Optimal. Leaf size=158 \[ \frac {2 (b+2 c x) \left (4 c (a B e+2 A c d)-4 b c (A e+B d)+b^2 B e\right )}{3 c \left (b^2-4 a c\right )^2 \sqrt {a+b x+c x^2}}+\frac {2 \left (-x \left (2 c (A c d-a B e)-b c (A e+B d)+b^2 B e\right )-b (a B e+A c d)+2 a c (A e+B d)\right )}{3 c \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2}} \]
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Rubi [A] time = 0.12, antiderivative size = 158, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.080, Rules used = {777, 613} \begin {gather*} \frac {2 (b+2 c x) \left (4 c (a B e+2 A c d)-4 b c (A e+B d)+b^2 B e\right )}{3 c \left (b^2-4 a c\right )^2 \sqrt {a+b x+c x^2}}+\frac {2 \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 )}{3 c \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 613
Rule 777
Rubi steps
\begin {align*} \int \frac {(A+B x) (d+e x)}{\left (a+b x+c x^2\right )^{5/2}} \, dx &=\frac {2 \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 )}{3 c \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2}}-\frac {\left (b^2 B e-4 b c (B d+A e)+4 c (2 A c d+a B e)\right ) \int \frac {1}{\left (a+b x+c x^2\right )^{3/2}} \, dx}{3 c \left (b^2-4 a c\right )}\\ &=\frac {2 \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 )}{3 c \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2}}+\frac {2 \left (b^2 B e-4 b c (B d+A e)+4 c (2 A c d+a B e)\right ) (b+2 c x)}{3 c \left (b^2-4 a c\right )^2 \sqrt {a+b x+c x^2}}\\ \end {align*}
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Mathematica [A] time = 0.31, size = 200, normalized size = 1.27 \begin {gather*} -\frac {2 \left (A \left (8 c \left (a^2 e-3 a c d x-2 c^2 d x^3\right )+2 b^2 \left (a e-3 c d x+6 c e x^2\right )+4 b c \left (-3 a d+3 a e x-6 c d x^2+2 c e x^3\right )+b^3 (d+3 e x)\right )+B \left (8 a^2 (c d-b e)+2 a \left (b^2 (d-6 e x)+6 b c x (d-e x)-4 c^2 e x^3\right )+b x \left (3 b^2 (d-e x)-2 b c x (e x-6 d)+8 c^2 d x^2\right )\right )\right )}{3 \left (b^2-4 a c\right )^2 (a+x (b+c x))^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 1.77, size = 248, normalized size = 1.57 \begin {gather*} -\frac {2 \left (8 a^2 A c e-8 a^2 b B e+8 a^2 B c d+2 a A b^2 e-12 a A b c d+12 a A b c e x-24 a A c^2 d x+2 a b^2 B d-12 a b^2 B e x+12 a b B c d x-12 a b B c e x^2-8 a B c^2 e x^3+A b^3 d+3 A b^3 e x-6 A b^2 c d x+12 A b^2 c e x^2-24 A b c^2 d x^2+8 A b c^2 e x^3-16 A c^3 d x^3+3 b^3 B d x-3 b^3 B e x^2+12 b^2 B c d x^2-2 b^2 B c e x^3+8 b B c^2 d x^3\right )}{3 \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 8.91, size = 353, normalized size = 2.23 \begin {gather*} -\frac {2 \, {\left (2 \, {\left (4 \, {\left (B b c^{2} - 2 \, A c^{3}\right )} d - {\left (B b^{2} c + 4 \, {\left (B a - A b\right )} c^{2}\right )} e\right )} x^{3} + 3 \, {\left (4 \, {\left (B b^{2} c - 2 \, A b c^{2}\right )} d - {\left (B b^{3} + 4 \, {\left (B a b - A b^{2}\right )} c\right )} e\right )} x^{2} + {\left (2 \, B a b^{2} + A b^{3} + 4 \, {\left (2 \, B a^{2} - 3 \, A a b\right )} c\right )} d - 2 \, {\left (4 \, B