Optimal. Leaf size=154 \[ -\frac {10 b^4 (d+e x)^{3/2} (b d-a e)}{3 e^6}+\frac {20 b^3 \sqrt {d+e x} (b d-a e)^2}{e^6}+\frac {20 b^2 (b d-a e)^3}{e^6 \sqrt {d+e x}}-\frac {10 b (b d-a e)^4}{3 e^6 (d+e x)^{3/2}}+\frac {2 (b d-a e)^5}{5 e^6 (d+e x)^{5/2}}+\frac {2 b^5 (d+e x)^{5/2}}{5 e^6} \]
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Rubi [A] time = 0.05, antiderivative size = 154, 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 = {27, 43} \begin {gather*} -\frac {10 b^4 (d+e x)^{3/2} (b d-a e)}{3 e^6}+\frac {20 b^3 \sqrt {d+e x} (b d-a e)^2}{e^6}+\frac {20 b^2 (b d-a e)^3}{e^6 \sqrt {d+e x}}-\frac {10 b (b d-a e)^4}{3 e^6 (d+e x)^{3/2}}+\frac {2 (b d-a e)^5}{5 e^6 (d+e x)^{5/2}}+\frac {2 b^5 (d+e x)^{5/2}}{5 e^6} \end {gather*}
Antiderivative was successfully verified.
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Rule 27
Rule 43
Rubi steps
\begin {align*} \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^2}{(d+e x)^{7/2}} \, dx &=\int \frac {(a+b x)^5}{(d+e x)^{7/2}} \, dx\\ &=\int \left (\frac {(-b d+a e)^5}{e^5 (d+e x)^{7/2}}+\frac {5 b (b d-a e)^4}{e^5 (d+e x)^{5/2}}-\frac {10 b^2 (b d-a e)^3}{e^5 (d+e x)^{3/2}}+\frac {10 b^3 (b d-a e)^2}{e^5 \sqrt {d+e x}}-\frac {5 b^4 (b d-a e) \sqrt {d+e x}}{e^5}+\frac {b^5 (d+e x)^{3/2}}{e^5}\right ) \, dx\\ &=\frac {2 (b d-a e)^5}{5 e^6 (d+e x)^{5/2}}-\frac {10 b (b d-a e)^4}{3 e^6 (d+e x)^{3/2}}+\frac {20 b^2 (b d-a e)^3}{e^6 \sqrt {d+e x}}+\frac {20 b^3 (b d-a e)^2 \sqrt {d+e x}}{e^6}-\frac {10 b^4 (b d-a e) (d+e x)^{3/2}}{3 e^6}+\frac {2 b^5 (d+e x)^{5/2}}{5 e^6}\\ \end {align*}
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Mathematica [A] time = 0.09, size = 123, normalized size = 0.80 \begin {gather*} \frac {2 \left (-25 b^4 (d+e x)^4 (b d-a e)+150 b^3 (d+e x)^3 (b d-a e)^2+150 b^2 (d+e x)^2 (b d-a e)^3-25 b (d+e x) (b d-a e)^4+3 (b d-a e)^5+3 b^5 (d+e x)^5\right )}{15 e^6 (d+e x)^{5/2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [B] time = 0.09, size = 315, normalized size = 2.05 \begin {gather*} \frac {2 \left (-3 a^5 e^5-25 a^4 b e^4 (d+e x)+15 a^4 b d e^4-30 a^3 b^2 d^2 e^3-150 a^3 b^2 e^3 (d+e x)^2+100 a^3 b^2 d e^3 (d+e x)+30 a^2 b^3 d^3 e^2-150 a^2 b^3 d^2 e^2 (d+e x)+150 a^2 b^3 e^2 (d+e x)^3+450 a^2 b^3 d e^2 (d+e x)^2-15 a b^4 d^4 e+100 a b^4 d^3 e (d+e x)-450 a b^4 d^2 e (d+e x)^2+25 a b^4 e (d+e x)^4-300 a b^4 d e (d+e x)^3+3 b^5 d^5-25 b^5 d^4 (d+e x)+150 b^5 d^3 (d+e x)^2+150 b^5 d^2 (d+e x)^3+3 b^5 (d+e x)^5-25 b^5 d (d+e x)^4\right )}{15 e^6 (d+e x)^{5/2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 0.43, size = 294, normalized size = 1.91 \begin {gather*} \frac {2 \, {\left (3 \, b^{5} e^{5} x^{5} + 256 \, b^{5} d^{5} - 640 \, a b^{4} d^{4} e + 480 \, a^{2} b^{3} d^{3} e^{2} - 80 \, a^{3} b^{2} d^{2} e^{3} - 10 \, a^{4} b d e^{4} - 3 \, a^{5} e^{5} - 5 \, {\left (2 \, b^{5} d e^{4} - 5 \, a b^{4} e^{5}\right )} x^{4} + 10 \, {\left (8 \, b^{5} d^{2} e^{3} - 20 \, a b^{4} d