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 = 0.0624928, antiderivative size = 106, normalized size of antiderivative = 1., 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} \[ -\frac{(d+e x)^4 (A b-a B)}{4 (a+b x)^3 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^2}-\frac{(d+e x)^3 (B d-A e)}{3 \left (a^2+2 a b x+b^2 x^2\right )^{3/2} (b d-a e)^2} \]
Antiderivative was successfully verified.
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Rule 769
Rule 646
Rule 37
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*}
Mathematica [A] time = 0.0815082, size = 142, normalized size = 1.34 \[ \frac{-A b \left (a^2 e^2+2 a b e (d+2 e x)+b^2 \left (3 d^2+8 d e x+6 e^2 x^2\right )\right )-B \left (2 a^2 b e (d+6 e x)+3 a^3 e^2+a b^2 \left (d^2+8 d e x+18 e^2 x^2\right )+4 b^3 x \left (d^2+3 d e x+3 e^2 x^2\right )\right )}{12 b^4 (a+b x)^3 \sqrt{(a+b x)^2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.008, size = 174, normalized size = 1.6 \begin{align*} -{\frac{ \left ( bx+a \right ) \left ( 12\,B{x}^{3}{b}^{3}{e}^{2}+6\,A{x}^{2}{b}^{3}{e}^{2}+18\,B{x}^{2}a{b}^{2}{e}^{2}+12\,B{x}^{2}{b}^{3}de+4\,Axa{b}^{2}{e}^{2}+8\,Ax{b}^{3}de+12\,Bx{a}^{2}b{e}^{2}+8\,Bxa{b}^{2}de+4\,Bx{b}^{3}{d}^{2}+Ab{a}^{2}{e}^{2}+2\,Aa{b}^{2}de+3\,A{b}^{3}{d}^{2}+3\,B{a}^{3}{e}^{2}+2\,B{a}^{2}bde+Ba{b}^{2}{d}^{2} \right ) }{12\,{b}^{4}} \left ( \left ( bx+a \right ) ^{2} \right ) ^{-{\frac{5}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 0.996534, size = 448, normalized size = 4.23 \begin{align*} -\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 a^{3} b e^{2}}{4 \,{\left (b^{2}\right )}^{\frac{9}{2}}{\left (x + \frac{a}{b}\right )}^{4}} + \frac{2 \, B a^{2} e^{2}}{3 \,{\left (b^{2}\right )}^{\frac{7}{2}}{\left (x + \frac{a}{b}\right )}^{3}} - \frac{B a e^{2}}{2 \,{\left (b^{2}\right )}^{\frac{5}{2}} b{\left (x + \frac{a}{b}\right )}^{2}} - \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{{\left (2 \, B d e + A e^{2}\right )} a^{2} b^{2}}{4 \,{\left (b^{2}\right )}^{\frac{9}{2}}{\left (x + \frac{a}{b}\right )}^{4}} - \frac{A d^{2}}{4 \,{\left (b^{2}\right )}^{\frac{5}{2}}{\left (x + \frac{a}{b}\right )}^{4}} + \frac{B a^{3} e^{2}}{2 \,{\left (b^{2}\right )}^{\frac{5}{2}} b^{3}{\left (x + \frac{a}{b}\right )}^{4}} + \frac{2 \,{\left (2 \, B d e + A e^{2}\right )} a b}{3 \,{\left (b^{2}\right )}^{\frac{7}{2}}{\left (x + \frac{a}{b}\right )}^{3}} - \frac{2 \, B d e + A e^{2}}{2 \,{\left (b^{2}\right )}^{\frac{5}{2}}{\left (x + \frac{a}{b}\right )}^{2}} + \frac{{\left (B d^{2} + 2 \, A d e\right )} a}{4 \,{\left (b^{2}\right )}^{\frac{5}{2}} b{\left (x + \frac{a}{b}\right )}^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.32759, size = 392, normalized size = 3.7 \begin{align*} -\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{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \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{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \mathit{sage}_{0} x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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