a^{2} b - A a b^{2} - 4 \, A a^{2} c\right )} e + 3 \, {\left ({\left (B b^{3} - 8 \, A a c^{2} + 2 \, {\left (2 \, B a b - A b^{2}\right )} c\right )} d - {\left (4 \, B a b^{2} - A b^{3} - 4 \, A a b c\right )} e\right )} x\right )} \sqrt {c x^{2} + b x + a}}{3 \, {\left (a^{2} b^{4} - 8 \, a^{3} b^{2} c + 16 \, a^{4} c^{2} + {\left (b^{4} c^{2} - 8 \, a b^{2} c^{3} + 16 \, a^{2} c^{4}\right )} x^{4} + 2 \, {\left (b^{5} c - 8 \, a b^{3} c^{2} + 16 \, a^{2} b c^{3}\right )} x^{3} + {\left (b^{6} - 6 \, a b^{4} c + 32 \, a^{3} c^{3}\right )} x^{2} + 2 \, {\left (a b^{5} - 8 \, a^{2} b^{3} c + 16 \, a^{3} b c^{2}\right )} x\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.33, size = 309, normalized size = 1.96 \begin {gather*} -\frac {2 \, {\left ({\left ({\left (\frac {2 \, {\left (4 \, B b c^{2} d - 8 \, A c^{3} d - B b^{2} c e - 4 \, B a c^{2} e + 4 \, A b c^{2} e\right )} x}{b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}} + \frac {3 \, {\left (4 \, B b^{2} c d - 8 \, A b c^{2} d - B b^{3} e - 4 \, B a b c e + 4 \, A b^{2} c e\right )}}{b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}}\right )} x + \frac {3 \, {\left (B b^{3} d + 4 \, B a b c d - 2 \, A b^{2} c d - 8 \, A a c^{2} d - 4 \, B a b^{2} e + A b^{3} e + 4 \, A a b c e\right )}}{b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}}\right )} x + \frac {2 \, B a b^{2} d + A b^{3} d + 8 \, B a^{2} c d - 12 \, A a b c d - 8 \, B a^{2} b e + 2 \, A a b^{2} e + 8 \, A a^{2} c e}{b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}}\right )}}{3 \, {\left (c x^{2} + b x + a\right )}^{\frac {3}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 256, normalized size = 1.62 \begin {gather*} -\frac {2 \left (8 A b \,c^{2} e \,x^{3}-16 A \,c^{3} d \,x^{3}-8 B a \,c^{2} e \,x^{3}-2 B \,b^{2} c e \,x^{3}+8 B b \,c^{2} d \,x^{3}+12 A \,b^{2} c e \,x^{2}-24 A b \,c^{2} d \,x^{2}-12 B a b c e \,x^{2}-3 B \,b^{3} e \,x^{2}+12 B \,b^{2} c d \,x^{2}+12 A a b c e x -24 A a \,c^{2} d x +3 A \,b^{3} e x -6 A \,b^{2} c d x -12 B a \,b^{2} e x +12 B a b c d x +3 B \,b^{3} d x +8 A \,a^{2} c e +2 A a \,b^{2} e -12 A a b c d +A \,b^{3} d -8 B \,a^{2} b e +8 B \,a^{2} c d +2 B a \,b^{2} d \right )}{3 \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )} \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 [B] time = 2.82, size = 246, normalized size = 1.56 \begin {gather*} -\frac {2\,\left (A\,b^3\,d+2\,A\,a\,b^2\,e+2\,B\,a\,b^2\,d+8\,A\,a^2\,c\,e-8\,B\,a^2\,b\,e+8\,B\,a^2\,c\,d+3\,A\,b^3\,e\,x+3\,B\,b^3\,d\,x-16\,A\,c^3\,d\,x^3-3\,B\,b^3\,e\,x^2-24\,A\,b\,c^2\,d\,x^2+12\,A\,b^2\,c\,e\,x^2+12\,B\,b^2\,c\,d\,x^2+8\,A\,b\,c^2\,e\,x^3-8\,B\,a\,c^2\,e\,x^3+8\,B\,b\,c^2\,d\,x^3-2\,B\,b^2\,c\,e\,x^3-12\,A\,a\,b\,c\,d-24\,A\,a\,c^2\,d\,x-6\,A\,b^2\,c\,d\,x-12\,B\,a\,b^2\,e\,x-12\,B\,a\,b\,c\,e\,x^2+12\,A\,a\,b\,c\,e\,x+12\,B\,a\,b\,c\,d\,x\right )}{3\,{\left (4\,a\,c-b^2\right )}^2\,{\left (c\,x^2+b\,x+a\right )}^{3/2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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