Optimal. Leaf size=135 \[ -\frac {-2 a B e+A b e+b B d}{3 b^3 (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {(A b-a B) (b d-a e)}{4 b^3 (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {B e}{2 b^3 (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}} \]
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Rubi [A] time = 0.10, antiderivative size = 135, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.065, Rules used = {770, 77} \begin {gather*} -\frac {-2 a B e+A b e+b B d}{3 b^3 (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {(A b-a B) (b d-a e)}{4 b^3 (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {B e}{2 b^3 (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 77
Rule 770
Rubi steps
\begin {align*} \int \frac {(A+B x) (d+e x)}{\left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx &=\frac {\left (b^4 \left (a b+b^2 x\right )\right ) \int \frac {(A+B x) (d+e x)}{\left (a b+b^2 x\right )^5} \, dx}{\sqrt {a^2+2 a b x+b^2 x^2}}\\ &=\frac {\left (b^4 \left (a b+b^2 x\right )\right ) \int \left (\frac {(A b-a B) (b d-a e)}{b^7 (a+b x)^5}+\frac {b B d+A b e-2 a B e}{b^7 (a+b x)^4}+\frac {B e}{b^7 (a+b x)^3}\right ) \, dx}{\sqrt {a^2+2 a b x+b^2 x^2}}\\ &=-\frac {(A b-a B) (b d-a e)}{4 b^3 (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {b B d+A b e-2 a B e}{3 b^3 (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {B e}{2 b^3 (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 75, normalized size = 0.56 \begin {gather*} \frac {-B \left (a^2 e+a b (d+4 e x)+2 b^2 x (2 d+3 e x)\right )-A b (a e+3 b d+4 b e x)}{12 b^3 (a+b x)^3 \sqrt {(a+b x)^2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [B] time = 1.59, size = 497, normalized size = 3.68 \begin {gather*} \frac {-2 \left (-3 a^6 b B e+3 a^5 A b^2 e+3 a^5 b^2 B d-3 a^4 A b^3 d-a^2 b^5 B e x^4-a A b^6 e x^4-a b^6 B d x^4-4 a b^6 B e x^5-3 A b^7 d x^4-4 A b^7 e x^5-4 b^7 B d x^5-6 b^7 B e x^6\right )-2 \sqrt {b^2} \sqrt {a^2+2 a b x+b^2 x^2} \left (-3 a^5 B e+3 a^4 A b e+3 a^4 b B d+3 a^4 b B e x-3 a^3 A b^2 d-3 a^3 A b^2 e x-3 a^3 b^2 B d x-3 a^3 b^2 B e x^2+3 a^2 A b^3 d x+3 a^2 A b^3 e x^2+3 a^2 b^3 B d x^2+3 a^2 b^3 B e x^3-3 a A b^4 d x^2-3 a A b^4 e x^3-3 a b^4 B d x^3-2 a b^4 B e x^4+3 A b^5 d x^3+4 A b^5 e x^4+4 b^5 B d x^4+6 b^5 B e x^5\right )}{3 b^3 x^4 \sqrt {a^2+2 a b x+b^2 x^2} \left (-8 a^3 b^5-24 a^2 b^6 x-24 a b^7 x^2-8 b^8 x^3\right )+3 b^3 \sqrt {b^2} x^4 \left (8 a^4 b^4+32 a^3 b^5 x+48 a^2 b^6 x^2+32 a b^7 x^3+8 b^8 x^4\right )} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.44, size = 106, normalized size = 0.79 \begin {gather*} -\frac {6 \, B b^{2} e x^{2} + {\left (B a b + 3 \, A b^{2}\right )} d + {\left (B a^{2} + A a b\right )} e + 4 \, {\left (B b^{2} d + {\left (B a b + A b^{2}\right )} e\right )} x}{12 \, {\left (b^{7} x^{4} + 4 \, a b^{6} x^{3} + 6 \, a^{2} b^{5} x^{2} + 4 \, a^{3} b^{4} x + a^{4} b^{3}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \mathit {sage}_{0} x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 77, normalized size = 0.57 \begin {gather*} -\frac {\left (b x +a \right ) \left (6 B \,b^{2} e \,x^{2}+4 A \,b^{2} e x +4 B a b e x +4 B \,b^{2} d x +A a b e +3 A \,b^{2} d +B \,a^{2} e +B a b d \right )}{12 \left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}} b^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.68, size = 121, normalized size = 0.90 \begin {gather*} -\frac {B d + A e}{3 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} b^{2}} - \frac {B e}{2 \, b^{5} {\left (x + \frac {a}{b}\right )}^{2}} + \frac {2 \, B a e}{3 \, b^{6} {\left (x + \frac {a}{b}\right )}^{3}} - \frac {A d}{4 \, b^{5} {\left (x + \frac {a}{b}\right )}^{4}} - \frac {B a^{2} e}{4 \, b^{7} {\left (x + \frac {a}{b}\right )}^{4}} + \frac {{\left (B d + A e\right )} a}{4 \, b^{6} {\left (x + \frac {a}{b}\right )}^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.24, size = 87, normalized size = 0.64 \begin {gather*} -\frac {\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}\,\left (3\,A\,b^2\,d+B\,a^2\,e+4\,A\,b^2\,e\,x+4\,B\,b^2\,d\,x+6\,B\,b^2\,e\,x^2+A\,a\,b\,e+B\,a\,b\,d+4\,B\,a\,b\,e\,x\right )}{12\,b^3\,{\left (a+b\,x\right )}^5} \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 (\left (a + b x\right )^{2}\right )^{\frac {5}{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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