Optimal. Leaf size=234 \[ \frac {\sqrt {a+b x+c x^2} \left (B \left (-4 c e (4 a e+9 b d)+15 b^2 e^2+16 c^2 d^2\right )+2 c e x (6 A c e-5 b B e+4 B c d)+6 A c e (8 c d-3 b e)\right )}{24 c^3}-\frac {\tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right ) \left (4 b c \left (-3 a B e^2+4 A c d e+2 B c d^2\right )-8 c^2 \left (-a A e^2-2 a B d e+2 A c d^2\right )-6 b^2 c e (A e+2 B d)+5 b^3 B e^2\right )}{16 c^{7/2}}+\frac {B (d+e x)^2 \sqrt {a+b x+c x^2}}{3 c} \]
________________________________________________________________________________________
Rubi [A] time = 0.25, antiderivative size = 234, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.148, Rules used = {832, 779, 621, 206} \begin {gather*} \frac {\sqrt {a+b x+c x^2} \left (B \left (-4 c e (4 a e+9 b d)+15 b^2 e^2+16 c^2 d^2\right )+2 c e x (6 A c e-5 b B e+4 B c d)+6 A c e (8 c d-3 b e)\right )}{24 c^3}-\frac {\tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right ) \left (4 b c \left (-3 a B e^2+4 A c d e+2 B c d^2\right )-8 c^2 \left (-a A e^2-2 a B d e+2 A c d^2\right )-6 b^2 c e (A e+2 B d)+5 b^3 B e^2\right )}{16 c^{7/2}}+\frac {B (d+e x)^2 \sqrt {a+b x+c x^2}}{3 c} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 206
Rule 621
Rule 779
Rule 832
Rubi steps
\begin {align*} \int \frac {(A+B x) (d+e x)^2}{\sqrt {a+b x+c x^2}} \, dx &=\frac {B (d+e x)^2 \sqrt {a+b x+c x^2}}{3 c}+\frac {\int \frac {(d+e x) \left (\frac {1}{2} (-b B d+6 A c d-4 a B e)+\frac {1}{2} (4 B c d-5 b B e+6 A c e) x\right )}{\sqrt {a+b x+c x^2}} \, dx}{3 c}\\ &=\frac {B (d+e x)^2 \sqrt {a+b x+c x^2}}{3 c}+\frac {\left (6 A c e (8 c d-3 b e)+B \left (16 c^2 d^2+15 b^2 e^2-4 c e (9 b d+4 a e)\right )+2 c e (4 B c d-5 b B e+6 A c e) x\right ) \sqrt {a+b x+c x^2}}{24 c^3}-\frac {\left (5 b^3 B e^2-6 b^2 c e (2 B d+A e)-8 c^2 \left (2 A c d^2-2 a B d e-a A e^2\right )+4 b c \left (2 B c d^2+4 A c d e-3 a B e^2\right )\right ) \int \frac {1}{\sqrt {a+b x+c x^2}} \, dx}{16 c^3}\\ &=\frac {B (d+e x)^2 \sqrt {a+b x+c x^2}}{3 c}+\frac {\left (6 A c e (8 c d-3 b e)+B \left (16 c^2 d^2+15 b^2 e^2-4 c e (9 b d+4 a e)\right )+2 c e (4 B c d-5 b B e+6 A c e) x\right ) \sqrt {a+b x+c x^2}}{24 c^3}-\frac {\left (5 b^3 B e^2-6 b^2 c e (2 B d+A e)-8 c^2 \left (2 A c d^2-2 a B d e-a A e^2\right )+4 b c \left (2 B c d^2+4 A c d e-3 a B e^2\right )\right ) \operatorname {Subst}\left (\int \frac {1}{4 c-x^2} \, dx,x,\frac {b+2 c x}{\sqrt {a+b x+c x^2}}\right )}{8 c^3}\\ &=\frac {B (d+e x)^2 \sqrt {a+b x+c x^2}}{3 c}+\frac {\left (6 A c e (8 c d-3 b e)+B \left (16 c^2 d^2+15 b^2 e^2-4 c e (9 b d+4 a e)\right )+2 c e (4 B c d-5 b B e+6 A c e) x\right ) \sqrt {a+b x+c x^2}}{24 c^3}-\frac {\left (5 b^3 B e^2-6 b^2 c e (2 B d+A e)-8 c^2 \left (2 A c d^2-2 a B d e-a A e^2\right )+4 b c \left (2 B c d^2+4 A c d e-3 a B e^2\right )\right ) \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{16 c^{7/2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.20, size = 225, normalized size = 0.96 \begin {gather*} \frac {\frac {\sqrt {a+x (b+c x)} \left (B \left (-2 c e (8 a e+18 b d+5 b e x)+15 b^2 e^2+8 c^2 d (2 d+e x)\right )+6 A c e (-3 b e+8 c d+2 c e x)\right )}{8 c^2}-\frac {3 \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+x (b+c x)}}\right ) \left (4 b c \left (-3 a B e^2+4 A c d e+2 B c d^2\right )+8 c^2 \left (a A e^2+2 a B d e-2 A c d^2\right )-6 b^2 c e (A e+2 B d)+5 b^3 B e^2\right )}{16 c^{5/2}}+B (d+e x)^2 \sqrt {a+x (b+c x)}}{3 c} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [A] time = 0.87, size = 234, normalized size = 1.00 \begin {gather*} \frac {\sqrt {a+b x+c x^2} \left (-16 a B c e^2-18 A b c e^2+48 A c^2 d e+12 A c^2 e^2 x+15 b^2 B e^2-36 b B c d e-10 b B c e^2 x+24 B c^2 d^2+24 B c^2 d e x+8 B c^2 e^2 x^2\right )}{24 c^3}+\frac {\log \left (-2 \sqrt {c} \sqrt {a+b x+c x^2}+b+2 c x\right ) \left (8 a A c^2 e^2-12 a b B c e^2+16 a B c^2 d e-6 A b^2 c e^2+16 A b c^2 d e-16 A c^3 d^2+5 b^3 B e^2-12 b^2 B c d e+8 b B c^2 d^2\right )}{16 c^{7/2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.51, size = 487, normalized size = 2.08 \begin {gather*} \left [\frac {3 \, {\left (8 \, {\left (B b c^{2} - 2 \, A c^{3}\right )} d^{2} - 4 \, {\left (3 \, B b^{2} c - 4 \, {\left (B a + A b\right )} c^{2}\right )} d e + {\left (5 \, B b^{3} + 8 \, A a c^{2} - 6 \, {\left (2 \, B a b + A b^{2}\right )} c\right )} e^{2}\right )} \sqrt {c} \log \left (-8 \, c^{2} x^{2} - 8 \, b c x - b^{2} + 4 \, \sqrt {c x^{2} + b x + a} {\left (2 \, c x + b\right )} \sqrt {c} - 4 \, a c\right ) + 4 \, {\left (8 \, B c^{3} e^{2} x^{2} + 24 \, B c^{3} d^{2} - 12 \, {\left (3 \, B b c^{2} - 4 \, A c^{3}\right )} d e + {\left (15 \, B b^{2} c - 2 \, {\left (8 \, B a + 9 \, A b\right )} c^{2}\right )} e^{2} + 2 \, {\left (12 \, B c^{3} d e - {\left (5 \, B b c^{2} - 6 \, A c^{3}\right )} e^{2}\right )} x\right )} \sqrt {c x^{2} + b x + a}}{96 \, c^{4}}, \frac {3 \, {\left (8 \, {\left (B b c^{2} - 2 \, A c^{3}\right )} d^{2} - 4 \, {\left (3 \, B b^{2} c - 4 \, {\left (B a + A b\right )} c^{2}\right )} d e + {\left (5 \, B b^{3} + 8 \, A a c^{2} - 6 \, {\left (2 \, B a b + A b^{2}\right )} c\right )} e^{2}\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {c x^{2} + b x + a} {\left (2 \, c x + b\right )} \sqrt {-c}}{2 \, {\left (c^{2} x^{2} + b c x + a c\right )}}\right ) + 2 \, {\left (8 \, B c^{3} e^{2} x^{2} + 24 \, B c^{3} d^{2} - 12 \, {\left (3 \, B b c^{2} - 4 \, A c^{3}\right )} d e + {\left (15 \, B b^{2} c - 2 \, {\left (8 \, B a + 9 \, A b\right )} c^{2}\right )} e^{2} + 2 \, {\left (12 \, B