Optimal. Leaf size=93 \[ \frac {\tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a}}\right ) (A c d-a (B e+C d))}{\sqrt {a} c^{3/2}}+\frac {\log \left (a+c x^2\right ) (-a C e+A c e+B c d)}{2 c^2}+\frac {x (B e+C d)}{c}+\frac {C e x^2}{2 c} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.12, antiderivative size = 93, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.160, Rules used = {1629, 635, 205, 260} \[ \frac {\log \left (a+c x^2\right ) (-a C e+A c e+B c d)}{2 c^2}+\frac {\tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a}}\right ) (A c d-a (B e+C d))}{\sqrt {a} c^{3/2}}+\frac {x (B e+C d)}{c}+\frac {C e x^2}{2 c} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 205
Rule 260
Rule 635
Rule 1629
Rubi steps
\begin {align*} \int \frac {(d+e x) \left (A+B x+C x^2\right )}{a+c x^2} \, dx &=\int \left (\frac {C d+B e}{c}+\frac {C e x}{c}+\frac {A c d-a (C d+B e)+(B c d+A c e-a C e) x}{c \left (a+c x^2\right )}\right ) \, dx\\ &=\frac {(C d+B e) x}{c}+\frac {C e x^2}{2 c}+\frac {\int \frac {A c d-a (C d+B e)+(B c d+A c e-a C e) x}{a+c x^2} \, dx}{c}\\ &=\frac {(C d+B e) x}{c}+\frac {C e x^2}{2 c}+\frac {(B c d+A c e-a C e) \int \frac {x}{a+c x^2} \, dx}{c}+\frac {(A c d-a (C d+B e)) \int \frac {1}{a+c x^2} \, dx}{c}\\ &=\frac {(C d+B e) x}{c}+\frac {C e x^2}{2 c}+\frac {(A c d-a (C d+B e)) \tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a}}\right )}{\sqrt {a} c^{3/2}}+\frac {(B c d+A c e-a C e) \log \left (a+c x^2\right )}{2 c^2}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.09, size = 86, normalized size = 0.92 \[ \frac {\log \left (a+c x^2\right ) (-a C e+A c e+B c d)-\frac {2 \sqrt {c} \tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a}}\right ) (a B e+a C d-A c d)}{\sqrt {a}}+c x (2 B e+2 C d+C e x)}{2 c^2} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.58, size = 206, normalized size = 2.22 \[ \left [\frac {C a c e x^{2} - {\left (B a e + {\left (C a - A c\right )} d\right )} \sqrt {-a c} \log \left (\frac {c x^{2} + 2 \, \sqrt {-a c} x - a}{c x^{2} + a}\right ) + 2 \, {\left (C a c d + B a c e\right )} x + {\left (B a c d - {\left (C a^{2} - A a c\right )} e\right )} \log \left (c x^{2} + a\right )}{2 \, a c^{2}}, \frac {C a c e x^{2} - 2 \, {\left (B a e + {\left (C a - A c\right )} d\right )} \sqrt {a c} \arctan \left (\frac {\sqrt {a c} x}{a}\right ) + 2 \, {\left (C a c d + B a c e\right )} x + {\left (B a c d - {\left (C a^{2} - A a c\right )} e\right )} \log \left (c x^{2} + a\right )}{2 \, a c^{2}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.16, size = 91, normalized size = 0.98 \[ -\frac {{\left (C a d - A c d + B a e\right )} \arctan \left (\frac {c x}{\sqrt {a c}}\right )}{\sqrt {a c} c} + \frac {{\left (B c d - C a e + A c e\right )} \log \left (c x^{2} + a\right )}{2 \, c^{2}} + \frac {C c x^{2} e + 2 \, C c d x + 2 \, B c x e}{2 \, c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.01, size = 133, normalized size = 1.43 \[ \frac {A d \arctan \left (\frac {c x}{\sqrt {a c}}\right )}{\sqrt {a c}}-\frac {B a e \arctan \left (\frac {c x}{\sqrt {a c}}\right )}{\sqrt {a c}\, c}-\frac {C a d \arctan \left (\frac {c x}{\sqrt {a c}}\right )}{\sqrt {a c}\, c}+\frac {C e \,x^{2}}{2 c}+\frac {A e \ln \left (c \,x^{2}+a \right )}{2 c}+\frac {B d \ln \left (c \,x^{2}+a \right )}{2 c}+\frac {B e x}{c}-\frac {C a e \ln \left (c \,x^{2}+a \right )}{2 c^{2}}+\frac {C d x}{c} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.97, size = 86, normalized size = 0.92 \[ -\frac {{\left (B a e + {\left (C a - A c\right )} d\right )} \arctan \left (\frac {c x}{\sqrt {a c}}\right )}{\sqrt {a c} c} + \frac {C e x^{2} + 2 \, {\left (C d + B e\right )} x}{2 \, c} + \frac {{\left (B c d - {\left (C a - A c\right )} e\right )} \log \left (c x^{2} + a\right )}{2 \, c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 3.78, size = 97, normalized size = 1.04 \[ \frac {x\,\left (B\,e+C\,d\right )}{c}-\frac {\mathrm {atan}\left (\frac {\sqrt {c}\,x}{\sqrt {a}}\right )\,\left (B\,a\,e-A\,c\,d+C\,a\,d\right )}{\sqrt {a}\,c^{3/2}}+\frac {C\,e\,x^2}{2\,c}+\frac {\ln \left (c\,x^2+a\right )\,\left (4\,A\,a\,c^3\,e+4\,B\,a\,c^3\,d-4\,C\,a^2\,c^2\,e\right )}{8\,a\,c^4} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [B] time = 1.66, size = 337, normalized size = 3.62 \[ \frac {C e x^{2}}{2 c} + x \left (\frac {B e}{c} + \frac {C d}{c}\right ) + \left (- \frac {- A c e - B c d + C a e}{2 c^{2}} - \frac {\sqrt {- a c^{5}} \left (- A c d + B a e + C a d\right )}{2 a c^{4}}\right ) \log {\left (x + \frac {A a c e + B a c d - C a^{2} e - 2 a c^{2} \left (- \frac {- A c e - B c d + C a e}{2 c^{2}} - \frac {\sqrt {- a c^{5}} \left (- A c d + B a e + C a d\right )}{2 a c^{4}}\right )}{- A c^{2} d + B a c e + C a c d} \right )} + \left (- \frac {- A c e - B c d + C a e}{2 c^{2}} + \frac {\sqrt {- a c^{5}} \left (- A c d + B a e + C a d\right )}{2 a c^{4}}\right ) \log {\left (x + \frac {A a c e + B a c d - C a^{2} e - 2 a c^{2} \left (- \frac {- A c e - B c d + C a e}{2 c^{2}} + \frac {\sqrt {- a c^{5}} \left (- A c d + B a e + C a d\right )}{2 a c^{4}}\right )}{- A c^{2} d + B a c e + C a c d} \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________