Optimal. Leaf size=280 \[ \frac {\sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^2 (-a B e-3 A b e+4 b B d)}{e^5 (a+b x) (d+e x)}-\frac {\sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^3 (B d-A e)}{2 e^5 (a+b x) (d+e x)^2}+\frac {3 b \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e) \log (d+e x) (-a B e-A b e+2 b B d)}{e^5 (a+b x)}-\frac {b^2 x \sqrt {a^2+2 a b x+b^2 x^2} (-3 a B e-A b e+3 b B d)}{e^4 (a+b x)}+\frac {b^3 B x^2 \sqrt {a^2+2 a b x+b^2 x^2}}{2 e^3 (a+b x)} \]
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Rubi [A] time = 0.23, antiderivative size = 280, 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 {b^2 x \sqrt {a^2+2 a b x+b^2 x^2} (-3 a B e-A b e+3 b B d)}{e^4 (a+b x)}+\frac {\sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^2 (-a B e-3 A b e+4 b B d)}{e^5 (a+b x) (d+e x)}-\frac {\sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^3 (B d-A e)}{2 e^5 (a+b x) (d+e x)^2}+\frac {3 b \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e) \log (d+e x) (-a B e-A b e+2 b B d)}{e^5 (a+b x)}+\frac {b^3 B x^2 \sqrt {a^2+2 a b x+b^2 x^2}}{2 e^3 (a+b x)} \end {gather*}
Antiderivative was successfully verified.
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Rule 77
Rule 770
Rubi steps
\begin {align*} \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^{3/2}}{(d+e x)^3} \, dx &=\frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \frac {\left (a b+b^2 x\right )^3 (A+B x)}{(d+e x)^3} \, dx}{b^2 \left (a b+b^2 x\right )}\\ &=\frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \left (\frac {b^5 (-3 b B d+A b e+3 a B e)}{e^4}+\frac {b^6 B x}{e^3}-\frac {b^3 (b d-a e)^3 (-B d+A e)}{e^4 (d+e x)^3}+\frac {b^3 (b d-a e)^2 (-4 b B d+3 A b e+a B e)}{e^4 (d+e x)^2}-\frac {3 b^4 (b d-a e) (-2 b B d+A b e+a B e)}{e^4 (d+e x)}\right ) \, dx}{b^2 \left (a b+b^2 x\right )}\\ &=-\frac {b^2 (3 b B d-A b e-3 a B e) x \sqrt {a^2+2 a b x+b^2 x^2}}{e^4 (a+b x)}+\frac {b^3 B x^2 \sqrt {a^2+2 a b x+b^2 x^2}}{2 e^3 (a+b x)}-\frac {(b d-a e)^3 (B d-A e) \sqrt {a^2+2 a b x+b^2 x^2}}{2 e^5 (a+b x) (d+e x)^2}+\frac {(b d-a e)^2 (4 b B d-3 A b e-a B e) \sqrt {a^2+2 a b x+b^2 x^2}}{e^5 (a+b x) (d+e x)}+\frac {3 b (b d-a e) (2 b B d-A b e-a B e) \sqrt {a^2+2 a b x+b^2 x^2} \log (d+e x)}{e^5 (a+b x)}\\ \end {align*}
