Optimal. Leaf size=166 \[ -\frac {2 \left (\left (b^2-4 a c\right ) (c d-b e)-c e x \left (b^2-4 a c\right )\right )}{\left (b^2-4 a c\right ) \sqrt {a+b x+c x^2} \left (a e^2-b d e+c d^2\right )}-\frac {e (2 c d-b e) \tanh ^{-1}\left (\frac {-2 a e+x (2 c d-b e)+b d}{2 \sqrt {a+b x+c x^2} \sqrt {a e^2-b d e+c d^2}}\right )}{\left (a e^2-b d e+c d^2\right )^{3/2}} \]
________________________________________________________________________________________
Rubi [A] time = 0.13, antiderivative size = 166, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {822, 12, 724, 206} \begin {gather*} -\frac {2 \left (\left (b^2-4 a c\right ) (c d-b e)-c e x \left (b^2-4 a c\right )\right )}{\left (b^2-4 a c\right ) \sqrt {a+b x+c x^2} \left (a e^2-b d e+c d^2\right )}-\frac {e (2 c d-b e) \tanh ^{-1}\left (\frac {-2 a e+x (2 c d-b e)+b d}{2 \sqrt {a+b x+c x^2} \sqrt {a e^2-b d e+c d^2}}\right )}{\left (a e^2-b d e+c d^2\right )^{3/2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 12
Rule 206
Rule 724
Rule 822
Rubi steps
\begin {align*} \int \frac {b+2 c x}{(d+e x) \left (a+b x+c x^2\right )^{3/2}} \, dx &=-\frac {2 \left (\left (b^2-4 a c\right ) (c d-b e)-c \left (b^2-4 a c\right ) e x\right )}{\left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) \sqrt {a+b x+c x^2}}-\frac {2 \int \frac {\left (b^2-4 a c\right ) e (2 c d-b e)}{2 (d+e x) \sqrt {a+b x+c x^2}} \, dx}{\left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right )}\\ &=-\frac {2 \left (\left (b^2-4 a c\right ) (c d-b e)-c \left (b^2-4 a c\right ) e x\right )}{\left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) \sqrt {a+b x+c x^2}}-\frac {(e (2 c d-b e)) \int \frac {1}{(d+e x) \sqrt {a+b x+c x^2}} \, dx}{c d^2-b d e+a e^2}\\ &=-\frac {2 \left (\left (b^2-4 a c\right ) (c d-b e)-c \left (b^2-4 a c\right ) e x\right )}{\left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) \sqrt {a+b x+c x^2}}+\frac {(2 e (2 c d-b e)) \operatorname {Subst}\left (\int \frac {1}{4 c d^2-4 b d e+4 a e^2-x^2} \, dx,x,\frac {-b d+2 a e-(2 c d-b e) x}{\sqrt {a+b x+c x^2}}\right )}{c d^2-b d e+a e^2}\\ &=-\frac {2 \left (\left (b^2-4 a c\right ) (c d-b e)-c \left (b^2-4 a c\right ) e x\right )}{\left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) \sqrt {a+b x+c x^2}}-\frac {e (2 c d-b e) \tanh ^{-1}\left (\frac {b d-2 a e+(2 c d-b e) x}{2 \sqrt {c d^2-b d e+a e^2} \sqrt {a+b x+c x^2}}\right )}{\left (c d^2-b d e+a e^2\right )^{3/2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.19, size = 133, normalized size = 0.80 \begin {gather*} \frac {2 (b e-c d+c e x)}{\sqrt {a+x (b+c x)} \left (e (a e-b d)+c d^2\right )}+\frac {e (2 c d-b e) \tanh ^{-1}\left (\frac {2 a e-b d+b e x-2 c d x}{2 \sqrt {a+x (b+c x)} \sqrt {e (a e-b d)+c d^2}}\right )}{\left (e (a e-b d)+c d^2\right )^{3/2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [A] time = 0.91, size = 139, normalized size = 0.84 \begin {gather*} -\frac {2 (-b e+c d-c e x)}{\sqrt {a+b x+c x^2} \left (a e^2-b d e+c d^2\right )}-\frac {2 \left (2 c d e-b e^2\right ) \tan ^{-1}\left (\frac {-e \sqrt {a+b x+c x^2}+\sqrt {c} d+\sqrt {c} e x}{\sqrt {-a e^2+b d e-c d^2}}\right )}{\left (-a e^2+b d e-c d^2\right )^{3/2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [B] time = 0.92, size = 918, normalized size = 5.53 \begin {gather*} \left [-\frac {{\left (2 \, a c d e - a b e^{2} + {\left (2 \, c^{2} d e - b c e^{2}\right )} x^{2} + {\left (2 \, b c d e - b^{2} e^{2}\right )} x\right )} \sqrt {c