Optimal. Leaf size=210 \[ \frac {e \sqrt {a+b x+c x^2} \left (-2 c (4 a B e+A b e+b B d)+4 A c^2 d+3 b^2 B e\right )}{c^2 \left (b^2-4 a c\right )}+\frac {2 (d+e x) \left (-x \left (2 c (A c d-a B e)-b c (A e+B d)+b^2 B e\right )-b (a B e+A c d)+2 a c (A e+B d)\right )}{c \left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}}+\frac {e \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right ) (2 A c e-3 b B e+4 B c d)}{2 c^{5/2}} \]
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Rubi [A] time = 0.19, antiderivative size = 210, 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 = {818, 640, 621, 206} \begin {gather*} \frac {e \sqrt {a+b x+c x^2} \left (-2 c (4 a B e+A b e+b B d)+4 A c^2 d+3 b^2 B e\right )}{c^2 \left (b^2-4 a c\right )}+\frac {2 (d+e x) \left (-x \left (2 c (A c d-a B e)-b c (A e+B d)+b^2 B e\right )-b (a B e+A c d)+2 a c (A e+B d)\right )}{c \left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}}+\frac {e \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right ) (2 A c e-3 b B e+4 B c d)}{2 c^{5/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 206
Rule 621
Rule 640
Rule 818
Rubi steps
\begin {align*} \int \frac {(A+B x) (d+e x)^2}{\left (a+b x+c x^2\right )^{3/2}} \, dx &=\frac {2 (d+e x) \left (2 a c (B d+A e)-b (A c d+a B e)-\left (b^2 B e-b c (B d+A e)+2 c (A c d-a B e)\right ) x\right )}{c \left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}}+\frac {2 \int \frac {\frac {1}{2} e \left (b^2 B d-4 a c (2 B d+A e)+2 b (A c d+a B e)\right )+\frac {1}{2} e \left (4 A c^2 d+3 b^2 B e-2 c (b B d+A b e+4 a B e)\right ) x}{\sqrt {a+b x+c x^2}} \, dx}{c \left (b^2-4 a c\right )}\\ &=\frac {2 (d+e x) \left (2 a c (B d+A e)-b (A c d+a B e)-\left (b^2 B e-b c (B d+A e)+2 c (A c d-a B e)\right ) x\right )}{c \left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}}+\frac {e \left (4 A c^2 d+3 b^2 B e-2 c (b B d+A b e+4 a B e)\right ) \sqrt {a+b x+c x^2}}{c^2 \left (b^2-4 a c\right )}+\frac {(e (4 B c d-3 b B e+2 A c e)) \int \frac {1}{\sqrt {a+b x+c x^2}} \, dx}{2 c^2}\\ &=\frac {2 (d+e x) \left (2 a c (B d+A e)-b (A c d+a B e)-\left (b^2 B e-b c (B d+A e)+2 c (A c d-a B e)\right ) x\right )}{c \left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}}+\frac {e \left (4 A c^2 d+3 b^2 B e-2 c (b B d+A b e+4 a B e)\right ) \sqrt {a+b x+c x^2}}{c^2 \left (b^2-4 a c\right )}+\frac {(e (4 B c d-3 b B e+2 A c e)) \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 )}{c^2}\\ &=\frac {2 (d+e x) \left (2 a c (B d+A e)-b (A c d+a B e)-\left (b^2 B e-b c (B d+A e)+2 c (A c d-a B e)\right ) x\right )}{c \left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}}+\frac {e \left (4 A c^2 d+3 b^2 B e-2 c (b B d+A b e+4 a B e)\right ) \sqrt {a+b x+c x^2}}{c^2 \left (b^2-4 a c\right )}+\frac {e (4 B c d-3 b B e+2 A c e) \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{2 c^{5/2}}\\ \end {align*}
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Mathematica [A] time = 0.45, size = 238, normalized size = 1.13 \begin {gather*} \frac {\frac {2 \sqrt {c} \left (B \left (8 a^2 c e^2+a \left (-3 b^2 e^2+2 b c e (2 d+5 e x)-4 c^2 \left (d^2+2 d e x-e^2 x^2\right )\right )-b x \left (3 b^2 e^2+b c e (e x-4 d)+2 c^2 d^2\right )\right )+2 A c \left (a b e^2-2 a c e (2 d+e x)+b^2 e^2 x+b c d (d-2 e x)+2 c^2 d^2 x\right )\right )}{\sqrt {a+x (b+c x)}}+e \left (b^2-4 a c\right ) \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+x (b+c x)}}\right ) (-2 A c e+3 b B e-4 B c d)}{2 