Optimal. Leaf size=41 \[ \frac {\left (a^2+2 a b x+b^2 x^2\right )^{7/2}}{7 (d+e x)^7 (b d-a e)} \]
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Rubi [A] time = 0.02, antiderivative size = 41, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.030, Rules used = {767} \begin {gather*} \frac {\left (a^2+2 a b x+b^2 x^2\right )^{7/2}}{7 (d+e x)^7 (b d-a e)} \end {gather*}
Antiderivative was successfully verified.
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Rule 767
Rubi steps
\begin {align*} \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^8} \, dx &=\frac {\left (a^2+2 a b x+b^2 x^2\right )^{7/2}}{7 (b d-a e) (d+e x)^7}\\ \end {align*}
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Mathematica [B] time = 0.11, size = 289, normalized size = 7.05 \begin {gather*} -\frac {\sqrt {(a+b x)^2} \left (a^6 e^6+a^5 b e^5 (d+7 e x)+a^4 b^2 e^4 \left (d^2+7 d e x+21 e^2 x^2\right )+a^3 b^3 e^3 \left (d^3+7 d^2 e x+21 d e^2 x^2+35 e^3 x^3\right )+a^2 b^4 e^2 \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 )+a b^5 e \left (d^5+7 d^4 e x+21 d^3 e^2 x^2+35 d^2 e^3 x^3+35 d e^4 x^4+21 e^5 x^5\right )+b^6 \left (d^6+7 d^5 e x+21 d^4 e^2 x^2+35 d^3 e^3 x^3+35 d^2 e^4 x^4+21 d e^5 x^5+7 e^6 x^6\right )\right )}{7 e^7 (a+b x) (d+e x)^7} \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 [B] time = 0.42, size = 398, normalized size = 9.71 \begin {gather*} -\frac {7 \, b^{6} e^{6} x^{6} + b^{6} d^{6} + a b^{5} d^{5} e + a^{2} b^{4} d^{4} e^{2} + a^{3} b^{3} d^{3} e^{3} + a^{4} b^{2} d^{2} e^{4} + a^{5} b d e^{5} + a^{6} e^{6} + 21 \, {\left (b^{6} d e^{5} + a b^{5} e^{6}\right )} x^{5} + 35 \, {\left (b^{6} d^{2} e^{4} + a b^{5} d e^{5} + a^{2} b^{4} e^{6}\right )} x^{4} + 35 \, {\left (b^{6} d^{3} e^{3} + a b^{5} d^{2} e^{4} + a^{2} b^{4} d e^{5} + a^{3} b^{3} e^{6}\right )} x^{3} + 21 \, {\left (b^{6} d^{4} e^{2} + a b^{5} d^{3} e^{3} + a^{2} b^{4} d^{2} e^{4} + a^{3} b^{3} d e^{5} + a^{4} b^{2} e^{6}\right )} x^{2} + 7 \, {\left (b^{6} d^{5} e + a b^{5} d^{4} e^{2} + a^{2} b^{4} d^{3} e^{3} + a^{3} b^{3} d^{2} e^{4} + a^{4} b^{2} d e^{5} + a^{5} b e^{6}\right )} x}{7 \, {\left (e^{14} x^{7} + 7 \, d e^{13} x^{6} + 21 \, d^{2} e^{12} x^{5} + 35 \, d^{3} e^{11} x^{4} + 35 \, d^{4} e^{10} x^{3} + 21 \, d^{5} e^{9} x^{2} + 7 \, d^{6} e^{8} x + d^{7} e^{7}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.19, size = 514, normalized size = 12.54 \begin {gather*} -\frac {{\left (7 \, b^{6} x^{6} e^{6} \mathrm {sgn}\left (b x + a\right ) + 21 \, b^{6} d x^{5} e^{5} \mathrm {sgn}\left (b x + a\right ) + 35 \, b^{6} d^{2} x^{4} e^{4} \mathrm {sgn}\left (b x + a\right ) + 35 \, b^{6} d^{3} x^{3} e^{3} \mathrm {sgn}\left (b x + a\right ) + 21 \, b^{6} d^{4} x^{2} e^{2} \mathrm {sgn}\left (b x + a\right ) + 7 \, b^{6} d^{5} x e \mathrm {sgn}\left (b x + a\right ) + b^{6} d^{6} \mathrm {sgn}\left (b x + a\right ) + 21 \, a b^{5} x^{5} e^{6} \mathrm {sgn}\left (b x + a\right ) + 35 \, a b^{5} d x^{4} e^{5} \mathrm {sgn}\left (b x + a\right ) + 35 \, a b^{5} d^{2} x^{3} e^{4} \mathrm {sgn}\left (b x + a\right ) + 21 \, a b^{5} d^{3} x^{2} e^{3} \mathrm {sgn}\left (b x + a\right ) + 7 \, a b^{5} d^{4} x e^{2} \mathrm {sgn}\left (b x + a\right ) + a b^{5} d^{5} e \mathrm {sgn}\left (b x + a\right ) + 35 \, a^{2} b^{4} x^{4} e^{6} \mathrm {sgn}\left (b x + a\right ) + 35 \, a^{2} b^{4} d x^{3} e^{5} \mathrm {sgn}\left (b x + a\right ) + 21 \, a^{2} b^{4} d^{2} x^{2} e^{4} \mathrm {sgn}\left (b x + a\right ) + 7 \, a^{2} b^{4} d^{3} x e^{3} \mathrm {sgn}\left (b x + a\right ) + a^{2} b^{4} d^{4} e^{2} \mathrm {sgn}\left (b x + a\right ) + 35 \, a^{3} b^{3} x^{3} e^{6} \mathrm {sgn}\left (b x + a\right ) + 21 \, a^{3} b^{3} d x^{2} e^{5} \mathrm {sgn}\left (b x + a\right ) + 7 \, a^{3} b^{3} d^{2} x e^{4} \mathrm {sgn}\left (b x + a\right ) + a^{3} b^{3} d^{3} e^{3} \mathrm {sgn}\left (b x + a\right ) + 21 \, a^{4} b^{2} x^{2} e^{6} \mathrm {sgn}\left (b x + a\right ) + 7 \, a^{4} b^{2} d x e^{5} \mathrm {sgn}\left (b x + a\right ) + a^{4} b^{2} d^{2} e^{4} \mathrm {sgn}\left (b x + a\right ) + 7 \, a^{5} b x e^{6} \mathrm {sgn}\left (b x + a\right ) + a^{5} b d e^{5} \mathrm {sgn}\left (b x + a\right ) + a^{6} e^{6} \mathrm {sgn}\left (b x + a\right )\right )} e^{\left (-7\right )}}{7 \, {\left (x e + d\right )}^{7}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.05, size = 386, normalized size = 9.41 \begin {gather*} -\frac {\left (7 b^{6} e^{6} x^{6}+21 a \,b^{5} e^{6} x^{5}+21 b^{6} d \,e^{5} x^{5}+35 a^{2} b^{4} e^{6} x^{4}+35 a \,b^{5} d \,e^{5} x^{4}+35 b^{6} d^{2} e^{4} x^{4}+35 a^{3} b^{3} e^{6} x^{3}+35 a^{2} b^{4} d \,e^{5} x^{3}+35 a \,b^{5} d^{2} e^{4} x^{3}+35 b^{6} d^{3} e^{3} x^{3}+21 a^{4} b^{2} e^{6} x^{2}+21 a^{3} b^{3} d \,e^{5} x^{2}+21 a^{2} b^{4} d^{2} e^{4} x^{2}+21 a \,b^{5} d^{3} e^{3} x^{2}+21 b^{6} d^{4} e^{2} x^{2}+7 a^{5} b \,e^{6} x +7 a^{4} b^{2} d \,e^{5} x +7 a^{3} b^{3} d^{2} e^{4} x +7 a^{2} b^{4} d^{3} e^{3} x +7 a \,b^{5} d^{4} e^{2} x +7 b^{6} d^{5} e x +a^{6} e^{6}+a^{5} b d \,e^{5}+a^{4} b^{2} d^{2} e^{4}+a^{3} b^{3} d^{3} e^{3}+a^{2} b^{4} d^{4} e^{2}+a \,b^{5} d^{5} e +b^{6} d^{6}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}}}{7 \left (e x +d \right )^{7} \left (b x +a \right )^{5} e^{7}} \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 = 