Optimal. Leaf size=359 \[ -\frac {5 b^2 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^4}{3 e^7 (a+b x) (d+e x)^9}+\frac {3 b \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^5}{5 e^7 (a+b x) (d+e x)^{10}}-\frac {\sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^6}{11 e^7 (a+b x) (d+e x)^{11}}-\frac {b^6 \sqrt {a^2+2 a b x+b^2 x^2}}{5 e^7 (a+b x) (d+e x)^5}+\frac {b^5 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)}{e^7 (a+b x) (d+e x)^6}-\frac {15 b^4 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^2}{7 e^7 (a+b x) (d+e x)^7}+\frac {5 b^3 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^3}{2 e^7 (a+b x) (d+e x)^8} \]
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Rubi [A] time = 0.20, antiderivative size = 359, 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^6 \sqrt {a^2+2 a b x+b^2 x^2}}{5 e^7 (a+b x) (d+e x)^5}+\frac {b^5 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)}{e^7 (a+b x) (d+e x)^6}-\frac {15 b^4 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^2}{7 e^7 (a+b x) (d+e x)^7}+\frac {5 b^3 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^3}{2 e^7 (a+b x) (d+e x)^8}-\frac {5 b^2 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^4}{3 e^7 (a+b x) (d+e x)^9}+\frac {3 b \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^5}{5 e^7 (a+b x) (d+e x)^{10}}-\frac {\sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^6}{11 e^7 (a+b x) (d+e x)^{11}} \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 )^{5/2}}{(d+e x)^{12}} \, 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 )^5}{(d+e x)^{12}} \, dx}{b^4 \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)^6}{(d+e x)^{12}} \, 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)^6}{e^6 (d+e x)^{12}}-\frac {6 b (b d-a e)^5}{e^6 (d+e x)^{11}}+\frac {15 b^2 (b d-a e)^4}{e^6 (d+e x)^{10}}-\frac {20 b^3 (b d-a e)^3}{e^6 (d+e x)^9}+\frac {15 b^4 (b d-a e)^2}{e^6 (d+e x)^8}-\frac {6 b^5 (b d-a e)}{e^6 (d+e x)^7}+\frac {b^6}{e^6 (d+e x)^6}\right ) \, dx}{a b+b^2 x}\\ &=-\frac {(b d-a e)^6 \sqrt {a^2+2 a b x+b^2 x^2}}{11 e^7 (a+b x) (d+e x)^{11}}+\frac {3 b (b d-a e)^5 \sqrt {a^2+2 a b x+b^2 x^2}}{5 e^7 (a+b x) (d+e x)^{10}}-\frac {5 b^2 (b d-a e)^4 \sqrt {a^2+2 a b x+b^2 x^2}}{3 e^7 (a+b x) (d+e x)^9}+\frac {5 b^3 (b d-a e)^3 \sqrt {a^2+2 a b x+b^2 x^2}}{2 e^7 (a+b x) (d+e x)^8}-\frac {15 b^4 (b d-a e)^2 \sqrt {a^2+2 a b x+b^2 x^2}}{7 e^7 (a+b x) (d+e x)^7}+\frac {b^5 (b d-a e) \sqrt {a^2+2 a b x+b^2 x^2}}{e^7 (a+b x) (d+e x)^6}-\frac {b^6 \sqrt {a^2+2 a b x+b^2 x^2}}{5 e^7 (a+b x) (d+e x)^5}\\ \end {align*}
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Mathematica [A] time = 0.11, size = 295, normalized size = 0.82 \begin {gather*} -\frac {\sqrt {(a+b x)^2} \left (210 a^6 e^6+126 a^5 b e^5 (d+11 e x)+70 a^4 b^2 e^4 \left (d^2+11 d e x+55 e^2 x^2\right )+35 a^3 b^3 e^3 \left (d^3+11 d^2 e x+55 d e^2 x^2+165 e^3 x^3\right )+15 a^2 b^4 e^2 \left (d^4+11 d^3 e x+55 d^2 e^2 x^2+165 d e^3 x^3+330 e^4 x^4\right )+5 a b^5 e \left (d^5+11 d^4 e x+55 d^3 e^2 x^2+165 d^2 e^3 x^3+330 d e^4 x^4+462 e^5 x^5\right )+b^6 \left (d^6+11 d^5 e x+55 d^4 e^2 x^2+165 d^3 e^3 x^3+330 d^2 e^4 x^4+462 d e^5 x^5+462 e^6 x^6\right )\right )}{2310 e^7 (a+b x) (d+e x)^{11}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 