c^{5/2} \left (4 a c-b^2\right )} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 1.35, size = 281, normalized size = 1.34 \begin {gather*} \frac {\log \left (-2 c^{5/2} \sqrt {a+b x+c x^2}+b c^2+2 c^3 x\right ) \left (-2 A c e^2+3 b B e^2-4 B c d e\right )}{2 c^{5/2}}-\frac {-8 a^2 B c e^2-2 a A b c e^2+8 a A c^2 d e+4 a A c^2 e^2 x+3 a b^2 B e^2-4 a b B c d e-10 a b B c e^2 x+4 a B c^2 d^2+8 a B c^2 d e x-4 a B c^2 e^2 x^2-2 A b^2 c e^2 x-2 A b c^2 d^2+4 A b c^2 d e x-4 A c^3 d^2 x+3 b^3 B e^2 x-4 b^2 B c d e x+b^2 B c e^2 x^2+2 b B c^2 d^2 x}{c^2 \left (4 a c-b^2\right ) \sqrt {a+b x+c x^2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 2.51, size = 945, normalized size = 4.50 \begin {gather*} \left [\frac {{\left (4 \, {\left (B a b^{2} c - 4 \, B a^{2} c^{2}\right )} d e - {\left (3 \, B a b^{3} + 8 \, A a^{2} c^{2} - 2 \, {\left (6 \, B a^{2} b + A a b^{2}\right )} c\right )} e^{2} + {\left (4 \, {\left (B b^{2} c^{2} - 4 \, B a c^{3}\right )} d e - {\left (3 \, B b^{3} c + 8 \, A a c^{3} - 2 \, {\left (6 \, B a b + A b^{2}\right )} c^{2}\right )} e^{2}\right )} x^{2} + {\left (4 \, {\left (B b^{3} c - 4 \, B a b c^{2}\right )} d e - {\left (3 \, B b^{4} + 8 \, A a b c^{2} - 2 \, {\left (6 \, B a b^{2} + A b^{3}\right )} c\right )} e^{2}\right )} x\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 (2 \, {\left (2 \, B a - A b\right )} c^{3} d^{2} + {\left (B b^{2} c^{2} - 4 \, B a c^{3}\right )} e^{2} x^{2} - 4 \, {\left (B a b c^{2} - 2 \, A a c^{3}\right )} d e + {\left (3 \, B a b^{2} c - 2 \, {\left (4 \, B a^{2} + A a b\right )} c^{2}\right )} e^{2} + {\left (2 \, {\left (B b c^{3} - 2 \, A c^{4}\right )} d^{2} - 4 \, {\left (B b^{2} c^{2} - {\left (2 \, B a + A b\right )} c^{3}\right )} d e + {\left (3 \, B b^{3} c + 4 \, A a c^{3} - 2 \, {\left (5 \, B a b + A b^{2}\right )} c^{2}\right )} e^{2}\right )} x\right )} \sqrt {c x^{2} + b x + a}}{4 \, {\left (a b^{2} c^{3} - 4 \, a^{2} c^{4} + {\left (b^{2} c^{4} - 4 \, a c^{5}\right )} x^{2} + {\left (b^{3} c^{3} - 4 \, a b c^{4}\right )} x\right )}}, -\frac {{\left (4 \, {\left (B a b^{2} c - 4 \, B a^{2} c^{2}\right )} d e - {\left (3 \, B a b^{3} + 8 \, A a^{2} c^{2} - 2 \, {\left (6 \, B a^{2} b + A a b^{2}\right )} c\right )} e^{2} + {\left (4 \, {\left (B b^{2} c^{2} - 4 \, B a c^{3}\right )} d e - {\left (3 \, B b^{3} c + 8 \, A a c^{3} - 2 \, {\left (6 \, B a b + A b^{2}\right )} c^{2}\right )} e^{2}\right )} x^{2} + {\left (4 \, {\left (B b^{3} c - 4 \, B a b c^{2}\right )} d e - {\left (3 \, B b^{4} + 8 \, A a b c^{2} - 2 \, {\left (6 \, B a b^{2} + A b^{3}\right )} c\right )} e^{2}\right )} x\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 (2 \, {\left (2 \, B a - A b\right )} c^{3} d^{2} + {\left (B b^{2} c^{2} - 4 \, B a c^{3}\right )} e^{2} x^{2} - 4 \, {\left (B a b c^{2} - 2 \, A a c^{3}\right )} d e + {\left (3 \, B a b^{2} c - 2 \, {\left (4 \, B a^{2} + A a b\right )} c^{2}\right )} e^{2} + {\left (2 \, {\left (B b c^{3} - 2 \, A c^{4}\right )} d^{2} - 4 \, {\left (B b^{2} c^{2} - {\left (2 \, B a + A b\right )} c^{3}\right )} d e + {\left (3 \, B b^{3} c + 4 \, A a c^{3} - 2 \, {\left (5 \, B a b + A b^{2}\right )} c^{2}\right )} e^{2}\right )} x\right )} \sqrt {c x^{2} + b x + a}}{2 \, {\left (a b^{2} c^{3} - 4 \, a^{2} c^{4} + {\left (b^{2} c^{4} - 4 \, a c^{5}\right )} x^{2} + {\left (b^{3} c^{3} - 4 \, a b c^{4}\right )} x\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.31, size = 294, normalized size = 1.40 \begin {gather*} \frac {{\left (\frac {{\left (B b^{2} c e^{2} - 4 \, B a c^{2} e^{2}\right )} x}{b^{2} c^{2} - 4 \, a c^{3}} + \frac {2 \, B b c^{2} d^{2} - 4 \, A c^{3} d^{2} - 4 \, B b^{2} c d e + 8 \, B a c^{2} d e + 4 \, A b c^{2} d e + 3 \, B b^{3} e^{2} - 10 \, B a b c e^{2} - 2 \, A b^{2} c e^{2} + 4 \, A a c^{2} e^{2}}{b^{2} c^{2} - 4 \, a c^{3}}\right )} x + \frac {4 \, B a c^{2} d^{2} - 2 \, A b c^{2} d^{2} - 4 \, B a b c d e + 8 \, A a c^{2} d e + 3 \, B a b^{2} e^{2} - 8 \, B a^{2} c e^{2} - 2 \, A a b c e^{2}}{b^{2} c^{2} - 4 \, a c^{3}}}{\sqrt {c x^{2} + b x + a}} - \frac {{\left (4 \, B c d e - 3 \, B b e^{2} + 2 \, A c e^{2}\right )} \log \left ({\left | -2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} \sqrt {c} - b \right |}\right )}{2 \, c^{\frac {5}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.05, size = 779, normalized size = 3.71 \begin {gather*} \frac {A \,b^{2} e^{2} x}{\left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}\, c}-\frac {4 A b d e x}{\left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}}+\frac {4 B a b \,e^{2} x}{\left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}\, c}-\frac {3 B \,b^{3} e^{2} x}{2 \left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}\, c^{2}}+\frac {2 B \,b^{2} d e x}{\left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}\, c}-\frac {2 B b \,d^{2} x}{\left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}}+\frac {A \,b^{3} e^{2}}{2 \left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}\, c^{2}}-\frac {2 A \,b^{2} d e}{\left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}\, c}+\frac {2 B a \,b^{2} e^{2}}{\left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}\, c^{2}}-\frac {3 B \,b^{4} e^{2}}{4 \left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}\, c^{3}}+\frac {B \,b^{3} d e}{\left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}\, c^{2}}-\frac {B \,b^{2} d^{2}}{\left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}\, c}+\frac {B \,e^{2} x^{2}}{\sqrt {c \,x^{2}+b x +a}\, c}-\frac {A \,e^{2} x}{\sqrt {c \,x^{2}+b x +a}\, c}+\frac {2 \left (2 c x +b \right ) A \,d^{2}}{\left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}}+\frac {3 B b \,e^{2} x}{2 \sqrt {c \,x^{2}+b x +a}\, c^{2}}-\frac {2 B d e x}{\sqrt {c \,x^{2}+b x +a}\, c}+\frac {A \,e^{2} \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{c^{\frac {3}{2}}}-\frac {3 B b \,e^{2} \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{2 c^{\frac {5}{2}}}+\frac {2 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 b \,e^{2}}{2 \sqrt {c \,x^{2}+b x +a}\, c^{2}}-\frac {2 A d e}{\sqrt {c \,x^{2}+b x +a}\, c}+\frac {2 B a \,e^{2}}{\sqrt {c \,x^{2}+b x +a}\, c^{2}}-\frac {3 B \,b^{2} e^{2}}{4 \sqrt {c \,x^{2}+b x +a}\, c^{3}}+\frac {B b d e}{\sqrt {c \,x^{2}+b x +a}\, c^{2}}-\frac {B \,d^{2}}{\sqrt {c \,x^{2}+b x +a}\, c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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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.
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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}{{\left (c\,x^2+b\,x+a\right )}^{3/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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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}}{\left (a + b x + c x^{2}\right )^{\frac {3}{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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