2.38, size = 1010, normalized size = 24.63 \begin {gather*} \frac {\left (\frac {-6\,a^5\,b\,e^5+15\,a^4\,b^2\,d\,e^4-20\,a^3\,b^3\,d^2\,e^3+15\,a^2\,b^4\,d^3\,e^2-6\,a\,b^5\,d^4\,e+b^6\,d^5}{6\,e^7}+\frac {d\,\left (\frac {15\,a^4\,b^2\,e^5-20\,a^3\,b^3\,d\,e^4+15\,a^2\,b^4\,d^2\,e^3-6\,a\,b^5\,d^3\,e^2+b^6\,d^4\,e}{6\,e^7}-\frac {d\,\left (\frac {20\,a^3\,b^3\,e^5-15\,a^2\,b^4\,d\,e^4+6\,a\,b^5\,d^2\,e^3-b^6\,d^3\,e^2}{6\,e^7}-\frac {d\,\left (\frac {d\,\left (\frac {b^6\,d}{6\,e^3}-\frac {b^5\,\left (6\,a\,e-b\,d\right )}{6\,e^3}\right )}{e}+\frac {b^4\,\left (15\,a^2\,e^2-6\,a\,b\,d\,e+b^2\,d^2\right )}{6\,e^4}\right )}{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 )}^6}-\frac {\left (\frac {15\,a^2\,b^4\,e^2-24\,a\,b^5\,d\,e+10\,b^6\,d^2}{3\,e^7}+\frac {d\,\left (\frac {b^6\,d}{3\,e^6}-\frac {2\,b^5\,\left (3\,a\,e-2\,b\,d\right )}{3\,e^6}\right )}{e}\right )\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{\left (a+b\,x\right )\,{\left (d+e\,x\right )}^3}-\frac {\left (\frac {a^6}{7\,e}-\frac {d\,\left (\frac {d\,\left (\frac {d\,\left (\frac {d\,\left (\frac {d\,\left (\frac {6\,a\,b^5}{7\,e}-\frac {b^6\,d}{7\,e^2}\right )}{e}-\frac {15\,a^2\,b^4}{7\,e}\right )}{e}+\frac {20\,a^3\,b^3}{7\,e}\right )}{e}-\frac {15\,a^4\,b^2}{7\,e}\right )}{e}+\frac {6\,a^5\,b}{7\,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 {15\,a^4\,b^2\,e^4-40\,a^3\,b^3\,d\,e^3+45\,a^2\,b^4\,d^2\,e^2-24\,a\,b^5\,d^3\,e+5\,b^6\,d^4}{5\,e^7}+\frac {d\,\left (\frac {-20\,a^3\,b^3\,e^4+30\,a^2\,b^4\,d\,e^3-18\,a\,b^5\,d^2\,e^2+4\,b^6\,d^3\,e}{5\,e^7}+\frac {d\,\left (\frac {d\,\left (\frac {b^6\,d}{5\,e^4}-\frac {2\,b^5\,\left (3\,a\,e-b\,d\right )}{5\,e^4}\right )}{e}+\frac {3\,b^4\,\left (5\,a^2\,e^2-4\,a\,b\,d\,e+b^2\,d^2\right )}{5\,e^5}\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 )}^5}+\frac {\left (\frac {5\,b^6\,d-6\,a\,b^5\,e}{2\,e^7}+\frac {b^6\,d}{2\,e^7}\right )\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{\left (a+b\,x\right )\,{\left (d+e\,x\right )}^2}+\frac {\left (\frac {-20\,a^3\,b^3\,e^3+45\,a^2\,b^4\,d\,e^2-36\,a\,b^5\,d^2\,e+10\,b^6\,d^3}{4\,e^7}+\frac {d\,\left (\frac {d\,\left (\frac {b^6\,d}{4\,e^5}-\frac {3\,b^5\,\left (2\,a\,e-b\,d\right )}{4\,e^5}\right )}{e}+\frac {3\,b^4\,\left (5\,a^2\,e^2-6\,a\,b\,d\,e+2\,b^2\,d^2\right )}{4\,e^6}\right )}{e}\right )\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{\left (a+b\,x\right )\,{\left (d+e\,x\right )}^4}-\frac {b^6\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{e^7\,\left (a+b\,x\right )\,\left (d+e\,x\right )} \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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