Optimal. Leaf size=298 \[ \frac {b^2 \sqrt {a^2+2 a b x+b^2 x^2} (-3 a B e-A b e+4 b B d)}{5 e^5 (a+b x) (d+e x)^5}-\frac {b \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e) (-a B e-A b e+2 b B d)}{2 e^5 (a+b x) (d+e x)^6}+\frac {\sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^2 (-a B e-3 A b e+4 b B d)}{7 e^5 (a+b x) (d+e x)^7}-\frac {\sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^3 (B d-A e)}{8 e^5 (a+b x) (d+e x)^8}-\frac {b^3 B \sqrt {a^2+2 a b x+b^2 x^2}}{4 e^5 (a+b x) (d+e x)^4} \]
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Rubi [A] time = 0.18, antiderivative size = 298, 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 \sqrt {a^2+2 a b x+b^2 x^2} (-3 a B e-A b e+4 b B d)}{5 e^5 (a+b x) (d+e x)^5}-\frac {b \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e) (-a B e-A b e+2 b B d)}{2 e^5 (a+b x) (d+e x)^6}+\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)}{7 e^5 (a+b x) (d+e x)^7}-\frac {\sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^3 (B d-A e)}{8 e^5 (a+b x) (d+e x)^8}-\frac {b^3 B \sqrt {a^2+2 a b x+b^2 x^2}}{4 e^5 (a+b x) (d+e x)^4} \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)^9} \, 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)^9} \, 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^3 (b d-a e)^3 (-B d+A e)}{e^4 (d+e x)^9}+\frac {b^3 (b d-a e)^2 (-4 b B d+3 A b e+a B e)}{e^4 (d+e x)^8}-\frac {3 b^4 (b d-a e) (-2 b B d+A b e+a B e)}{e^4 (d+e x)^7}+\frac {b^5 (-4 b B d+A b e+3 a B e)}{e^4 (d+e x)^6}+\frac {b^6 B}{e^4 (d+e x)^5}\right ) \, dx}{b^2 \left (a b+b^2 x\right )}\\ &=-\frac {(b d-a e)^3 (B d-A e) \sqrt {a^2+2 a b x+b^2 x^2}}{8 e^5 (a+b x) (d+e x)^8}+\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}}{7 e^5 (a+b x) (d+e x)^7}-\frac {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}}{2 e^5 (a+b x) (d+e x)^6}+\frac {b^2 (4 b B d-A b e-3 a B e) \sqrt {a^2+2 a b x+b^2 x^2}}{5 e^5 (a+b x) (d+e x)^5}-\frac {b^3 B \sqrt {a^2+2 a b x+b^2 x^2}}{4 e^5 (a+b x) (d+e x)^4}\\ \end {align*}
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Mathematica [A] time = 0.11, size = 229, normalized size = 0.77 \begin {gather*} -\frac {\sqrt {(a+b x)^2} \left (5 a^3 e^3 (7 A e+B (d+8 e x))+5 a^2 b e^2 \left (3 A e (d+8 e x)+B \left (d^2+8 d e x+28 e^2 x^2\right )\right )+a b^2 e \left (5 A e \left (d^2+8 d e x+28 e^2 x^2\right )+3 B \left (d^3+8 d^2 e x+28 d e^2 x^2+56 e^3 x^3\right )\right )+b^3 \left (A e \left (d^3+8 d^2 e x+28 d e^2 x^2+56 e^3 x^3\right )+B \left (d^4+8 d^3 e x+28 d^2 e^2 x^2+56 d e^3 x^3+70 e^4 x^4\right )\right )\right )}{280 e^5 (a+b x) (d+e x)^8} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 180.03, size = 0, normalized size = 0.00 \begin {gather*} \text {\$Aborted} \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 0.44, size = 335, normalized size = 1.12 \begin {gather*} -\frac {70 \, B b^{3} e^{4} x^{4} + B b^{3} d^{4} + 35 \, A a^{3} e^{4} + {\left (3 \, B a b^{2} + A b^{3}\right )} d^{3} e + 5 \, {\left (B a^{2} b + A a b^{2}\right )} d^{2} e^{2} + 5 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} d e^{3} + 56 \, {\left (B b^{3} d e^{3} + {\left (3 \, B a b^{2} + A b^{3}\right )} e^{4}\right )} x^{3} + 28 \, {\left (B b^{3} d^{2} e^{2} + {\left (3 \, B a b^{2} + A b^{3}\right )} d e^{3} + 5 \, {\left (B a^{2} b + A a b^{2}\right )} e^{4}\right )} x^{2} + 8 \, {\left (B b^{3} d^{3} e + {\left (3 \, B a b^{2} + A b^{3}\right )} d^{2} e^{2} + 5 \, {\left (B a^{2} b + A a b^{2}\right )} d e^{3} + 5 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} e^{4}\right )} x}{280 \, {\left (e^{13} x^{8} + 8 \, d e^{12} x^{7} + 28 \, d^{2} e^{11} x^{6} + 56 \, d^{3} e^{10} x^{5} + 70 \, d^{4} e^{9} x^{4} + 56 \, d^{5} e^{8} x^{3} + 28 \, d^{6} e^{7} x^{2} + 8 \, d^{7} e^{6} x + d^{8} e^{5}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.19, size = 425, normalized size = 1.43 \begin {gather*} -\frac {{\left (70 \, B b^{3} x^{4} e^{4} \mathrm {sgn}\left (b x + a\right ) + 56 \, B b^{3} d x^{3} e^{3} \mathrm {sgn}\left (b x + a\right ) + 28 \, B b^{3} d^{2} x^{2} e^{2} \mathrm {sgn}\left (b x + a\right ) + 8 \, B b^{3} d^{3} x e \mathrm {sgn}\left (b x + a\right ) + B b^{3} d^{4} \mathrm {sgn}\left (b x + a\right ) + 168 \, B a b^{2} x^{3} e^{4} \mathrm {sgn}\left (b x + a\right ) + 56 \, A b^{3} x^{3} e^{4} \mathrm {sgn}\left (b x + a\right ) + 84 \, B a b^{2} d x^{2} e^{3} \mathrm {sgn}\left (b x + a\right ) + 28 \, A b^{3} d x^{2} e^{3} \mathrm {sgn}\left (b x + a\right ) + 24 \, B a b^{2} d^{2} x e^{2} \mathrm {sgn}\left (b x + a\right ) + 8 \, A b^{3} d^{2} x e^{2} \mathrm {sgn}\left (b x + a\right ) + 3 \, B a b^{2} d^{3} e \mathrm {sgn}\left (b x + a\right ) + A b^{3} d^{3} e \mathrm {sgn}\left (b x + a\right ) + 140 \, B a^{2} b x^{2} e^{4} \mathrm {sgn}\left (b x + a\right ) + 140 \, A a b^{2} x^{2} e^{4} \mathrm {sgn}\left (b x + a\right ) + 40 \, B a^{2} b d x e^{3} \mathrm {sgn}\left (b x + a\right ) + 40 \, A a b^{2} d x e^{3} \mathrm {sgn}\left (b x + a\right ) + 5 \, B a^{2} b d^{2} e^{2} \mathrm {sgn}\left (b x + a\right ) + 5 \, A a b^{2} d^{2} e^{2} \mathrm {sgn}\left (b x + a\right ) + 40 \, B a^{3} x e^{4} \mathrm {sgn}\left (b x + a\right ) + 120 \, A a^{2} b x e^{4} \mathrm {sgn}\left (b x + a\right ) + 5 \, B a^{3} d e^{3} \mathrm {sgn}\left (b x + a\right ) + 15 \, A a^{2} b d e^{3} \mathrm {sgn}\left (b x + a\right ) + 35 \, A a^{3} e^{4} \mathrm {sgn}\left (b x + a\right )\right )} e^{\left (-5\right )}}{280 \, {\left (x e + d\right )}^{8}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 315, normalized size = 1.06 \begin {gather*} -\frac {\left (70 B \,b^{3} e^{4} x^{4}+56 A \,b^{3} e^{4} x^{3}+168 B a \,b^{2} e^{4} x^{3}+56 B \,b^{3} d \,e^{3} x^{3}+140 A a \,b^{2} e^{4} x^{2}+28 A \,b^{3} d \,e^{3} x^{2}+140 B \,a^{2} b \,e^{4} x^{2}+84 B a \,b^{2} d \,e^{3} x^{2}+28 B \,b^{3} d^{2} e^{2} x^{2}+120 A \,a^{2} b \,e^{4} x +40 A a \,b^{2} d \,e^{3} x +8 A \,b^{3} d^{2} e^{2} x +40 B \,a^{3} e^{4} x +40 B \,a^{2} b d \,e^{3} x +24 B a \,b^{2} d^{2} e^{2} x +8 B \,b^{3} d^{3} e x +35 A \,a^{3} e^{4}+15 A \,a^{2} b d \,e^{3}+5 A a \,b^{2} d^{2} e^{2}+A \,b^{3} d^{3} e +5 B \,a^{3} d \,e^{3}+5 B \,a^{2} b \,d^{2} e^{2}+3 B a \,b^{2} d^{3} e +B \,b^{3} d^{4}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {3}{2}}}{280 \left (e x +d \right )^{8} \left (b x +a \right )^{3} 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 [B] time = 2.27, size = 577, normalized size = 1.94 \begin {gather*} -\frac {\left (\frac {A\,b^3\,e-3\,B\,b^3\,d+3\,B\,a\,b^2\,e}{5\,e^5}-\frac {B\,b^3\,d}{5\,e^5}\right )\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{\left (a+b\,x\right )\,{\left (d+e\,x\right )}^5}-\frac {\left (\frac {A\,a^3}{8\,e}-\frac {d\,\left (\frac {B\,a^3+3\,A\,b\,a^2}{8\,e}+\frac {d\,\left (\frac {d\,\left (\frac {A\,b^3+3\,B\,a\,b^2}{8\,e}-\frac {B\,b^3\,d}{8\,e^2}\right )}{e}-\frac {3\,a\,b\,\left (A\,b+B\,a\right )}{8\,e}\right )}{e}\right )}{e}\right )\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{\left (a+b\,x\right )\,{\left (d+e\,x\right )}^8}-\frac {\left (\frac {B\,a^3\,e^3-3\,B\,a^2\,b\,d\,e^2+3\,A\,a^2\,b\,e^3+3\,B\,a\,b^2\,d^2\,e-3\,A\,a\,b^2\,d\,e^2-B\,b^3\,d^3+A\,b^3\,d^2\,e}{7\,e^5}-\frac {d\,\left (\frac {3\,B\,a^2\,b\,e^3-3\,B\,a\,b^2\,d\,e^2+3\,A\,a\,b^2\,e^3+B\,b^3\,d^2\,e-A\,b^3\,d\,e^2}{7\,e^5}-\frac {d\,\left (\frac {b^2\,\left (A\,b\,e+3\,B\,a\,e-B\,b\,d\right )}{7\,e^3}-\frac {B\,b^3\,d}{7\,e^3}\right )}{e}\right )}{e}\right )\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{\left (a+b\,x\right )\,{\left (d+e\,x\right )}^7}-\frac {\left (\frac {3\,B\,a^2\,b\,e^2-6\,B\,a\,b^2\,d\,e+3\,A\,a\,b^2\,e^2+3\,B\,b^3\,d^2-2\,A\,b^3\,d\,e}{6\,e^5}-\frac {d\,\left (\frac {b^2\,\left (A\,b\,e+3\,B\,a\,e-2\,B\,b\,d\right )}{6\,e^4}-\frac {B\,b^3\,d}{6\,e^4}\right )}{e}\right )\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{\left (a+b\,x\right )\,{\left (d+e\,x\right )}^6}-\frac {B\,b^3\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{4\,e^5\,\left (a+b\,x\right )\,{\left (d+e\,x\right )}^4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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