Optimal. Leaf size=362 \[ -\frac {15 b^2 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^4}{13 e^7 (a+b x) (d+e x)^{13}}+\frac {3 b \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^5}{7 e^7 (a+b x) (d+e x)^{14}}-\frac {\sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^6}{15 e^7 (a+b x) (d+e x)^{15}}-\frac {b^6 \sqrt {a^2+2 a b x+b^2 x^2}}{9 e^7 (a+b x) (d+e x)^9}+\frac {3 b^5 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)}{5 e^7 (a+b x) (d+e x)^{10}}-\frac {15 b^4 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^2}{11 e^7 (a+b x) (d+e x)^{11}}+\frac {5 b^3 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^3}{3 e^7 (a+b x) (d+e x)^{12}} \]
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Rubi [A] time = 0.20, antiderivative size = 362, 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}}{9 e^7 (a+b x) (d+e x)^9}+\frac {3 b^5 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)}{5 e^7 (a+b x) (d+e x)^{10}}-\frac {15 b^4 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^2}{11 e^7 (a+b x) (d+e x)^{11}}+\frac {5 b^3 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^3}{3 e^7 (a+b x) (d+e x)^{12}}-\frac {15 b^2 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^4}{13 e^7 (a+b x) (d+e x)^{13}}+\frac {3 b \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^5}{7 e^7 (a+b x) (d+e x)^{14}}-\frac {\sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^6}{15 e^7 (a+b x) (d+e x)^{15}} \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)^{16}} \, 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)^{16}} \, 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)^{16}} \, 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)^{16}}-\frac {6 b (b d-a e)^5}{e^6 (d+e x)^{15}}+\frac {15 b^2 (b d-a e)^4}{e^6 (d+e x)^{14}}-\frac {20 b^3 (b d-a e)^3}{e^6 (d+e x)^{13}}+\frac {15 b^4 (b d-a e)^2}{e^6 (d+e x)^{12}}-\frac {6 b^5 (b d-a e)}{e^6 (d+e x)^{11}}+\frac {b^6}{e^6 (d+e x)^{10}}\right ) \, dx}{a b+b^2 x}\\ &=-\frac {(b d-a e)^6 \sqrt {a^2+2 a b x+b^2 x^2}}{15 e^7 (a+b x) (d+e x)^{15}}+\frac {3 b (b d-a e)^5 \sqrt {a^2+2 a b x+b^2 x^2}}{7 e^7 (a+b x) (d+e x)^{14}}-\frac {15 b^2 (b d-a e)^4 \sqrt {a^2+2 a b x+b^2 x^2}}{13 e^7 (a+b x) (d+e x)^{13}}+\frac {5 b^3 (b d-a e)^3 \sqrt {a^2+2 a b x+b^2 x^2}}{3 e^7 (a+b x) (d+e x)^{12}}-\frac {15 b^4 (b d-a e)^2 \sqrt {a^2+2 a b x+b^2 x^2}}{11 e^7 (a+b x) (d+e x)^{11}}+\frac {3 b^5 (b d-a e) \sqrt {a^2+2 a b x+b^2 x^2}}{5 e^7 (a+b x) (d+e x)^{10}}-\frac {b^6 \sqrt {a^2+2 a b x+b^2 x^2}}{9 e^7 (a+b x) (d+e x)^9}\\ \end {align*}
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Mathematica [A] time = 0.12, size = 295, normalized size = 0.81 \begin {gather*} -\frac {\sqrt {(a+b x)^2} \left (3003 a^6 e^6+1287 a^5 b e^5 (d+15 e x)+495 a^4 b^2 e^4 \left (d^2+15 d e x+105 e^2 x^2\right )+165 a^3 b^3 e^3 \left (d^3+15 d^2 e x+105 d e^2 x^2+455 e^3 x^3\right )+45 a^2 b^4 e^2 \left (d^4+15 d^3 e x+105 d^2 e^2 x^2+455 d e^3 x^3+1365 e^4 x^4\right )+9 a b^5 e \left (d^5+15 d^4 e x+105 d^3 e^2 x^2+455 d^2 e^3 x^3+1365 d e^4 x^4+3003 e^5 x^5\right )+b^6 \left (d^6+15 d^5 e x+105 d^4 e^2 x^2+455 d^3 e^3 x^3+1365 d^2 e^4 x^4+3003 d e^5 x^5+5005 e^6 x^6\right )\right )}{45045 e^7 (a+b x) (d+e x)^{15}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 180.24, 