180.08, 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 = 463, normalized size = 1.29 \begin {gather*} -\frac {462 \, b^{6} e^{6} x^{6} + b^{6} d^{6} + 5 \, a b^{5} d^{5} e + 15 \, a^{2} b^{4} d^{4} e^{2} + 35 \, a^{3} b^{3} d^{3} e^{3} + 70 \, a^{4} b^{2} d^{2} e^{4} + 126 \, a^{5} b d e^{5} + 210 \, a^{6} e^{6} + 462 \, {\left (b^{6} d e^{5} + 5 \, a b^{5} e^{6}\right )} x^{5} + 330 \, {\left (b^{6} d^{2} e^{4} + 5 \, a b^{5} d e^{5} + 15 \, a^{2} b^{4} e^{6}\right )} x^{4} + 165 \, {\left (b^{6} d^{3} e^{3} + 5 \, a b^{5} d^{2} e^{4} + 15 \, a^{2} b^{4} d e^{5} + 35 \, a^{3} b^{3} e^{6}\right )} x^{3} + 55 \, {\left (b^{6} d^{4} e^{2} + 5 \, a b^{5} d^{3} e^{3} + 15 \, a^{2} b^{4} d^{2} e^{4} + 35 \, a^{3} b^{3} d e^{5} + 70 \, a^{4} b^{2} e^{6}\right )} x^{2} + 11 \, {\left (b^{6} d^{5} e + 5 \, a b^{5} d^{4} e^{2} + 15 \, a^{2} b^{4} d^{3} e^{3} + 35 \, a^{3} b^{3} d^{2} e^{4} + 70 \, a^{4} b^{2} d e^{5} + 126 \, a^{5} b e^{6}\right )} x}{2310 \, {\left (e^{18} x^{11} + 11 \, d e^{17} x^{10} + 55 \, d^{2} e^{16} x^{9} + 165 \, d^{3} e^{15} x^{8} + 330 \, d^{4} e^{14} x^{7} + 462 \, d^{5} e^{13} x^{6} + 462 \, d^{6} e^{12} x^{5} + 330 \, d^{7} e^{11} x^{4} + 165 \, d^{8} e^{10} x^{3} + 55 \, d^{9} e^{9} x^{2} + 11 \, d^{10} e^{8} x + d^{11} e^{7}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.20, size = 520, normalized size = 1.45 \begin {gather*} -\frac {{\left (462 \, b^{6} x^{6} e^{6} \mathrm {sgn}\left (b x + a\right ) + 462 \, b^{6} d x^{5} e^{5} \mathrm {sgn}\left (b x + a\right ) + 330 \, b^{6} d^{2} x^{4} e^{4} \mathrm {sgn}\left (b x + a\right ) + 165 \, b^{6} d^{3} x^{3} e^{3} \mathrm {sgn}\left (b x + a\right ) + 55 \, b^{6} d^{4} x^{2} e^{2} \mathrm {sgn}\left (b x + a\right ) + 11 \, b^{6} d^{5} x e \mathrm {sgn}\left (b x + a\right ) + b^{6} d^{6} \mathrm {sgn}\left (b x + a\right ) + 2310 \, a b^{5} x^{5} e^{6} \mathrm {sgn}\left (b x + a\right ) + 1650 \, a b^{5} d x^{4} e^{5} \mathrm {sgn}\left (b x + a\right ) + 825 \, a b^{5} d^{2} x^{3} e^{4} \mathrm {sgn}\left (b x + a\right ) + 275 \, a b^{5} d^{3} x^{2} e^{3} \mathrm {sgn}\left (b x + a\right ) + 55 \, a b^{5} d^{4} x e^{2} \mathrm {sgn}\left (b x + a\right ) + 5 \, a b^{5} d^{5} e \mathrm {sgn}\left (b x + a\right ) + 4950 \, a^{2} b^{4} x^{4} e^{6} \mathrm {sgn}\left (b x + a\right ) + 2475 \, a^{2} b^{4} d x^{3} e^{5} \mathrm {sgn}\left (b x + a\right ) + 825 \, a^{2} b^{4} d^{2} x^{2} e^{4} \mathrm {sgn}\left (b x + a\right ) + 165 \, a^{2} b^{4} d^{3} x e^{3} \mathrm {sgn}\left (b x + a\right ) + 15 \, a^{2} b^{4} d^{4} e^{2} \mathrm {sgn}\left (b x + a\right ) + 5775 \, a^{3} b^{3} x^{3} e^{6} \mathrm {sgn}\left (b x + a\right ) + 1925 \, a^{3} b^{3} d x^{2} e^{5} \mathrm {sgn}\left (b x + a\right ) + 385 \, a^{3} b^{3} d^{2} x e^{4} \mathrm {sgn}\left (b x + a\right ) + 35 \, a^{3} b^{3} d^{3} e^{3} \mathrm {sgn}\left (b x + a\right ) + 3850 \, a^{4} b^{2} x^{2} e^{6} \mathrm {sgn}\left (b x + a\right ) + 770 \, a^{4} b^{2} d x e^{5} \mathrm {sgn}\left (b x + a\right ) + 70 \, a^{4} b^{2} d^{2} e^{4} \mathrm {sgn}\left (b x + a\right ) + 1386 \, a^{5} b x e^{6} \mathrm {sgn}\left (b x + a\right ) + 126 \, a^{5} b d e^{5} \mathrm {sgn}\left (b x + a\right ) + 210 \, a^{6} e^{6} \mathrm {sgn}\left (b x + a\right )\right )} e^{\left (-7\right )}}{2310 \, {\left (x e + d\right )}^{11}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.07, size = 392, normalized size = 1.09 \begin {gather*} -\frac {\left (462 b^{6} e^{6} x^{6}+2310 a \,b^{5} e^{6} x^{5}+462 b^{6} d \,e^{5} x^{5}+4950 a^{2} b^{4} e^{6} x^{4}+1650 a \,b^{5} d \,e^{5} x^{4}+330 b^{6} d^{2} e^{4} x^{4}+5775 a^{3} b^{3} e^{6} x^{3}+2475 a^{2} b^{4} d \,e^{5} x^{3}+825 a \,b^{5} d^{2} e^{4} x^{3}+165 b^{6} d^{3} e^{3} x^{3}+3850 a^{4} b^{2} e^{6} x^{2}+1925 a^{3} b^{3} d \,e^{5} x^{2}+825 a^{2} b^{4} d^{2} e^{4} x^{2}+275 a \,b^{5} d^{3} e^{3} x^{2}+55 b^{6} d^{4} e^{2} x^{2}+1386 a^{5} b \,e^{6} x +770 a^{4} b^{2} d \,e^{5} x +385 a^{3} b^{3} d^{2} e^{4} x +165 a^{2} b^{4} d^{3} e^{3} x +55 a \,b^{5} d^{4} e^{2} x +11 b^{6} d^{5} e x +210 a^{6} e^{6}+126 a^{5} b d \,e^{5}+70 a^{4} b^{2} d^{2} e^{4}+35 a^{3} b^{3} d^{3} e^{3}+15 a^{2} b^{4} d^{4} e^{2}+5 a \,b^{5} d^{5} e +b^{6} d^{6}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}}}{2310 \left (e x +d \right )^{11} \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 = 2.53, size = 1010, normalized size = 2.81 \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}{10\,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}{10\,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}{10\,e^7}-\frac {d\,\left (\frac {d\,\left (\frac {b^6\,d}{10\,e^3}-\frac {b^5\,\left (6\,a\,e-b\,d\right )}{10\,e^3}\right )}{e}+\frac {b^4\,\left (15\,a^2\,e^2-6\,a\,b\,d\,e+b^2\,d^2\right )}{10\,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 )}^{10}}-\frac {\left (\frac {15\,a^2\,b^4\,e^2-24\,a\,b^5\,d\,e+10\,b^6\,d^2}{7\,e^7}+\frac {d\,\left (\frac {b^6\,d}{7\,e^6}-\frac {2\,b^5\,\left (3\,a\,e-2\,b\,d\right )}{7\,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 )}^7}-\frac {\left (\frac {a^6}{11\,e}-\frac {d\,\left (\frac {d\,\left (\frac {d\,\left (\frac {d\,\left (\frac {d\,\left (\frac {6\,a\,b^5}{11\,e}-\frac {b^6\,d}{11\,e^2}\right )}{e}-\frac {15\,a^2\,b^4}{11\,e}\right )}{e}+\frac {20\,a^3\,b^3}{11\,e}\right )}{e}-\frac {15\,a^4\,b^2}{11\,e}\right )}{e}+\frac {6\,a^5\,b}{11\,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 )}^{11}}-\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}{9\,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}{9\,e^7}+\frac {d\,\left (\frac {d\,\left (\frac {b^6\,d}{9\,e^4}-\frac {2\,b^5\,\left (3\,a\,e-b\,d\right )}{9\,e^4}\right )}{e}+\frac {b^4\,\left (5\,a^2\,e^2-4\,a\,b\,d\,e+b^2\,d^2\right )}{3\,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 )}^9}+\frac {\left (\frac {5\,b^6\,d-6\,a\,b^5\,e}{6\,e^7}+\frac {b^6\,d}{6\,e^7}\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 {-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}{8\,e^7}+\frac {d\,\left (\frac {d\,\left (\frac {b^6\,d}{8\,e^5}-\frac {3\,b^5\,\left (2\,a\,e-b\,d\right )}{8\,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 )}{8\,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 )}^8}-\frac {b^6\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{5\,e^7\,\left (a+b\,x\right )\,{\left (d+e\,x\right )}^5} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: HeuristicGCDFailed} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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