c^{3} d e - {\left (5 \, B b c^{2} - 6 \, A c^{3}\right )} e^{2}\right )} x\right )} \sqrt {c x^{2} + b x + a}}{48 \, c^{4}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.26, size = 231, normalized size = 0.99 \begin {gather*} \frac {1}{24} \, \sqrt {c x^{2} + b x + a} {\left (2 \, {\left (\frac {4 \, B x e^{2}}{c} + \frac {12 \, B c^{2} d e - 5 \, B b c e^{2} + 6 \, A c^{2} e^{2}}{c^{3}}\right )} x + \frac {24 \, B c^{2} d^{2} - 36 \, B b c d e + 48 \, A c^{2} d e + 15 \, B b^{2} e^{2} - 16 \, B a c e^{2} - 18 \, A b c e^{2}}{c^{3}}\right )} + \frac {{\left (8 \, B b c^{2} d^{2} - 16 \, A c^{3} d^{2} - 12 \, B b^{2} c d e + 16 \, B a c^{2} d e + 16 \, A b c^{2} d e + 5 \, B b^{3} e^{2} - 12 \, B a b c e^{2} - 6 \, A b^{2} c e^{2} + 8 \, A a c^{2} e^{2}\right )} \log \left ({\left | -2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} \sqrt {c} - b \right |}\right )}{16 \, c^{\frac {7}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [B] time = 0.06, size = 537, normalized size = 2.29 \begin {gather*} \frac {\sqrt {c \,x^{2}+b x +a}\, B \,e^{2} x^{2}}{3 c}-\frac {A a \,e^{2} \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{2 c^{\frac {3}{2}}}+\frac {3 A \,b^{2} e^{2} \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{8 c^{\frac {5}{2}}}-\frac {A b d e \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{c^{\frac {3}{2}}}+\frac {A \,d^{2} \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{\sqrt {c}}+\frac {3 B a b \,e^{2} \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{4 c^{\frac {5}{2}}}-\frac {B a d e \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{c^{\frac {3}{2}}}-\frac {5 B \,b^{3} e^{2} \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{16 c^{\frac {7}{2}}}+\frac {3 B \,b^{2} d e \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{4 c^{\frac {5}{2}}}-\frac {B b \,d^{2} \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{2 c^{\frac {3}{2}}}+\frac {\sqrt {c \,x^{2}+b x +a}\, A \,e^{2} x}{2 c}-\frac {5 \sqrt {c \,x^{2}+b x +a}\, B b \,e^{2} x}{12 c^{2}}+\frac {\sqrt {c \,x^{2}+b x +a}\, B d e x}{c}-\frac {3 \sqrt {c \,x^{2}+b x +a}\, A b \,e^{2}}{4 c^{2}}+\frac {2 \sqrt {c \,x^{2}+b x +a}\, A d e}{c}-\frac {2 \sqrt {c \,x^{2}+b x +a}\, B a \,e^{2}}{3 c^{2}}+\frac {5 \sqrt {c \,x^{2}+b x +a}\, B \,b^{2} e^{2}}{8 c^{3}}-\frac {3 \sqrt {c \,x^{2}+b x +a}\, B b d e}{2 c^{2}}+\frac {\sqrt {c \,x^{2}+b x +a}\, B \,d^{2}}{c} \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 = -0.00 \begin {gather*} \int \frac {\left (A+B\,x\right )\,{\left (d+e\,x\right )}^2}{\sqrt {c\,x^2+b\,x+a}} \,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 {\left (A + B x\right ) \left (d + e x\right )^{2}}{\sqrt {a + b x + c x^{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________