Optimal. Leaf size=111 \[ \frac {2 (b+2 c x) \left (-4 b c (A e+B d)+8 A c^2 d+b^2 B e\right )}{3 b^4 c \sqrt {b x+c x^2}}-\frac {2 \left (x \left (-b c (A e+B d)+2 A c^2 d+b^2 B e\right )+A b c d\right )}{3 b^2 c \left (b x+c x^2\right )^{3/2}} \]
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Rubi [A] time = 0.10, antiderivative size = 111, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {777, 613} \begin {gather*} \frac {2 (b+2 c x) \left (-4 b c (A e+B d)+8 A c^2 d+b^2 B e\right )}{3 b^4 c \sqrt {b x+c x^2}}-\frac {2 \left (x \left (-b c (A e+B d)+2 A c^2 d+b^2 B e\right )+A b c d\right )}{3 b^2 c \left (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 (b x+c x^2\right )^{5/2}} \, dx &=-\frac {2 \left (A b c d+\left (2 A c^2 d+b^2 B e-b c (B d+A e)\right ) x\right )}{3 b^2 c \left (b x+c x^2\right )^{3/2}}-\frac {\left (8 A c^2 d+b^2 B e-4 b c (B d+A e)\right ) \int \frac {1}{\left (b x+c x^2\right )^{3/2}} \, dx}{3 b^2 c}\\ &=-\frac {2 \left (A b c d+\left (2 A c^2 d+b^2 B e-b c (B d+A e)\right ) x\right )}{3 b^2 c \left (b x+c x^2\right )^{3/2}}+\frac {2 \left (8 A c^2 d+b^2 B e-4 b c (B d+A e)\right ) (b+2 c x)}{3 b^4 c \sqrt {b x+c x^2}}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 107, normalized size = 0.96 \begin {gather*} -\frac {2 \left (A \left (b^3 (d+3 e x)-6 b^2 c x (d-2 e x)+8 b c^2 x^2 (e x-3 d)-16 c^3 d x^3\right )+b 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 )}{3 b^4 (x (b+c x))^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.46, size = 149, normalized size = 1.34 \begin {gather*} \frac {2 \sqrt {b x+c x^2} \left (-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 b^4 x^2 (b+c x)^2} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.41, size = 152, normalized size = 1.37 \begin {gather*} -\frac {2 \, {\left (A b^{3} d + 2 \, {\left (4 \, {\left (B b c^{2} - 2 \, A c^{3}\right )} d - {\left (B b^{2} c - 4 \, A b 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 \, A b^{2} c\right )} e\right )} x^{2} + 3 \, {\left (A b^{3} e + {\left (B b^{3} - 2 \, A b^{2} c\right )} d\right )} x\right )} \sqrt {c x^{2} + b x}}{3 \, {\left (b^{4} c^{2} x^{4} + 2 \, b^{5} c x^{3} + b^{6} x^{2}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.30, size = 132, normalized size = 1.19 \begin {gather*} -\frac {2 \, {\left ({\left (x {\left (\frac {2 \, {\left (4 \, B b c^{2} d - 8 \, A c^{3} d - B b^{2} c e + 4 \, A b c^{2} e\right )} x}{b^{4}} + \frac {3 \, {\left (4 \, B b^{2} c d - 8 \, A b c^{2} d - B b^{3} e + 4 \, A b^{2} c e\right )}}{b^{4}}\right )} + \frac {3 \, {\left (B b^{3} d - 2 \, A b^{2} c d + A b^{3} e\right )}}{b^{4}}\right )} x + \frac {A d}{b}\right )}}{3 \, {\left (c x^{2} + b x\right )}^{\frac {3}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 141, normalized size = 1.27 \begin {gather*} -\frac {2 \left (c x +b \right ) \left (8 A b \,c^{2} e \,x^{3}-16 A \,c^{3} d \,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}-3 B \,b^{3} e \,x^{2}+12 B \,b^{2} c d \,x^{2}+3 A \,b^{3} e x -6 A \,b^{2} c d x +3 B \,b^{3} d x +A d \,b^{3}\right ) x}{3 \left (c \,x^{2}+b x \right )^{\frac {5}{2}} b^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.54, size = 211, normalized size = 1.90 \begin {gather*} -\frac {4 \, A c d x}{3 \, {\left (c x^{2} + b x\right )}^{\frac {3}{2}} b^{2}} + \frac {32 \, A c^{2} d x}{3 \, \sqrt {c x^{2} + b x} b^{4}} + \frac {4 \, B e x}{3 \, \sqrt {c x^{2} + b x} b^{2}} - \frac {2 \, B e x}{3 \, {\left (c x^{2} + b x\right )}^{\frac {3}{2}} c} - \frac {2 \, A d}{3 \, {\left (c x^{2} + b x\right )}^{\frac {3}{2}} b} + \frac {16 \, A c d}{3 \, \sqrt {c x^{2} + b x} b^{3}} + \frac {2 \, B e}{3 \, \sqrt {c x^{2} + b x} b c} + \frac {2 \, {\left (B d + A e\right )} x}{3 \, {\left (c x^{2} + b x\right )}^{\frac {3}{2}} b} - \frac {16 \, {\left (B d + A e\right )} c x}{3 \, \sqrt {c x^{2} + b x} b^{3}} - \frac {8 \, {\left (B d + A e\right )}}{3 \, \sqrt {c x^{2} + b x} b^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.73, size = 134, normalized size = 1.21 \begin {gather*} -\frac {2\,\left (A\,b^3\,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\,b\,c^2\,d\,x^3-2\,B\,b^2\,c\,e\,x^3-6\,A\,b^2\,c\,d\,x\right )}{3\,b^4\,{\left (c\,x^2+b\,x\right )}^{3/2}} \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 )}{\left (x \left (b + c x\right )\right )^{\frac {5}{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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