Optimal. Leaf size=298 \[ \frac {b^2 \sqrt {a^2+2 a b x+b^2 x^2} (-3 a B e-A b e+4 b B d)}{4 e^5 (a+b x) (d+e x)^4}-\frac {3 b \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e) (-a B e-A b e+2 b B d)}{5 e^5 (a+b x) (d+e x)^5}+\frac {\sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^2 (-a B e-3 A b e+4 b B d)}{6 e^5 (a+b x) (d+e x)^6}-\frac {\sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^3 (B d-A e)}{7 e^5 (a+b x) (d+e x)^7}-\frac {b^3 B \sqrt {a^2+2 a b x+b^2 x^2}}{3 e^5 (a+b x) (d+e x)^3} \]
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Rubi [A] time = 0.22, 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)}{4 e^5 (a+b x) (d+e x)^4}-\frac {3 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)}{5 e^5 (a+b x) (d+e x)^5}+\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)}{6 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)^3 (B d-A e)}{7 e^5 (a+b x) (d+e x)^7}-\frac {b^3 B \sqrt {a^2+2 a b x+b^2 x^2}}{3 e^5 (a+b x) (d+e x)^3} \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)^8} \, 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)^8} \, 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)^8}+\frac {b^3 (b d-a e)^2 (-4 b B d+3 A b e+a B e)}{e^4 (d+e x)^7}-\frac {3 b^4 (b d-a e) (-2 b B d+A b e+a B e)}{e^4 (d+e x)^6}+\frac {b^5 (-4 b B d+A b e+3 a B e)}{e^4 (d+e x)^5}+\frac {b^6 B}{e^4 (d+e x)^4}\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}}{7 e^5 (a+b x) (d+e x)^7}+\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}}{6 e^5 (a+b x) (d+e x)^6}-\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}}{5 e^5 (a+b x) (d+e x)^5}+\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}}{4 e^5 (a+b x) (d+e x)^4}-\frac {b^3 B \sqrt {a^2+2 a b x+b^2 x^2}}{3 e^5 (a+b x) (d+e x)^3}\\ \end {align*}
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Mathematica [A] time = 0.10, size = 233, normalized size = 0.78 \begin {gather*} -\frac {\sqrt {(a+b x)^2} \left (10 a^3 e^3 (6 A e+B (d+7 e x))+6 a^2 b e^2 \left (5 A e (d+7 e x)+2 B \left (d^2+7 d e x+21 e^2 x^2\right )\right )+3 a b^2 e \left (4 A e \left (d^2+7 d e x+21 e^2 x^2\right )+3 B \left (d^3+7 d^2 e x+21 d e^2 x^2+35 e^3 x^3\right )\right )+b^3 \left (3 A e \left (d^3+7 d^2 e x+21 d e^2 x^2+35 e^3 x^3\right )+4 B \left (d^4+7 d^3 e x+21 d^2 e^2 x^2+35 d e^3 x^3+35 e^4 x^4\right )\right )\right )}{420 e^5 (a+b x) (d+e x)^7} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 180.02, 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 = 0.41, size = 332, normalized size = 1.11 \begin {gather*} -\frac {140 \, B b^{3} e^{4} x^{4} + 4 \, B b^{3} d^{4} + 60 \, A a^{3} e^{4} + 3 \, {\left (3 \, B a b^{2} + A b^{3}\right )} d^{3} e + 12 \, {\left (B a^{2} b + A a b^{2}\right )} d^{2} e^{2} + 10 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} d e^{3} + 35 \, {\left (4 \, B b^{3} d e^{3} + 3 \, {\left (3 \, B a b^{2} + A b^{3}\right )} e^{4}\right )} x^{3} + 21 \, {\left (4 \, B b^{3} d^{2} e^{2} + 3 \, {\left (3 \, B a b^{2} + A b^{3}\right )} d e^{3} + 12 \, {\left (B a^{2} b + A a b^{2}\right )} e^{4}\right )} x^{2} + 7 \, {\left (4 \, B b^{3} d^{3} e + 3 \, {\left (3 \, B a b^{2} + A b^{3}\right )} d^{2} e^{2} + 12 \, {\left (B a^{2} b + A a b^{2}\right )} d e^{3} + 10 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} e^{4}\right )} x}{420 \, {\left (e^{12} x^{7} + 7 \, d e^{11} x^{6} + 21 \, d^{2} e^{10} x^{5} + 35 \, d^{3} e^{9} x^{4} + 35 \, d^{4} e^{8} x^{3} + 21 \, d^{5} e^{7} x^{2} + 7 \, d^{6} e^{6} x + d^{7} e^{5}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.23, size = 427, normalized size = 1.43 \begin {gather*} -\frac {{\left (140 \, B b^{3} x^{4} e^{4} \mathrm {sgn}\left (b x + a\right ) + 140 \, B b^{3} d x^{3} e^{3} \mathrm {sgn}\left (b x + a\right ) + 84 \, B b^{3} d^{2} x^{2} e^{2} \mathrm {sgn}\left (b x + a\right ) + 28 \, B b^{3} d^{3} x e \mathrm {sgn}\left (b x + a\right ) + 4 \, B b^{3} d^{4} \mathrm {sgn}\left (b x + a\right ) + 315 \, B a b^{2} x^{3} e^{4} \mathrm {sgn}\left (b x + a\right ) + 105 \, A b^{3} x^{3} e^{4} \mathrm {sgn}\left (b x + a\right ) + 189 \, B a b^{2} d x^{2} e^{3} \mathrm {sgn}\left (b x + a\right ) + 63 \, A b^{3} d x^{2} e^{3} \mathrm {sgn}\left (b x + a\right ) + 63 \, B a b^{2} d^{2} x e^{2} \mathrm {sgn}\left (b x + a\right ) + 21 \, A b^{3} d^{2} x e^{2} \mathrm {sgn}\left (b x + a\right ) + 9 \, B a b^{2} d^{3} e \mathrm {sgn}\left (b x + a\right ) + 3 \, A b^{3} d^{3} e \mathrm {sgn}\left (b x + a\right ) + 252 \, B a^{2} b x^{2} e^{4} \mathrm {sgn}\left (b x + a\right ) + 252 \, A a b^{2} x^{2} e^{4} \mathrm {sgn}\left (b x + a\right ) + 84 \, B a^{2} b d x e^{3} \mathrm {sgn}\left (b x + a\right ) + 84 \, A a b^{2} d x e^{3} \mathrm {sgn}\left (b x + a\right ) + 12 \, B a^{2} b d^{2} e^{2} \mathrm {sgn}\left (b x + a\right ) + 12 \, A a b^{2} d^{2} e^{2} \mathrm {sgn}\left (b x + a\right ) + 70 \, B a^{3} x e^{4} \mathrm {sgn}\left (b x + a\right ) + 210 \, A a^{2} b x e^{4} \mathrm {sgn}\left (b x + a\right ) + 10 \, B a^{3} d e^{3} \mathrm {sgn}\left (b x + a\right ) + 30 \, A a^{2} b d e^{3} \mathrm {sgn}\left (b x + a\right ) + 60 \, A a^{3} e^{4} \mathrm {sgn}\left (b x + a\right )\right )} e^{\left (-5\right )}}{420 \, {\left (x e + d\right )}^{7}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.07, size = 317, normalized size = 1.06 \begin {gather*} -\frac {\left (140 B \,b^{3} e^{4} x^{4}+105 A \,b^{3} e^{4} x^{3}+315 B a \,b^{2} e^{4} x^{3}+140 B \,b^{3} d \,e^{3} x^{3}+252 A a \,b^{2} e^{4} x^{2}+63 A \,b^{3} d \,e^{3} x^{2}+252 B \,a^{2} b \,e^{4} x^{2}+189 B a \,b^{2} d \,e^{3} x^{2}+84 B \,b^{3} d^{2} e^{2} x^{2}+210 A \,a^{2} b \,e^{4} x +84 A a \,b^{2} d \,e^{3} x +21 A \,b^{3} d^{2} e^{2} x +70 B \,a^{3} e^{4} x +84 B \,a^{2} b d \,e^{3} x +63 B a \,b^{2} d^{2} e^{2} x +28 B \,b^{3} d^{3} e x +60 A \,a^{3} e^{4}+30 A \,a^{2} b d \,e^{3}+12 A a \,b^{2} d^{2} e^{2}+3 A \,b^{3} d^{3} e +10 B \,a^{3} d \,e^{3}+12 B \,a^{2} b \,d^{2} e^{2}+9 B a \,b^{2} d^{3} e +4 B \,b^{3} d^{4}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {3}{2}}}{420 \left (e x +d \right )^{7} \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 = 2.22, 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}{4\,e^5}-\frac {B\,b^3\,d}{4\,e^5}\right )\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{\left (a+b\,x\right )\,{\left (d+e\,x\right )}^4}-\frac {\left (\frac {A\,a^3}{7\,e}-\frac {d\,\left (\frac {B\,a^3+3\,A\,b\,a^2}{7\,e}+\frac {d\,\left (\frac {d\,\left (\frac {A\,b^3+3\,B\,a\,b^2}{7\,e}-\frac {B\,b^3\,d}{7\,e^2}\right )}{e}-\frac {3\,a\,b\,\left (A\,b+B\,a\right )}{7\,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 )}^7}-\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}{6\,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}{6\,e^5}-\frac {d\,\left (\frac {b^2\,\left (A\,b\,e+3\,B\,a\,e-B\,b\,d\right )}{6\,e^3}-\frac {B\,b^3\,d}{6\,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 )}^6}-\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}{5\,e^5}-\frac {d\,\left (\frac {b^2\,\left (A\,b\,e+3\,B\,a\,e-2\,B\,b\,d\right )}{5\,e^4}-\frac {B\,b^3\,d}{5\,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 )}^5}-\frac {B\,b^3\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{3\,e^5\,\left (a+b\,x\right )\,{\left (d+e\,x\right )}^3} \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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