Optimal. Leaf size=249 \[ -\frac {(b d-a e)^2 (-4 a B e+3 A b e+b B d)}{b^5 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {(A b-a B) (b d-a e)^3}{2 b^5 (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {3 e (a+b x) (b d-a e) \log (a+b x) (-2 a B e+A b e+b B d)}{b^5 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {e^2 x (a+b x) (-3 a B e+A b e+3 b B d)}{b^4 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {B e^3 x^2 (a+b x)}{2 b^3 \sqrt {a^2+2 a b x+b^2 x^2}} \]
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Rubi [A] time = 0.22, antiderivative size = 249, 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 {e^2 x (a+b x) (-3 a B e+A b e+3 b B d)}{b^4 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {(b d-a e)^2 (-4 a B e+3 A b e+b B d)}{b^5 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {(A b-a B) (b d-a e)^3}{2 b^5 (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {3 e (a+b x) (b d-a e) \log (a+b x) (-2 a B e+A b e+b B d)}{b^5 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {B e^3 x^2 (a+b x)}{2 b^3 \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)^3}{\left (a^2+2 a b x+b^2 x^2\right )^{3/2}} \, dx &=\frac {\left (b^2 \left (a b+b^2 x\right )\right ) \int \frac {(A+B x) (d+e x)^3}{\left (a b+b^2 x\right )^3} \, dx}{\sqrt {a^2+2 a b x+b^2 x^2}}\\ &=\frac {\left (b^2 \left (a b+b^2 x\right )\right ) \int \left (\frac {e^2 (3 b B d+A b e-3 a B e)}{b^7}+\frac {B e^3 x}{b^6}+\frac {(A b-a B) (b d-a e)^3}{b^7 (a+b x)^3}+\frac {(b d-a e)^2 (b B d+3 A b e-4 a B e)}{b^7 (a+b x)^2}+\frac {3 e (b d-a e) (b B d+A b e-2 a B e)}{b^7 (a+b x)}\right ) \, dx}{\sqrt {a^2+2 a b x+b^2 x^2}}\\ &=-\frac {(b d-a e)^2 (b B d+3 A b e-4 a B e)}{b^5 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {(A b-a B) (b d-a e)^3}{2 b^5 (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {e^2 (3 b B d+A b e-3 a B e) x (a+b x)}{b^4 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {B e^3 x^2 (a+b x)}{2 b^3 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {3 e (b d-a e) (b B d+A b e-2 a B e) (a+b x) \log (a+b x)}{b^5 \sqrt {a^2+2 a b x+b^2 x^2}}\\ \end {align*}
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Mathematica [A] time = 0.16, size = 256, normalized size = 1.03 \begin {gather*} \frac {-A b \left (5 a^3 e^3+a^2 b e^2 (4 e x-9 d)+a b^2 e \left (3 d^2-12 d e x-4 e^2 x^2\right )+b^3 \left (d^3+6 d^2 e x-2 e^3 x^3\right )\right )+B \left (7 a^4 e^3+a^3 b e^2 (2 e x-15 d)+a^2 b^2 e \left (9 d^2-12 d e x-11 e^2 x^2\right )-a b^3 \left (d^3-12 d^2 e x-12 d e^2 x^2+4 e^3 x^3\right )+b^4 x \left (-2 d^3+6 d e^2 x^2+e^3 x^3\right )\right )+6 e (a+b x)^2 (b d-a e) \log (a+b x) (-2 a B e+A b e+b B d)}{2 b^5 (a+b x) \sqrt {(a+b x)^2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [B] time = 9.48, size = 7046, normalized size = 28.30 \begin {gather*} \text {Result too large to show} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 0.43, size = 442, normalized size = 1.78 \begin {gather*} \frac {B b^{4} e^{3} x^{4} - {\left (B a b^{3} + A b^{4}\right )} d^{3} + 3 \, {\left (3 \, B a^{2} b^{2} - A a b^{3}\right )} d^{2} e - 3 \, {\left (5 \, B a^{3} b - 3 \, A a^{2} b^{2}\right )} d e^{2} + {\left (7 \, B a^{4} - 5 \, A a^{3} b\right )} e^{3} + 2 \, {\left (3 \, B b^{4} d e^{2} - {\left (2 \, B a b^{3} - A b^{4}\right )} e^{3}\right )} x^{3} + {\left (12 \, B a b^{3} d e^{2} - {\left (11 \, B a^{2} b^{2} - 4 \, A a b^{3}\right )} e^{3}\right )} x^{2} - 2 \, {\left (B b^{4} d^{3} - 3 \, {\left (2 \, B a b^{3} - A b^{4}\right )} d^{2} e + 6 \, {\left (B a^{2} b^{2} - A a b^{3}\right )} d e^{2} - {\left (B a^{3} b - 2 \, A a^{2} b^{2}\right )} e^{3}\right )} x + 6 \, {\left (B a^{2} b^{2} d^{2} e - {\left (3 \, B a^{3} b - A a^{2} b^{2}\right )} d e^{2} + {\left (2 \, B a^{4} - A a^{3} b\right )} e^{3} + {\left (B b^{4} d^{2} e - {\left (3 \, B a b^{3} - A b^{4}\right )} d e^{2} + {\left (2 \, B a^{2} b^{2} - A a b^{3}\right )} e^{3}\right )} x^{2} + 2 \, {\left (B a b^{3} d^{2} e - {\left (3 \, B a^{2} b^{2} - A a b^{3}\right )} d e^{2} + {\left (2 \, B a^{3} b - A a^{2} b^{2}\right )} e^{3}\right )} x\right )} \log \left (b x + a\right )}{2 \, {\left (b^{7} x^{2} + 2 \, a b^{6} x + a^{2} b^{5}\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 [B] time = 0.14, size = 556, normalized size = 2.23 \begin {gather*} -\frac {\left (-B \,b^{4} e^{3} x^{4}+6 A a \,b^{3} e^{3} x^{2} \ln \left (b x +a \right )-6 A \,b^{4} d \,e^{2} x^{2} \ln \left (b x +a \right )-2 A \,b^{4} e^{3} x^{3}-12 B \,a^{2} b^{2} e^{3} x^{2} \ln \left (b x +a \right )+18 B a \,b^{3} d \,e^{2} x^{2} \ln \left (b x +a \right )+4 B a \,b^{3} e^{3} x^{3}-6 B \,b^{4} d^{2} e \,x^{2} \ln \left (b x +a \right )-6 B \,b^{4} d \,e^{2} x^{3}+12 A \,a^{2} b^{2} e^{3} x \ln \left (b x +a \right )-12 A a \,b^{3} d \,e^{2} x \ln \left (b x +a \right )-4 A a \,b^{3} e^{3} x^{2}-24 B \,a^{3} b \,e^{3} x \ln \left (b x +a \right )+36 B \,a^{2} b^{2} d \,e^{2} x \ln \left (b x +a \right )+11 B \,a^{2} b^{2} e^{3} x^{2}-12 B a \,b^{3} d^{2} e x \ln \left (b x +a \right )-12 B a \,b^{3} d \,e^{2} x^{2}+6 A \,a^{3} b \,e^{3} \ln \left (b x +a \right )-6 A \,a^{2} b^{2} d \,e^{2} \ln \left (b x +a \right )+4 A \,a^{2} b^{2} e^{3} x -12 A a \,b^{3} d \,e^{2} x +6 A \,b^{4} d^{2} e x -12 B \,a^{4} e^{3} \ln \left (b x +a \right )+18 B \,a^{3} b d \,e^{2} \ln \left (b x +a \right )-2 B \,a^{3} b \,e^{3} x -6 B \,a^{2} b^{2} d^{2} e \ln \left (b x +a \right )+12 B \,a^{2} b^{2} d \,e^{2} x -12 B a \,b^{3} d^{2} e x +2 B \,b^{4} d^{3} x +5 A \,a^{3} b \,e^{3}-9 A \,a^{2} b^{2} d \,e^{2}+3 A a \,b^{3} d^{2} e +A \,b^{4} d^{3}-7 B \,a^{4} e^{3}+15 B \,a^{3} b d \,e^{2}-9 B \,a^{2} b^{2} d^{2} e +B a \,b^{3} d^{3}\right ) \left (b x +a \right )}{2 \left (\left (b x +a \right )^{2}\right )^{\frac {3}{2}} b^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.51, size = 487, normalized size = 1.96 \begin {gather*} \frac {B e^{3} x^{3}}{2 \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} b^{2}} - \frac {5 \, B a e^{3} x^{2}}{2 \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} b^{3}} + \frac {6 \, B a^{2} e^{3} \log \left (x + \frac {a}{b}\right )}{b^{5}} - \frac {5 \, B a^{3} e^{3}}{\sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} b^{5}} + \frac {{\left (3 \, B d e^{2} + A e^{3}\right )} x^{2}}{\sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} b^{2}} + \frac {12 \, B a^{3} e^{3} x}{b^{6} {\left (x + \frac {a}{b}\right )}^{2}} - \frac {A d^{3}}{2 \, b^{3} {\left (x + \frac {a}{b}\right )}^{2}} + \frac {23 \, B a^{4} e^{3}}{2 \, b^{7} {\left (x + \frac {a}{b}\right )}^{2}} - \frac {3 \, {\left (3 \, B d e^{2} + A e^{3}\right )} a \log \left (x + \frac {a}{b}\right )}{b^{4}} + \frac {3 \, {\left (B d^{2} e + A d e^{2}\right )} \log \left (x + \frac {a}{b}\right )}{b^{3}} + \frac {2 \, {\left (3 \, B d e^{2} + A e^{3}\right )} a^{2}}{\sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} b^{4}} - \frac {B d^{3} + 3 \, A d^{2} e}{\sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} b^{2}} - \frac {6 \, {\left (3 \, B d e^{2} + A e^{3}\right )} a^{2} x}{b^{5} {\left (x + \frac {a}{b}\right )}^{2}} + \frac {6 \, {\left (B d^{2} e + A d e^{2}\right )} a x}{b^{4} {\left (x + \frac {a}{b}\right )}^{2}} - \frac {11 \, {\left (3 \, B d e^{2} + A e^{3}\right )} a^{3}}{2 \, b^{6} {\left (x + \frac {a}{b}\right )}^{2}} + \frac {9 \, {\left (B d^{2} e + A d e^{2}\right )} a^{2}}{2 \, b^{5} {\left (x + \frac {a}{b}\right )}^{2}} + \frac {{\left (B d^{3} + 3 \, A d^{2} e\right )} a}{2 \, b^{4} {\left (x + \frac {a}{b}\right )}^{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 = -0.00 \begin {gather*} \int \frac {\left (A+B\,x\right )\,{\left (d+e\,x\right )}^3}{{\left (a^2+2\,a\,b\,x+b^2\,x^2\right )}^{3/2}} \,d x \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 )^{3}}{\left (\left (a + b x\right )^{2}\right )^{\frac {3}{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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