,x,\sqrt [3]{(a e+c d x) (d+e x)}\right )}{2 d \left (c d^2+a e^2+2 c d e x\right )}\\ &=\frac {3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{2/3}}{4 c d}+\frac {\left (3 \left (d^2-\frac {a e^2}{c}\right ) \sqrt {\left (c d^2+a e^2+2 c d e x\right )^2}\right ) \text {Subst}\left (\int \frac {\left (1-\sqrt {3}\right ) \left (c d^2-a e^2\right )^{2/3}+2^{2/3} \sqrt [3]{c} \sqrt [3]{d} \sqrt [3]{e} x}{\sqrt {-4 a c d^2 e^2+\left (c d^2+a e^2\right )^2+4 c d e x^3}} \, dx,x,\sqrt [3]{(a e+c d x) (d+e x)}\right )}{2\ 2^{2/3} \sqrt [3]{c} d^{4/3} \sqrt [3]{e} \left (c d^2+a e^2+2 c d e x\right )}+\frac {\left (3 \left (c d^2-a e^2\right )^{2/3} \left (d^2-\frac {a e^2}{c}\right ) \sqrt {\left (c d^2+a e^2+2 c d e x\right )^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {-4 a c d^2 e^2+\left (c d^2+a e^2\right )^2+4 c d e x^3}} \, dx,x,\sqrt [3]{(a e+c d x) (d+e x)}\right )}{2 \sqrt [6]{2} \sqrt {2+\sqrt {3}} \sqrt [3]{c} d^{4/3} \sqrt [3]{e} \left (c d^2+a e^2+2 c d e x\right )}\\ &=\frac {3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{2/3}}{4 c d}+\frac {3 \left (c d^2-a e^2\right ) \sqrt {\left (c d^2+a e^2+2 c d e x\right )^2} \sqrt {\left (a e^2+c d (d+2 e x)\right )^2}}{2 \sqrt [3]{2} c^{5/3} d^{5/3} e^{2/3} \left (c d^2+a e^2+2 c d e x\right ) \left (\left (1+\sqrt {3}\right ) \left (c d^2-a e^2\right )^{2/3}+2^{2/3} \sqrt [3]{c} \sqrt [3]{d} \sqrt [3]{e} \sqrt [3]{(a e+c d x) (d+e x)}\right )}-\frac {3 \sqrt [4]{3} \sqrt {2-\sqrt {3}} \left (c d^2-a e^2\right )^{5/3} \sqrt {\left (c d^2+a e^2+2 c d e x\right )^2} \left (\left (c d^2-a e^2\right )^{2/3}+2^{2/3} \sqrt [3]{c} \sqrt [3]{d} \sqrt [3]{e} \sqrt [3]{(a e+c d x) (d+e x)}\right ) \sqrt {\frac {\left (c d^2-a e^2\right )^{4/3}-2^{2/3} \sqrt [3]{c} \sqrt [3]{d} \sqrt [3]{e} \left (c d^2-a e^2\right )^{2/3} \sqrt [3]{(a e+c d x) (d+e x)}+2 \sqrt [3]{2} c^{2/3} d^{2/3} e^{2/3} ((a e+c d x) (d+e x))^{2/3}}{\left (\left (1+\sqrt {3}\right ) \left (c d^2-a e^2\right )^{2/3}+2^{2/3} \sqrt [3]{c} \sqrt [3]{d} \sqrt [3]{e} \sqrt [3]{(a e+c d x) (d+e x)}\right )^2}} E\left (\sin ^{-1}\left (\frac {\left (1-\sqrt {3}\right ) \left (c d^2-a e^2\right )^{2/3}+2^{2/3} \sqrt [3]{c} \sqrt [3]{d} \sqrt [3]{e} \sqrt [3]{(a e+c d x) (d+e x)}}{\left (1+\sqrt {3}\right ) \left (c d^2-a e^2\right )^{2/3}+2^{2/3} \sqrt [3]{c} \sqrt [3]{d} \sqrt [3]{e} \sqrt [3]{(a e+c d x) (d+e x)}}\right )|-7-4 \sqrt {3}\right )}{4 \sqrt [3]{2} c^{5/3} d^{5/3} e^{2/3} \left (c d^2+a e^2+2 c d e x\right ) \sqrt {\frac {\left (c d^2-a e^2\right )^{2/3} \left (\left (c d^2-a e^2\right )^{2/3}+2^{2/3} \sqrt [3]{c} \sqrt [3]{d} \sqrt [3]{e} \sqrt [3]{(a e+c d x) (d+e x)}\right )}{\left (\left (1+\sqrt {3}\right ) \left (c d^2-a e^2\right )^{2/3}+2^{2/3} \sqrt [3]{c} \sqrt [3]{d} \sqrt [3]{e} \sqrt [3]{(a e+c d x) (d+e x)}\right )^2}} \sqrt {\left (a e^2+c d (d+2 e x)\right )^2}}+\frac {3^{3/4} \left (c d^2-a e^2\right )^{5/3} \sqrt {\left (c d^2+a e^2+2 c d e x\right )^2} \left (\left (c d^2-a e^2\right )^{2/3}+2^{2/3} \sqrt [3]{c} \sqrt [3]{d} \sqrt [3]{e} \sqrt [3]{(a e+c d x) (d+e x)}\right ) \sqrt {\frac {\left (c d^2-a e^2\right )^{4/3}-2^{2/3} \sqrt [3]{c} \sqrt [3]{d} \sqrt [3]{e} \left (c d^2-a e^2\right )^{2/3} \sqrt [3]{(a e+c d x) (d+e x)}+2 \sqrt [3]{2} c^{2/3} d^{2/3} e^{2/3} ((a e+c d x) (d+e x))^{2/3}}{\left (\left (1+\sqrt {3}\right ) \left (c d^2-a e^2\right )^{2/3}+2^{2/3} \sqrt [3]{c} \sqrt [3]{d} \sqrt [3]{e} \sqrt [3]{(a e+c d x) (d+e x)}\right )^2}} F\left (\sin ^{-1}\left (\frac {\left (1-\sqrt {3}\right ) \left (c d^2-a e^2\right )^{2/3}+2^{2/3} \sqrt [3]{c} \sqrt [3]{d} \sqrt [3]{e} \sqrt [3]{(a e+c d x) (d+e x)}}{\left (1+\sqrt {3}\right ) \left (c d^2-a e^2\right )^{2/3}+2^{2/3} \sqrt [3]{c} \sqrt [3]{d} \sqrt [3]{e} \sqrt [3]{(a e+c d x) (d+e x)}}\right )|-7-4 \sqrt {3}\right )}{2^{5/6} c^{5/3} d^{5/3} e^{2/3} \left (c d^2+a e^2+2 c d e x\right ) \sqrt {\frac {\left (c d^2-a e^2\right )^{2/3} \left (\left (c d^2-a e^2\right )^{2/3}+2^{2/3} \sqrt [3]{c} \sqrt [3]{d} \sqrt [3]{e} \sqrt [3]{(a e+c d x) (d+e x)}\right )}{\left (\left (1+\sqrt {3}\right ) \left (c d^2-a e^2\right )^{2/3}+2^{2/3} \sqrt [3]{c} \sqrt [3]{d} \sqrt [3]{e} \sqrt [3]{(a e+c d x) (d+e x)}\right )^2}} \sqrt {\left (a e^2+c d (d+2 e x)\right )^2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 4 in
optimal.
time = 10.06, size = 88, normalized size = 0.06 \begin {gather*} \frac {3 ((a e+c d x) (d+e x))^{2/3} \, _2F_1\left (-\frac {2}{3},\frac {2}{3};\frac {5}{3};\frac {e (a e+c d x)}{-c d^2+a e^2}\right )}{2 c d \left (\frac {c d (d+e x)}{c d^2-a e^2}\right )^{2/3}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F]
time = 0.33, size = 0, normalized size = 0.00 \[\int \frac {e x +d}{\left (a d e +\left (e^{2} a +c \,d^{2}\right ) x +c d e \,x^{2}\right )^{\frac {1}{3}}}\, dx\]
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 \frac {d + e x}{\sqrt [3]{\left (d + e x\right ) \left (a e + c d x\right )}}\, 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 = -0.00 \begin {gather*} \int \frac {d+e\,x}{{\left (c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e\right )}^{1/3}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________