d^{2} - b d e + a e^{2}} \log \left (\frac {8 \, a b d e - 8 \, a^{2} e^{2} - {\left (b^{2} + 4 \, a c\right )} d^{2} - {\left (8 \, c^{2} d^{2} - 8 \, b c d e + {\left (b^{2} + 4 \, a c\right )} e^{2}\right )} x^{2} - 4 \, \sqrt {c d^{2} - b d e + a e^{2}} \sqrt {c x^{2} + b x + a} {\left (b d - 2 \, a e + {\left (2 \, c d - b e\right )} x\right )} - 2 \, {\left (4 \, b c d^{2} + 4 \, a b e^{2} - {\left (3 \, b^{2} + 4 \, a c\right )} d e\right )} x}{e^{2} x^{2} + 2 \, d e x + d^{2}}\right ) + 4 \, {\left (c^{2} d^{3} - 2 \, b c d^{2} e - a b e^{3} + {\left (b^{2} + a c\right )} d e^{2} - {\left (c^{2} d^{2} e - b c d e^{2} + a c e^{3}\right )} x\right )} \sqrt {c x^{2} + b x + a}}{2 \, {\left (a c^{2} d^{4} - 2 \, a b c d^{3} e - 2 \, a^{2} b d e^{3} + a^{3} e^{4} + {\left (a b^{2} + 2 \, a^{2} c\right )} d^{2} e^{2} + {\left (c^{3} d^{4} - 2 \, b c^{2} d^{3} e - 2 \, a b c d e^{3} + a^{2} c e^{4} + {\left (b^{2} c + 2 \, a c^{2}\right )} d^{2} e^{2}\right )} x^{2} + {\left (b c^{2} d^{4} - 2 \, b^{2} c d^{3} e - 2 \, a b^{2} d e^{3} + a^{2} b e^{4} + {\left (b^{3} + 2 \, a b c\right )} d^{2} e^{2}\right )} x\right )}}, -\frac {{\left (2 \, a c d e - a b e^{2} + {\left (2 \, c^{2} d e - b c e^{2}\right )} x^{2} + {\left (2 \, b c d e - b^{2} e^{2}\right )} x\right )} \sqrt {-c d^{2} + b d e - a e^{2}} \arctan \left (-\frac {\sqrt {-c d^{2} + b d e - a e^{2}} \sqrt {c x^{2} + b x + a} {\left (b d - 2 \, a e + {\left (2 \, c d - b e\right )} x\right )}}{2 \, {\left (a c d^{2} - a b d e + a^{2} e^{2} + {\left (c^{2} d^{2} - b c d e + a c e^{2}\right )} x^{2} + {\left (b c d^{2} - b^{2} d e + a b e^{2}\right )} x\right )}}\right ) + 2 \, {\left (c^{2} d^{3} - 2 \, b c d^{2} e - a b e^{3} + {\left (b^{2} + a c\right )} d e^{2} - {\left (c^{2} d^{2} e - b c d e^{2} + a c e^{3}\right )} x\right )} \sqrt {c x^{2} + b x + a}}{a c^{2} d^{4} - 2 \, a b c d^{3} e - 2 \, a^{2} b d e^{3} + a^{3} e^{4} + {\left (a b^{2} + 2 \, a^{2} c\right )} d^{2} e^{2} + {\left (c^{3} d^{4} - 2 \, b c^{2} d^{3} e - 2 \, a b c d e^{3} + a^{2} c e^{4} + {\left (b^{2} c + 2 \, a c^{2}\right )} d^{2} e^{2}\right )} x^{2} + {\left (b c^{2} d^{4} - 2 \, b^{2} c d^{3} e - 2 \, a b^{2} d e^{3} + a^{2} b e^{4} + {\left (b^{3} + 2 \, a b c\right )} d^{2} e^{2}\right )} x}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [B] time = 0.31, size = 501, normalized size = 3.02 \begin {gather*} \frac {2 \, {\left (\frac {{\left (b^{2} c^{2} d^{2} e - 4 \, a c^{3} d^{2} e - b^{3} c d e^{2} + 4 \, a b c^{2} d e^{2} + a b^{2} c e^{3} - 4 \, a^{2} c^{2} e^{3}\right )} x}{b^{2} c^{2} d^{4} - 4 \, a c^{3} d^{4} - 2 \, b^{3} c d^{3} e + 8 \, a b c^{2} d^{3} e + b^{4} d^{2} e^{2} - 2 \, a b^{2} c d^{2} e^{2} - 8 \, a^{2} c^{2} d^{2} e^{2} - 2 \, a b^{3} d e^{3} + 8 \, a^{2} b c d e^{3} + a^{2} b^{2} e^{4} - 4 \, a^{3} c e^{4}} - \frac {b^{2} c^{2} d^{3} - 4 \, a c^{3} d^{3} - 2 \, b^{3} c d^{2} e + 8 \, a b c^{2} d^{2} e + b^{4} d e^{2} - 3 \, a b^{2} c d e^{2} - 4 \, a^{2} c^{2} d e^{2} - a b^{3} e^{3} + 4 \, a^{2} b c e^{3}}{b^{2} c^{2} d^{4} - 4 \, a c^{3} d^{4} - 2 \, b^{3} c d^{3} e + 8 \, a b c^{2} d^{3} e + b^{4} d^{2} e^{2} - 2 \, a b^{2} c d^{2} e^{2} - 8 \, a^{2} c^{2} d^{2} e^{2} - 2 \, a b^{3} d e^{3} + 8 \, a^{2} b c d e^{3} + a^{2} b^{2} e^{4} - 4 \, a^{3} c e^{4}}\right )}}{\sqrt {c x^{2} + b x + a}} - \frac {2 \, {\left (2 \, c d e - b e^{2}\right )} \arctan \left (-\frac {{\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} e + \sqrt {c} d}{\sqrt {-c d^{2} + b d e - a e^{2}}}\right )}{{\left (c d^{2} - b d e + a e^{2}\right )} \sqrt {-c d^{2} + b d e - a e^{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [B] time = 0.06, size = 1084, normalized size = 6.53 \begin {gather*} -\frac {2 b^{2} c e x}{\left (a \,e^{2}-b d e +c \,d^{2}\right ) \left (4 a c -b^{2}\right ) \sqrt {\left (x +\frac {d}{e}\right )^{2} c +\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}+\frac {a \,e^{2}-b d e +c \,d^{2}}{e^{2}}}}+\frac {8 b \,c^{2} d x}{\left (a \,e^{2}-b d e +c \,d^{2}\right ) \left (4 a c -b^{2}\right ) \sqrt {\left (x +\frac {d}{e}\right )^{2} c +\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}+\frac {a \,e^{2}-b d e +c \,d^{2}}{e^{2}}}}-\frac {8 c^{3} d^{2} x}{\left (a \,e^{2}-b d e +c \,d^{2}\right ) \left (4 a c -b^{2}\right ) \sqrt {\left (x +\frac {d}{e}\right )^{2} c +\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}+\frac {a \,e^{2}-b d e +c \,d^{2}}{e^{2}}}\, e}-\frac {b^{3} e}{\left (a \,e^{2}-b d e +c \,d^{2}\right ) \left (4 a c -b^{2}\right ) \sqrt {\left (x +\frac {d}{e}\right )^{2} c +\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}+\frac {a \,e^{2}-b d e +c \,d^{2}}{e^{2}}}}+\frac {4 b^{2} c d}{\left (a \,e^{2}-b d e +c \,d^{2}\right ) \left (4 a c -b^{2}\right ) \sqrt {\left (x +\frac {d}{e}\right )^{2} c +\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}+\frac {a \,e^{2}-b d e +c \,d^{2}}{e^{2}}}}-\frac {4 b \,c^{2} d^{2}}{\left (a \,e^{2}-b d e +c \,d^{2}\right ) \left (4 a c -b^{2}\right ) \sqrt {\left (x +\frac {d}{e}\right )^{2} c +\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}+\frac {a \,e^{2}-b d e +c \,d^{2}}{e^{2}}}\, e}-\frac {b e \ln \left (\frac {\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}+\frac {2 a \,e^{2}-2 b d e +2 c \,d^{2}}{e^{2}}+2 \sqrt {\frac {a \,e^{2}-b d e +c \,d^{2}}{e^{2}}}\, \sqrt {\left (x +\frac {d}{e}\right )^{2} c +\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}+\frac {a \,e^{2}-b d e +c \,d^{2}}{e^{2}}}}{x +\frac {d}{e}}\right )}{\left (a \,e^{2}-b d e +c \,d^{2}\right ) \sqrt {\frac {a \,e^{2}-b d e +c \,d^{2}}{e^{2}}}}+\frac {2 c d \ln \left (\frac {\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}+\frac {2 a \,e^{2}-2 b d e +2 c \,d^{2}}{e^{2}}+2 \sqrt {\frac {a \,e^{2}-b d e +c \,d^{2}}{e^{2}}}\, \sqrt {\left (x +\frac {d}{e}\right )^{2} c +\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}+\frac {a \,e^{2}-b d e +c \,d^{2}}{e^{2}}}}{x +\frac {d}{e}}\right )}{\left (a \,e^{2}-b d e +c \,d^{2}\right ) \sqrt {\frac {a \,e^{2}-b d e +c \,d^{2}}{e^{2}}}}+\frac {b e}{\left (a \,e^{2}-b d e +c \,d^{2}\right ) \sqrt {\left (x +\frac {d}{e}\right )^{2} c +\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}+\frac {a \,e^{2}-b d e +c \,d^{2}}{e^{2}}}}-\frac {2 c d}{\left (a \,e^{2}-b d e +c \,d^{2}\right ) \sqrt {\left (x +\frac {d}{e}\right )^{2} c +\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}+\frac {a \,e^{2}-b d e +c \,d^{2}}{e^{2}}}}+\frac {4 \left (2 c x +b \right ) c}{\left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}\, e} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \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 \frac {b+2\,c\,x}{\left (d+e\,x\right )\,{\left (c\,x^2+b\,x+a\right )}^{3/2}} \,d x \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 {b + 2 c x}{\left (d + e x\right ) \left (a + b x + c x^{2}\right )^{\frac {3}{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________