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Mathematica [A] time = 0.15, size = 256, normalized size = 0.91 \begin {gather*} \frac {\sqrt {(a+b x)^2} \left (-a^3 e^3 (A e+B (d+2 e x))-3 a^2 b e^2 (A e (d+2 e x)-B d (3 d+4 e x))+3 a b^2 e \left (A d e (3 d+4 e x)+B \left (-5 d^3-4 d^2 e x+4 d e^2 x^2+2 e^3 x^3\right )\right )+6 b (d+e x)^2 (b d-a e) \log (d+e x) (-a B e-A b e+2 b B d)+b^3 \left (A e \left (-5 d^3-4 d^2 e x+4 d e^2 x^2+2 e^3 x^3\right )+B \left (7 d^4+2 d^3 e x-11 d^2 e^2 x^2-4 d e^3 x^3+e^4 x^4\right )\right )\right )}{2 e^5 (a+b x) (d+e x)^2} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 6.36, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^{3/2}}{(d+e x)^3} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 0.43, size = 420, normalized size = 1.50 \begin {gather*} \frac {B b^{3} e^{4} x^{4} + 7 \, B b^{3} d^{4} - A a^{3} e^{4} - 5 \, {\left (3 \, B a b^{2} + A b^{3}\right )} d^{3} e + 9 \, {\left (B a^{2} b + A a b^{2}\right )} d^{2} e^{2} - {\left (B a^{3} + 3 \, A a^{2} b\right )} d e^{3} - 2 \, {\left (2 \, B b^{3} d e^{3} - {\left (3 \, B a b^{2} + A b^{3}\right )} e^{4}\right )} x^{3} - {\left (11 \, B b^{3} d^{2} e^{2} - 4 \, {\left (3 \, B a b^{2} + A b^{3}\right )} d e^{3}\right )} x^{2} + 2 \, {\left (B b^{3} d^{3} e - 2 \, {\left (3 \, B a b^{2} + A b^{3}\right )} d^{2} e^{2} + 6 \, {\left (B a^{2} b + A a b^{2}\right )} d e^{3} - {\left (B a^{3} + 3 \, A a^{2} b\right )} e^{4}\right )} x + 6 \, {\left (2 \, B b^{3} d^{4} - {\left (3 \, B a b^{2} + A b^{3}\right )} d^{3} e + {\left (B a^{2} b + A a b^{2}\right )} d^{2} e^{2} + {\left (2 \, B b^{3} d^{2} e^{2} - {\left (3 \, B a b^{2} + A b^{3}\right )} d e^{3} + {\left (B a^{2} b + A a b^{2}\right )} e^{4}\right )} x^{2} + 2 \, {\left (2 \, B b^{3} d^{3} e - {\left (3 \, B a b^{2} + A b^{3}\right )} d^{2} e^{2} + {\left (B a^{2} b + A a b^{2}\right )} d e^{3}\right )} x\right )} \log \left (e x + d\right )}{2 \, {\left (e^{7} x^{2} + 2 \, d e^{6} x + d^{2} e^{5}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.24, size = 417, normalized size = 1.49 \begin {gather*} 3 \, {\left (2 \, B b^{3} d^{2} \mathrm {sgn}\left (b x + a\right ) - 3 \, B a b^{2} d e \mathrm {sgn}\left (b x + a\right ) - A b^{3} d e \mathrm {sgn}\left (b x + a\right ) + B a^{2} b e^{2} \mathrm {sgn}\left (b x + a\right ) + A a b^{2} e^{2} \mathrm {sgn}\left (b x + a\right )\right )} e^{\left (-5\right )} \log \left ({\left | x e + d \right |}\right ) + \frac {1}{2} \, {\left (B b^{3} x^{2} e^{3} \mathrm {sgn}\left (b x + a\right ) - 6 \, B b^{3} d x e^{2} \mathrm {sgn}\left (b x + a\right ) + 6 \, B a b^{2} x e^{3} \mathrm {sgn}\left (b x + a\right ) + 2 \, A b^{3} x e^{3} \mathrm {sgn}\left (b x + a\right )\right )} e^{\left (-6\right )} + \frac {{\left (7 \, B b^{3} d^{4} \mathrm {sgn}\left (b x + a\right ) - 15 \, B a b^{2} d^{3} e \mathrm {sgn}\left (b x + a\right ) - 5 \, A b^{3} d^{3} e \mathrm {sgn}\left (b x + a\right ) + 9 \, B a^{2} b d^{2} e^{2} \mathrm {sgn}\left (b x + a\right ) + 9 \, A a b^{2} d^{2} e^{2} \mathrm {sgn}\left (b x + a\right ) - B a^{3} d e^{3} \mathrm {sgn}\left (b x + a\right ) - 3 \, A a^{2} b d e^{3} \mathrm {sgn}\left (b x + a\right ) - A a^{3} e^{4} \mathrm {sgn}\left (b x + a\right ) + 2 \, {\left (4 \, B b^{3} d^{3} e \mathrm {sgn}\left (b x + a\right ) - 9 \, B a b^{2} d^{2} e^{2} \mathrm {sgn}\left (b x + a\right ) - 3 \, A b^{3} d^{2} e^{2} \mathrm {sgn}\left (b x + a\right ) + 6 \, B a^{2} b d e^{3} \mathrm {sgn}\left (b x + a\right ) + 6 \, A a b^{2} d e^{3} \mathrm {sgn}\left (b x + a\right ) - B a^{3} e^{4} \mathrm {sgn}\left (b x + a\right ) - 3 \, A a^{2} b e^{4} \mathrm {sgn}\left (b x + a\right )\right )} x\right )} e^{\left (-5\right )}}{2 \, {\left (x e + d\right )}^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.07, size = 566, normalized size = 2.02 \begin {gather*} \frac {\left (\left (b x +a \right )^{2}\right )^{\frac {3}{2}} \left (B \,b^{3} e^{4} x^{4}+6 A a \,b^{2} e^{4} x^{2} \ln \left (e x +d \right )-6 A \,b^{3} d \,e^{3} x^{2} \ln \left (e x +d \right )+2 A \,b^{3} e^{4} x^{3}+6 B \,a^{2} b \,e^{4} x^{2} \ln \left (e x +d \right )-18 B a \,b^{2} d \,e^{3} x^{2} \ln \left (e x +d \right )+6 B a \,b^{2} e^{4} x^{3}+12 B \,b^{3} d^{2} e^{2} x^{2} \ln \left (e x +d \right )-4 B \,b^{3} d \,e^{3} x^{3}+12 A a \,b^{2} d \,e^{3} x \ln \left (e x +d \right )-12 A \,b^{3} d^{2} e^{2} x \ln \left (e x +d \right )+4 A \,b^{3} d \,e^{3} x^{2}+12 B \,a^{2} b d \,e^{3} x \ln \left (e x +d \right )-36 B a \,b^{2} d^{2} e^{2} x \ln \left (e x +d \right )+12 B a \,b^{2} d \,e^{3} x^{2}+24 B \,b^{3} d^{3} e x \ln \left (e x +d \right )-11 B \,b^{3} d^{2} e^{2} x^{2}-6 A \,a^{2} b \,e^{4} x +6 A a \,b^{2} d^{2} e^{2} \ln \left (e x +d \right )+12 A a \,b^{2} d \,e^{3} x -6 A \,b^{3} d^{3} e \ln \left (e x +d \right )-4 A \,b^{3} d^{2} e^{2} x -2 B \,a^{3} e^{4} x +6 B \,a^{2} b \,d^{2} e^{2} \ln \left (e x +d \right )+12 B \,a^{2} b d \,e^{3} x -18 B a \,b^{2} d^{3} e \ln \left (e x +d \right )-12 B a \,b^{2} d^{2} e^{2} x +12 B \,b^{3} d^{4} \ln \left (e x +d \right )+2 B \,b^{3} d^{3} e x -A \,a^{3} e^{4}-3 A \,a^{2} b d \,e^{3}+9 A a \,b^{2} d^{2} e^{2}-5 A \,b^{3} d^{3} e -B \,a^{3} d \,e^{3}+9 B \,a^{2} b \,d^{2} e^{2}-15 B a \,b^{2} d^{3} e +7 B \,b^{3} d^{4}\right )}{2 \left (b x +a \right )^{3} \left (e x +d \right )^{2} e^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \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 (a^2+2\,a\,b\,x+b^2\,x^2\right )}^{3/2}}{{\left (d+e\,x\right )}^3} \,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 (\left (a + b x\right )^{2}\right )^{\frac {3}{2}}}{\left (d + e x\right )^{3}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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