Optimal. Leaf size=106 \[ -\frac {(d+e x)^4 (A b-a B)}{4 (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^2}-\frac {(d+e x)^3 (B d-A e)}{3 \left (a^2+2 a b x+b^2 x^2\right )^{3/2} (b d-a e)^2} \]
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Rubi [A] time = 0.06, antiderivative size = 106, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {769, 646, 37} \begin {gather*} -\frac {(d+e x)^4 (A b-a B)}{4 (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^2}-\frac {(d+e x)^3 (B d-A e)}{3 \left (a^2+2 a b x+b^2 x^2\right )^{3/2} (b d-a e)^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 37
Rule 646
Rule 769
Rubi steps
\begin {align*} \int \frac {(A+B x) (d+e x)^2}{\left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx &=-\frac {(B d-A e) (d+e x)^3}{3 (b d-a e)^2 \left (a^2+2 a b x+b^2 x^2\right )^{3/2}}+\frac {(A b-a B) \int \frac {(d+e x)^3}{\left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx}{b d-a e}\\ &=-\frac {(B d-A e) (d+e x)^3}{3 (b d-a e)^2 \left (a^2+2 a b x+b^2 x^2\right )^{3/2}}+\frac {\left (b^4 (A b-a B) \left (a b+b^2 x\right )\right ) \int \frac {(d+e x)^3}{\left (a b+b^2 x\right )^5} \, dx}{(b d-a e) \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=-\frac {(B d-A e) (d+e x)^3}{3 (b d-a e)^2 \left (a^2+2 a b x+b^2 x^2\right )^{3/2}}-\frac {(A b-a B) (d+e x)^4}{4 (b d-a e)^2 (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}\\ \end {align*}
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Mathematica [A] time = 0.08, size = 142, normalized size = 1.34 \begin {gather*} \frac {-A b \left (a^2 e^2+2 a b e (d+2 e x)+b^2 \left (3 d^2+8 d e x+6 e^2 x^2\right )\right )-B \left (3 a^3 e^2+2 a^2 b e (d+6 e x)+a b^2 \left (d^2+8 d e x+18 e^2 x^2\right )+4 b^3 x \left (d^2+3 d e x+3 e^2 x^2\right )\right )}{12 b^4 (a+b x)^3 \sqrt {(a+b x)^2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [B] time = 2.58, size = 837, normalized size = 7.90 \begin {gather*} \frac {-2 \sqrt {b^2} \sqrt {a^2+2 b x a+b^2 x^2} \left (3 B e^2 a^6-3 A b e^2 a^5-6 b B d e a^5-3 b B e^2 x a^5+3 b^2 B d^2 a^4+3 b^2 B e^2 x^2 a^4+6 A b^2 d e a^4+3 A b^2 e^2 x a^4+6 b^2 B d e x a^4-3 b^3 B e^2 x^3 a^3-3 A b^3 d^2 a^3-3 A b^3 e^2 x^2 a^3-6 b^3 B d e x^2 a^3-3 b^3 B d^2 x a^3-6 A b^3 d e x a^3+6 b^4 B e^2 x^4 a^2+3 A b^4 e^2 x^3 a^2+6 b^4 B d e x^3 a^2+3 b^4 B d^2 x^2 a^2+6 A b^4 d e x^2 a^2+3 A b^4 d^2 x a^2+6 b^5 B e^2 x^5 a-2 A b^5 e^2 x^4 a-4 b^5 B d e x^4 a-3 b^5 B d^2 x^3 a-6 A b^5 d e x^3 a-3 A b^5 d^2 x^2 a+12 b^6 B e^2 x^6+6 A b^6 e^2 x^5+12 b^6 B d e x^5+4 b^6 B d^2 x^4+8 A b^6 d e x^4+3 A b^6 d^2 x^3\right )-2 \left (-12 B e^2 x^7 b^8-6 A e^2 x^6 b^8-12 B d e x^6 b^8-4 B d^2 x^5 b^8-8 A d e x^5 b^8-3 A d^2 x^4 b^8-18 a B e^2 x^6 b^7-4 a A e^2 x^5 b^7-8 a B d e x^5 b^7-a B d^2 x^4 b^7-2 a A d e x^4 b^7-12 a^2 B e^2 x^5 b^6-a^2 A e^2 x^4 b^6-2 a^2 B d e x^4 b^6-3 a^3 B e^2 x^4 b^5-3 a^4 A d^2 b^4+3 a^5 B d^2 b^3+6 a^5 A d e b^3-3 a^6 A e^2 b^2-6 a^6 B d e b^2+3 a^7 B e^2 b\right )}{3 b^4 \sqrt {a^2+2 b x a+b^2 x^2} \left (-8 x^3 b^8-24 a x^2 b^7-24 a^2 x b^6-8 a^3 b^5\right ) x^4+3 b^4 \sqrt {b^2} \left (8 x^4 b^8+32 a x^3 b^7+48 a^2 x^2 b^6+32 a^3 x b^5+8 a^4 b^4\right ) x^4} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 0.46, size = 189, normalized size = 1.78 \begin {gather*} -\frac {12 \, B b^{3} e^{2} x^{3} + {\left (B a b^{2} + 3 \, A b^{3}\right )} d^{2} + 2 \, {\left (B a^{2} b + A a b^{2}\right )} d e + {\left (3 \, B a^{3} + A a^{2} b\right )} e^{2} + 6 \, {\left (2 \, B b^{3} d e + {\left (3 \, B a b^{2} + A b^{3}\right )} e^{2}\right )} x^{2} + 4 \, {\left (B b^{3} d^{2} + 2 \, {\left (B a b^{2} + A b^{3}\right )} d e + {\left (3 \, B a^{2} b + A a b^{2}\right )} e^{2}\right )} x}{12 \, {\left (b^{8} x^{4} + 4 \, a b^{7} x^{3} + 6 \, a^{2} b^{6} x^{2} + 4 \, a^{3} b^{5} x + a^{4} b^{4}\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 = 174, normalized size = 1.64 \begin {gather*} -\frac {\left (b x +a \right ) \left (12 B \,b^{3} e^{2} x^{3}+6 A \,b^{3} e^{2} x^{2}+18 B a \,b^{2} e^{2} x^{2}+12 B \,b^{3} d e \,x^{2}+4 A a \,b^{2} e^{2} x +8 A \,b^{3} d e x +12 B \,a^{2} b \,e^{2} x +8 B a \,b^{2} d e x +4 B \,b^{3} d^{2} x +A \,a^{2} b \,e^{2}+2 A a \,b^{2} d e +3 A \,b^{3} d^{2}+3 B \,a^{3} e^{2}+2 B \,a^{2} b d e +B a \,b^{2} d^{2}\right )}{12 \left (\left (b x +a \right )^{2}\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 = 0.63, size = 279, normalized size = 2.63 \begin {gather*} -\frac {B e^{2} x^{2}}{{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} b^{2}} - \frac {2 \, B a^{2} e^{2}}{3 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} b^{4}} - \frac {B d^{2} + 2 \, A d e}{3 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} b^{2}} - \frac {B a e^{2}}{2 \, b^{6} {\left (x + \frac {a}{b}\right )}^{2}} + \frac {2 \, B a^{2} e^{2}}{3 \, b^{7} {\left (x + \frac {a}{b}\right )}^{3}} - \frac {A d^{2}}{4 \, b^{5} {\left (x + \frac {a}{b}\right )}^{4}} + \frac {B a^{3} e^{2}}{4 \, b^{8} {\left (x + \frac {a}{b}\right )}^{4}} - \frac {2 \, B d e + A e^{2}}{2 \, b^{5} {\left (x + \frac {a}{b}\right )}^{2}} + \frac {2 \, {\left (2 \, B d e + A e^{2}\right )} a}{3 \, b^{6} {\left (x + \frac {a}{b}\right )}^{3}} - \frac {{\left (2 \, B d e + A e^{2}\right )} a^{2}}{4 \, b^{7} {\left (x + \frac {a}{b}\right )}^{4}} + \frac {{\left (B d^{2} + 2 \, A d 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 = 2.36, size = 302, normalized size = 2.85 \begin {gather*} -\frac {\left (\frac {A\,d^2}{4\,b}-\frac {a\,\left (\frac {B\,d^2+2\,A\,e\,d}{4\,b}-\frac {a\,\left (\frac {A\,e^2+2\,B\,d\,e}{4\,b}-\frac {B\,a\,e^2}{4\,b^2}\right )}{b}\right )}{b}\right )\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{{\left (a+b\,x\right )}^5}-\frac {\left (\frac {A\,b\,e^2-2\,B\,a\,e^2+2\,B\,b\,d\,e}{2\,b^4}-\frac {B\,a\,e^2}{2\,b^4}\right )\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{{\left (a+b\,x\right )}^3}-\frac {\left (\frac {B\,a^2\,e^2-2\,B\,a\,b\,d\,e-A\,a\,b\,e^2+B\,b^2\,d^2+2\,A\,b^2\,d\,e}{3\,b^4}-\frac {a\,\left (\frac {e\,\left (A\,b\,e-B\,a\,e+2\,B\,b\,d\right )}{3\,b^3}-\frac {B\,a\,e^2}{3\,b^3}\right )}{b}\right )\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{{\left (a+b\,x\right )}^4}-\frac {B\,e^2\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{b^4\,{\left (a+b\,x\right )}^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 )^{2}}{\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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