Optimal. Leaf size=252 \[ -\frac {b^2 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^2}{e^5 (a+b x) (d+e x)^6}+\frac {4 b \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^3}{7 e^5 (a+b x) (d+e x)^7}-\frac {\sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^4}{8 e^5 (a+b x) (d+e x)^8}-\frac {b^4 \sqrt {a^2+2 a b x+b^2 x^2}}{4 e^5 (a+b x) (d+e x)^4}+\frac {4 b^3 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)}{5 e^5 (a+b x) (d+e x)^5} \]
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Rubi [A] time = 0.13, antiderivative size = 252, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {770, 21, 43} \begin {gather*} -\frac {b^4 \sqrt {a^2+2 a b x+b^2 x^2}}{4 e^5 (a+b x) (d+e x)^4}+\frac {4 b^3 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)}{5 e^5 (a+b x) (d+e x)^5}-\frac {b^2 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^2}{e^5 (a+b x) (d+e x)^6}+\frac {4 b \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^3}{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)^4}{8 e^5 (a+b x) (d+e x)^8} \end {gather*}
Antiderivative was successfully verified.
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Rule 21
Rule 43
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 {(a+b x) \left (a b+b^2 x\right )^3}{(d+e x)^9} \, dx}{b^2 \left (a b+b^2 x\right )}\\ &=\frac {\left (b \sqrt {a^2+2 a b x+b^2 x^2}\right ) \int \frac {(a+b x)^4}{(d+e x)^9} \, dx}{a b+b^2 x}\\ &=\frac {\left (b \sqrt {a^2+2 a b x+b^2 x^2}\right ) \int \left (\frac {(-b d+a e)^4}{e^4 (d+e x)^9}-\frac {4 b (b d-a e)^3}{e^4 (d+e x)^8}+\frac {6 b^2 (b d-a e)^2}{e^4 (d+e x)^7}-\frac {4 b^3 (b d-a e)}{e^4 (d+e x)^6}+\frac {b^4}{e^4 (d+e x)^5}\right ) \, dx}{a b+b^2 x}\\ &=-\frac {(b d-a e)^4 \sqrt {a^2+2 a b x+b^2 x^2}}{8 e^5 (a+b x) (d+e x)^8}+\frac {4 b (b d-a e)^3 \sqrt {a^2+2 a b x+b^2 x^2}}{7 e^5 (a+b x) (d+e x)^7}-\frac {b^2 (b d-a e)^2 \sqrt {a^2+2 a b x+b^2 x^2}}{e^5 (a+b x) (d+e x)^6}+\frac {4 b^3 (b d-a e) \sqrt {a^2+2 a b x+b^2 x^2}}{5 e^5 (a+b x) (d+e x)^5}-\frac {b^4 \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 = 0.06, size = 162, normalized size = 0.64 \begin {gather*} -\frac {\sqrt {(a+b x)^2} \left (35 a^4 e^4+20 a^3 b e^3 (d+8 e x)+10 a^2 b^2 e^2 \left (d^2+8 d e x+28 e^2 x^2\right )+4 a b^3 e \left (d^3+8 d^2 e x+28 d e^2 x^2+56 e^3 x^3\right )+b^4 \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 )}{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 = 0.43, size = 258, normalized size = 1.02 \begin {gather*} -\frac {70 \, b^{4} e^{4} x^{4} + b^{4} d^{4} + 4 \, a b^{3} d^{3} e + 10 \, a^{2} b^{2} d^{2} e^{2} + 20 \, a^{3} b d e^{3} + 35 \, a^{4} e^{4} + 56 \, {\left (b^{4} d e^{3} + 4 \, a b^{3} e^{4}\right )} x^{3} + 28 \, {\left (b^{4} d^{2} e^{2} + 4 \, a b^{3} d e^{3} + 