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 = 507, normalized size = 1.40 \begin {gather*} -\frac {5005 \, b^{6} e^{6} x^{6} + b^{6} d^{6} + 9 \, a b^{5} d^{5} e + 45 \, a^{2} b^{4} d^{4} e^{2} + 165 \, a^{3} b^{3} d^{3} e^{3} + 495 \, a^{4} b^{2} d^{2} e^{4} + 1287 \, a^{5} b d e^{5} + 3003 \, a^{6} e^{6} + 3003 \, {\left (b^{6} d e^{5} + 9 \, a b^{5} e^{6}\right )} x^{5} + 1365 \, {\left (b^{6} d^{2} e^{4} + 9 \, a b^{5} d e^{5} + 45 \, a^{2} b^{4} e^{6}\right )} x^{4} + 455 \, {\left (b^{6} d^{3} e^{3} + 9 \, a b^{5} d^{2} e^{4} + 45 \, a^{2} b^{4} d e^{5} + 165 \, a^{3} b^{3} e^{6}\right )} x^{3} + 105 \, {\left (b^{6} d^{4} e^{2} + 9 \, a b^{5} d^{3} e^{3} + 45 \, a^{2} b^{4} d^{2} e^{4} + 165 \, a^{3} b^{3} d e^{5} + 495 \, a^{4} b^{2} e^{6}\right )} x^{2} + 15 \, {\left (b^{6} d^{5} e + 9 \, a b^{5} d^{4} e^{2} + 45 \, a^{2} b^{4} d^{3} e^{3} + 165 \, a^{3} b^{3} d^{2} e^{4} + 495 \, a^{4} b^{2} d e^{5} + 1287 \, a^{5} b e^{6}\right )} x}{45045 \, {\left (e^{22} x^{15} + 15 \, d e^{21} x^{14} + 105 \, d^{2} e^{20} x^{13} + 455 \, d^{3} e^{19} x^{12} + 1365 \, d^{4} e^{18} x^{11} + 3003 \, d^{5} e^{17} x^{10} + 5005 \, d^{6} e^{16} x^{9} + 6435 \, d^{7} e^{15} x^{8} + 6435 \, d^{8} e^{14} x^{7} + 5005 \, d^{9} e^{13} x^{6} + 3003 \, d^{10} e^{12} x^{5} + 1365 \, d^{11} e^{11} x^{4} + 455 \, d^{12} e^{10} x^{3} + 105 \, d^{13} e^{9} x^{2} + 15 \, d^{14} e^{8} x + d^{15} e^{7}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.19, size = 520, normalized size = 1.44 \begin {gather*} -\frac {{\left (5005 \, b^{6} x^{6} e^{6} \mathrm {sgn}\left (b x + a\right ) + 3003 \, b^{6} d x^{5} e^{5} \mathrm {sgn}\left (b x + a\right ) + 1365 \, b^{6} d^{2} x^{4} e^{4} \mathrm {sgn}\left (b x + a\right ) + 455 \, b^{6} d^{3} x^{3} e^{3} \mathrm {sgn}\left (b x + a\right ) + 105 \, b^{6} d^{4} x^{2} e^{2} \mathrm {sgn}\left (b x + a\right ) + 15 \, b^{6} d^{5} x e \mathrm {sgn}\left (b x + a\right ) + b^{6} d^{6} \mathrm {sgn}\left (b x + a\right ) + 27027 \, a b^{5} x^{5} e^{6} \mathrm {sgn}\left (b x + a\right ) + 12285 \, a b^{5} d x^{4} e^{5} \mathrm {sgn}\left (b x + a\right ) + 4095 \, a b^{5} d^{2} x^{3} e^{4} \mathrm {sgn}\left (b x + a\right ) + 945 \, a b^{5} d^{3} x^{2} e^{3} \mathrm {sgn}\left (b x + a\right ) + 135 \, a b^{5} d^{4} x e^{2} \mathrm {sgn}\left (b x + a\right ) + 9 \, a b^{5} d^{5} e \mathrm {sgn}\left (b x + a\right ) + 61425 \, a^{2} b^{4} x^{4} e^{6} \mathrm {sgn}\left (b x + a\right ) + 20475 \, a^{2} b^{4} d x^{3} e^{5} \mathrm {sgn}\left (b x + a\right ) + 4725 \, a^{2} b^{4} d^{2} x^{2} e^{4} \mathrm {sgn}\left (b x + a\right ) + 675 \, a^{2} b^{4} d^{3} x e^{3} \mathrm {sgn}\left (b x + a\right ) + 45 \, a^{2} b^{4} d^{4} e^{2} \mathrm {sgn}\left (b x + a\right ) + 75075 \, a^{3} b^{3} x^{3} e^{6} \mathrm {sgn}\left (b x + a\right ) + 17325 \, a^{3} b^{3} d x^{2} e^{5} \mathrm {sgn}\left (b x + a\right ) + 2475 \, a^{3} b^{3} d^{2} x e^{4} \mathrm {sgn}\left (b x + a\right ) + 165 \, a^{3} b^{3} d^{3} e^{3} \mathrm {sgn}\left (b x + a\right ) + 51975 \, a^{4} b^{2} x^{2} e^{6} \mathrm {sgn}\left (b x + a\right ) + 7425 \, a^{4} b^{2} d x e^{5} \mathrm {sgn}\left (b x + a\right ) + 495 \, a^{4} b^{2} d^{2} e^{4} \mathrm {sgn}\left (b x + a\right ) + 19305 \, a^{5} b x e^{6} \mathrm {sgn}\left (b x + a\right ) + 1287 \, a^{5} b d e^{5} \mathrm {sgn}\left (b x + a\right ) + 3003 \, a^{6} e^{6} \mathrm {sgn}\left (b x + a\right )\right )} e^{\left (-7\right )}}{45045 \, {\left (x e + d\right )}^{15}} \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 = 1.08 \begin {gather*} -\frac {\left (5005 b^{6} e^{6} x^{6}+27027 a \,b^{5} e^{6} x^{5}+3003 b^{6} d \,e^{5} x^{5}+61425 a^{2} b^{4} e^{6} x^{4}+12285 a \,b^{5} d \,e^{5} x^{4}+1365 b^{6} d^{2} e^{4} x^{4}+75075 a^{3} b^{3} e^{6} x^{3}+20475 a^{2} b^{4} d \,e^{5} x^{3}+4095 a \,b^{5} d^{2} e^{4} x^{3}+455 b^{6} d^{3} e^{3} x^{3}+51975 a^{4} b^{2} e^{6} x^{2}+17325 a^{3} b^{3} d \,e^{5} x^{2}+4725 a^{2} b^{4} d^{2} e^{4} x^{2}+945 a \,b^{5} d^{3} e^{3} x^{2}+105 b^{6} d^{4} e^{2} x^{2}+19305 a^{5} b \,e^{6} x +7425 a^{4} b^{2} d \,e^{5} x +2475 a^{3} b^{3} d^{2} e^{4} x +675 a^{2} b^{4} d^{3} e^{3} x +135 a \,b^{5} d^{4} e^{2} x +15 b^{6} d^{5} e x +3003 a^{6} e^{6}+1287 a^{5} b d \,e^{5}+495 a^{4} b^{2} d^{2} e^{4}+165 a^{3} b^{3} d^{3} e^{3}+45 a^{2} b^{4} d^{4} e^{2}+9 a \,b^{5} d^{5} e +b^{6} d^{6}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}}}{45045 \left (e x +d \right )^{15} \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 = 2.45, size = 1010, normalized size = 2.79 \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}{14\,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}{14\,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}{14\,e^7}-\frac {d\,\left (\frac {d\,\left (\frac {b^6\,d}{14\,e^3}-\frac {b^5\,\left (6\,a\,e-b\,d\right )}{14\,e^3}\right )}{e}+\frac {b^4\,\left (15\,a^2\,e^2-6\,a\,b\,d\,e+b^2\,d^2\right )}{14\,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 )}^{14}}-\frac {\left (\frac {15\,a^2\,b^4\,e^2-24\,a\,b^5\,d\,e+10\,b^6\,d^2}{11\,e^7}+\frac {d\,\left (\frac {b^6\,d}{11\,e^6}-\frac {2\,b^5\,\left (3\,a\,e-2\,b\,d\right )}{11\,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 )}^{11}}-\frac {\left (\frac {a^6}{15\,e}-\frac {d\,\left (\frac {d\,\left (\frac {d\,\left (\frac {d\,\left (\frac {d\,\left (\frac {2\,a\,b^5}{5\,e}-\frac {b^6\,d}{15\,e^2}\right )}{e}-\frac {a^2\,b^4}{e}\right )}{e}+\frac {4\,a^3\,b^3}{3\,e}\right )}{e}-\frac {a^4\,b^2}{e}\right )}{e}+\frac {2\,a^5\,b}{5\,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 )}^{15}}-\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}{13\,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}{13\,e^7}+\frac {d\,\left (\frac {d\,\left (\frac {b^6\,d}{13\,e^4}-\frac {2\,b^5\,\left (3\,a\,e-b\,d\right )}{13\,e^4}\right )}{e}+\frac {3\,b^4\,\left (5\,a^2\,e^2-4\,a\,b\,d\,e+b^2\,d^2\right )}{13\,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 )}^{13}}+\frac {\left (\frac {5\,b^6\,d-6\,a\,b^5\,e}{10\,e^7}+\frac {b^6\,d}{10\,e^7}\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 {-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}{12\,e^7}+\frac {d\,\left (\frac {d\,\left (\frac {b^6\,d}{12\,e^5}-\frac {b^5\,\left (2\,a\,e-b\,d\right )}{4\,e^5}\right )}{e}+\frac {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 )}^{12}}-\frac {b^6\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{9\,e^7\,\left (a+b\,x\right )\,{\left (d+e\,x\right )}^9} \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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