Optimal. Leaf size=191 \[ \frac {(a+b x) (d+e x)^2 (A b-a B)}{2 b^2 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {(a+b x) (A b-a B) (b d-a e)^2 \log (a+b x)}{b^4 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {e x (a+b x) (A b-a B) (b d-a e)}{b^3 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {B (a+b x) (d+e x)^3}{3 b e \sqrt {a^2+2 a b x+b^2 x^2}} \]
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Rubi [A] time = 0.11, antiderivative size = 191, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.061, Rules used = {770, 77} \begin {gather*} \frac {(a+b x) (d+e x)^2 (A b-a B)}{2 b^2 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {e x (a+b x) (A b-a B) (b d-a e)}{b^3 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {(a+b x) (A b-a B) (b d-a e)^2 \log (a+b x)}{b^4 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {B (a+b x) (d+e x)^3}{3 b e \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)^2}{\sqrt {a^2+2 a b x+b^2 x^2}} \, dx &=\frac {\left (a b+b^2 x\right ) \int \frac {(A+B x) (d+e x)^2}{a b+b^2 x} \, dx}{\sqrt {a^2+2 a b x+b^2 x^2}}\\ &=\frac {\left (a b+b^2 x\right ) \int \left (\frac {(A b-a B) e (b d-a e)}{b^4}+\frac {(A b-a B) (b d-a e)^2}{b^4 (a+b x)}+\frac {(A b-a B) e (d+e x)}{b^3}+\frac {B (d+e x)^2}{b^2}\right ) \, dx}{\sqrt {a^2+2 a b x+b^2 x^2}}\\ &=\frac {(A b-a B) e (b d-a e) x (a+b x)}{b^3 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {(A b-a B) (a+b x) (d+e x)^2}{2 b^2 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {B (a+b x) (d+e x)^3}{3 b e \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {(A b-a B) (b d-a e)^2 (a+b x) \log (a+b x)}{b^4 \sqrt {a^2+2 a b x+b^2 x^2}}\\ \end {align*}
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Mathematica [A] time = 0.08, size = 118, normalized size = 0.62 \begin {gather*} \frac {(a+b x) \left (b x \left (6 a^2 B e^2-3 a b e (2 A e+4 B d+B e x)+b^2 \left (3 A e (4 d+e x)+2 B \left (3 d^2+3 d e x+e^2 x^2\right )\right )\right )+6 (A b-a B) (b d-a e)^2 \log (a+b x)\right )}{6 b^4 \sqrt {(a+b x)^2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [B] time = 1.11, size = 610, normalized size = 3.19 \begin {gather*} \frac {-6 a^2 B e^2 x+6 a A b e^2 x+12 a b B d e x+3 a b B e^2 x^2-12 A b^2 d e x-3 A b^2 e^2 x^2-6 b^2 B d^2 x-6 b^2 B d e x^2-2 b^2 B e^2 x^3}{12 \left (b^2\right )^{3/2}}+\frac {\sqrt {a^2+2 a b x+b^2 x^2} \left (11 a^2 B e^2-9 a A b e^2-18 a b B d e-5 a b B e^2 x+12 A b^2 d e+3 A b^2 e^2 x+6 b^2 B d^2+6 b^2 B d e x+2 b^2 B e^2 x^2\right )}{12 b^4}+\frac {\log \left (\sqrt {a^2+2 a b x+b^2 x^2}-a-\sqrt {b^2} x\right ) \left (a^3 \sqrt {b^2} B e^2+a^3 b B e^2-a^2 A b^2 e^2-a^2 A \sqrt {b^2} b e^2-2 a^2 b^2 B d e-2 a^2 \sqrt {b^2} b B d e+2 a A b^3 d e+2 a A \left (b^2\right )^{3/2} d e+a b^3 B d^2+a \left (b^2\right )^{3/2} B d^2-A b^4 d^2-A \sqrt {b^2} b^3 d^2\right )}{2 b^4 \sqrt {b^2}}+\frac {\log \left (\sqrt {a^2+2 a b x+b^2 x^2}+a-\sqrt {b^2} x\right ) \left (-a^3 \sqrt {b^2} B e^2+a^3 b B e^2-a^2 A b^2 e^2+a^2 A \sqrt {b^2} b e^2-2 a^2 b^2 B d e+2 a^2 \sqrt {b^2} b B d e+2 a A b^3 d e-2 a A \left (b^2\right )^{3/2} d e+a b^3 B d^2-a \left (b^2\right )^{3/2} B d^2-A b^4 d^2+A \sqrt {b^2} b^3 d^2\right )}{2 b^4 \sqrt {b^2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.43, size = 158, normalized size = 0.83 \begin {gather*} \frac {2 \, B b^{3} e^{2} x^{3} + 3 \, {\left (2 \, B b^{3} d e - {\left (B a b^{2} - A b^{3}\right )} e^{2}\right )} x^{2} + 6 \, {\left (B b^{3} d^{2} - 2 \, {\left (B a b^{2} - A b^{3}\right )} d e + {\left (B a^{2} b - A a b^{2}\right )} e^{2}\right )} x - 6 \, {\left ({\left (B a b^{2} - A b^{3}\right )} d^{2} - 2 \, {\left (B a^{2} b - A a b^{2}\right )} d e + {\left (B a^{3} - A a^{2} b\right )} e^{2}\right )} \log \left (b x + a\right )}{6 \, b^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.22, size = 254, normalized size = 1.33 \begin {gather*} \frac {2 \, B b^{2} x^{3} e^{2} \mathrm {sgn}\left (b x + a\right ) + 6 \, B b^{2} d x^{2} e \mathrm {sgn}\left (b x + a\right ) + 6 \, B b^{2} d^{2} x \mathrm {sgn}\left (b x + a\right ) - 3 \, B a b x^{2} e^{2} \mathrm {sgn}\left (b x + a\right ) + 3 \, A b^{2} x^{2} e^{2} \mathrm {sgn}\left (b x + a\right ) - 12 \, B a b d x e \mathrm {sgn}\left (b x + a\right ) + 12 \, A b^{2} d x e \mathrm {sgn}\left (b x + a\right ) + 6 \, B a^{2} x e^{2} \mathrm {sgn}\left (b x + a\right ) - 6 \, A a b x e^{2} \mathrm {sgn}\left (b x + a\right )}{6 \, b^{3}} - \frac {{\left (B a b^{2} d^{2} \mathrm {sgn}\left (b x + a\right ) - A b^{3} d^{2} \mathrm {sgn}\left (b x + a\right ) - 2 \, B a^{2} b d e \mathrm {sgn}\left (b x + a\right ) + 2 \, A a b^{2} d e \mathrm {sgn}\left (b x + a\right ) + B a^{3} e^{2} \mathrm {sgn}\left (b x + a\right ) - A a^{2} b e^{2} \mathrm {sgn}\left (b x + a\right )\right )} \log \left ({\left | b x + a \right |}\right )}{b^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 212, normalized size = 1.11 \begin {gather*} \frac {\left (b x +a \right ) \left (2 B \,b^{3} e^{2} x^{3}+3 A \,b^{3} e^{2} x^{2}-3 B a \,b^{2} e^{2} x^{2}+6 B \,b^{3} d e \,x^{2}+6 A \,a^{2} b \,e^{2} \ln \left (b x +a \right )-12 A a \,b^{2} d e \ln \left (b x +a \right )-6 A a \,b^{2} e^{2} x +6 A \,b^{3} d^{2} \ln \left (b x +a \right )+12 A \,b^{3} d e x -6 B \,a^{3} e^{2} \ln \left (b x +a \right )+12 B \,a^{2} b d e \ln \left (b x +a \right )+6 B \,a^{2} b \,e^{2} x -6 B a \,b^{2} d^{2} \ln \left (b x +a \right )-12 B a \,b^{2} d e x +6 B \,b^{3} d^{2} x \right )}{6 \sqrt {\left (b x +a \right )^{2}}\, b^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.62, size = 244, normalized size = 1.28 \begin {gather*} -\frac {5 \, B a e^{2} x^{2}}{6 \, b^{2}} + \frac {\sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} B e^{2} x^{2}}{3 \, b^{2}} + \frac {5 \, B a^{2} e^{2} x}{3 \, b^{3}} + \frac {A d^{2} \log \left (x + \frac {a}{b}\right )}{b} - \frac {B a^{3} e^{2} \log \left (x + \frac {a}{b}\right )}{b^{4}} + \frac {{\left (2 \, B d e + A e^{2}\right )} x^{2}}{2 \, b} - \frac {2 \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} B a^{2} e^{2}}{3 \, b^{4}} - \frac {{\left (2 \, B d e + A e^{2}\right )} a x}{b^{2}} + \frac {{\left (2 \, B d e + A e^{2}\right )} a^{2} \log \left (x + \frac {a}{b}\right )}{b^{3}} - \frac {{\left (B d^{2} + 2 \, A d e\right )} a \log \left (x + \frac {a}{b}\right )}{b^{2}} + \frac {\sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} {\left (B d^{2} + 2 \, A d e\right )}}{b^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {\left (A+B\,x\right )\,{\left (d+e\,x\right )}^2}{\sqrt {{\left (a+b\,x\right )}^2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.49, size = 117, normalized size = 0.61 \begin {gather*} \frac {B e^{2} x^{3}}{3 b} + x^{2} \left (\frac {A e^{2}}{2 b} - \frac {B a e^{2}}{2 b^{2}} + \frac {B d e}{b}\right ) + x \left (- \frac {A a e^{2}}{b^{2}} + \frac {2 A d e}{b} + \frac {B a^{2} e^{2}}{b^{3}} - \frac {2 B a d e}{b^{2}} + \frac {B d^{2}}{b}\right ) - \frac {\left (- A b + B a\right ) \left (a e - b d\right )^{2} \log {\left (a + b x \right )}}{b^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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