10 \, a^{2} b^{2} e^{4}\right )} x^{2} + 8 \, {\left (b^{4} d^{3} e + 4 \, a b^{3} d^{2} e^{2} + 10 \, a^{2} b^{2} d e^{3} + 20 \, a^{3} b 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 = 0.20, size = 264, normalized size = 1.05 \begin {gather*} -\frac {{\left (70 \, b^{4} x^{4} e^{4} \mathrm {sgn}\left (b x + a\right ) + 56 \, b^{4} d x^{3} e^{3} \mathrm {sgn}\left (b x + a\right ) + 28 \, b^{4} d^{2} x^{2} e^{2} \mathrm {sgn}\left (b x + a\right ) + 8 \, b^{4} d^{3} x e \mathrm {sgn}\left (b x + a\right ) + b^{4} d^{4} \mathrm {sgn}\left (b x + a\right ) + 224 \, a b^{3} x^{3} e^{4} \mathrm {sgn}\left (b x + a\right ) + 112 \, a b^{3} d x^{2} e^{3} \mathrm {sgn}\left (b x + a\right ) + 32 \, a b^{3} d^{2} x e^{2} \mathrm {sgn}\left (b x + a\right ) + 4 \, a b^{3} d^{3} e \mathrm {sgn}\left (b x + a\right ) + 280 \, a^{2} b^{2} x^{2} e^{4} \mathrm {sgn}\left (b x + a\right ) + 80 \, a^{2} b^{2} d x e^{3} \mathrm {sgn}\left (b x + a\right ) + 10 \, a^{2} b^{2} d^{2} e^{2} \mathrm {sgn}\left (b x + a\right ) + 160 \, a^{3} b x e^{4} \mathrm {sgn}\left (b x + a\right ) + 20 \, a^{3} b d e^{3} \mathrm {sgn}\left (b x + a\right ) + 35 \, a^{4} 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 = 0.07, size = 201, normalized size = 0.80 \begin {gather*} -\frac {\left (70 b^{4} e^{4} x^{4}+224 a \,b^{3} e^{4} x^{3}+56 b^{4} d \,e^{3} x^{3}+280 a^{2} b^{2} e^{4} x^{2}+112 a \,b^{3} d \,e^{3} x^{2}+28 b^{4} d^{2} e^{2} x^{2}+160 a^{3} b \,e^{4} x +80 a^{2} b^{2} d \,e^{3} x +32 a \,b^{3} d^{2} e^{2} x +8 b^{4} d^{3} e x +35 a^{4} e^{4}+20 a^{3} b d \,e^{3}+10 a^{2} b^{2} d^{2} e^{2}+4 a \,b^{3} d^{3} e +b^{4} 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 = 2.17, size = 449, normalized size = 1.78 \begin {gather*} \frac {\left (\frac {-4\,a^3\,b\,e^3+6\,a^2\,b^2\,d\,e^2-4\,a\,b^3\,d^2\,e+b^4\,d^3}{7\,e^5}+\frac {d\,\left (\frac {d\,\left (\frac {b^4\,d}{7\,e^3}-\frac {b^3\,\left (4\,a\,e-b\,d\right )}{7\,e^3}\right )}{e}+\frac {b^2\,\left (6\,a^2\,e^2-4\,a\,b\,d\,e+b^2\,d^2\right )}{7\,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 )}^7}-\frac {\left (\frac {a^4}{8\,e}-\frac {d\,\left (\frac {d\,\left (\frac {d\,\left (\frac {a\,b^3}{2\,e}-\frac {b^4\,d}{8\,e^2}\right )}{e}-\frac {3\,a^2\,b^2}{4\,e}\right )}{e}+\frac {a^3\,b}{2\,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 {6\,a^2\,b^2\,e^2-8\,a\,b^3\,d\,e+3\,b^4\,d^2}{6\,e^5}+\frac {d\,\left (\frac {b^4\,d}{6\,e^4}-\frac {b^3\,\left (2\,a\,e-b\,d\right )}{3\,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 {\left (\frac {3\,b^4\,d-4\,a\,b^3\,e}{5\,e^5}+\frac {b^4\,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 {b^4\,\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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