e^{4} + 15 \, a^{2} b^{3} e^{5}\right )} x^{3} + 30 \, {\left (16 \, b^{5} d^{3} e^{2} - 40 \, a b^{4} d^{2} e^{3} + 30 \, a^{2} b^{3} d e^{4} - 5 \, a^{3} b^{2} e^{5}\right )} x^{2} + 5 \, {\left (128 \, b^{5} d^{4} e - 320 \, a b^{4} d^{3} e^{2} + 240 \, a^{2} b^{3} d^{2} e^{3} - 40 \, a^{3} b^{2} d e^{4} - 5 \, a^{4} b e^{5}\right )} x\right )} \sqrt {e x + d}}{15 \, {\left (e^{9} x^{3} + 3 \, d e^{8} x^{2} + 3 \, d^{2} e^{7} x + d^{3} e^{6}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.20, size = 333, normalized size = 2.16 \begin {gather*} \frac {2}{15} \, {\left (3 \, {\left (x e + d\right )}^{\frac {5}{2}} b^{5} e^{24} - 25 \, {\left (x e + d\right )}^{\frac {3}{2}} b^{5} d e^{24} + 150 \, \sqrt {x e + d} b^{5} d^{2} e^{24} + 25 \, {\left (x e + d\right )}^{\frac {3}{2}} a b^{4} e^{25} - 300 \, \sqrt {x e + d} a b^{4} d e^{25} + 150 \, \sqrt {x e + d} a^{2} b^{3} e^{26}\right )} e^{\left (-30\right )} + \frac {2 \, {\left (150 \, {\left (x e + d\right )}^{2} b^{5} d^{3} - 25 \, {\left (x e + d\right )} b^{5} d^{4} + 3 \, b^{5} d^{5} - 450 \, {\left (x e + d\right )}^{2} a b^{4} d^{2} e + 100 \, {\left (x e + d\right )} a b^{4} d^{3} e - 15 \, a b^{4} d^{4} e + 450 \, {\left (x e + d\right )}^{2} a^{2} b^{3} d e^{2} - 150 \, {\left (x e + d\right )} a^{2} b^{3} d^{2} e^{2} + 30 \, a^{2} b^{3} d^{3} e^{2} - 150 \, {\left (x e + d\right )}^{2} a^{3} b^{2} e^{3} + 100 \, {\left (x e + d\right )} a^{3} b^{2} d e^{3} - 30 \, a^{3} b^{2} d^{2} e^{3} - 25 \, {\left (x e + d\right )} a^{4} b e^{4} + 15 \, a^{4} b d e^{4} - 3 \, a^{5} e^{5}\right )} e^{\left (-6\right )}}{15 \, {\left (x e + d\right )}^{\frac {5}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.06, size = 273, normalized size = 1.77 \begin {gather*} -\frac {2 \left (-3 b^{5} e^{5} x^{5}-25 a \,b^{4} e^{5} x^{4}+10 b^{5} d \,e^{4} x^{4}-150 a^{2} b^{3} e^{5} x^{3}+200 a \,b^{4} d \,e^{4} x^{3}-80 b^{5} d^{2} e^{3} x^{3}+150 a^{3} b^{2} e^{5} x^{2}-900 a^{2} b^{3} d \,e^{4} x^{2}+1200 a \,b^{4} d^{2} e^{3} x^{2}-480 b^{5} d^{3} e^{2} x^{2}+25 a^{4} b \,e^{5} x +200 a^{3} b^{2} d \,e^{4} x -1200 a^{2} b^{3} d^{2} e^{3} x +1600 a \,b^{4} d^{3} e^{2} x -640 b^{5} d^{4} e x +3 a^{5} e^{5}+10 a^{4} b d \,e^{4}+80 a^{3} b^{2} d^{2} e^{3}-480 a^{2} b^{3} d^{3} e^{2}+640 a \,b^{4} d^{4} e -256 b^{5} d^{5}\right )}{15 \left (e x +d \right )^{\frac {5}{2}} e^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.54, size = 265, normalized size = 1.72 \begin {gather*} \frac {2 \, {\left (\frac {3 \, {\left (e x + d\right )}^{\frac {5}{2}} b^{5} - 25 \, {\left (b^{5} d - a b^{4} e\right )} {\left (e x + d\right )}^{\frac {3}{2}} + 150 \, {\left (b^{5} d^{2} - 2 \, a b^{4} d e + a^{2} b^{3} e^{2}\right )} \sqrt {e x + d}}{e^{5}} + \frac {3 \, b^{5} d^{5} - 15 \, a b^{4} d^{4} e + 30 \, a^{2} b^{3} d^{3} e^{2} - 30 \, a^{3} b^{2} d^{2} e^{3} + 15 \, a^{4} b d e^{4} - 3 \, a^{5} e^{5} + 150 \, {\left (b^{5} d^{3} - 3 \, a b^{4} d^{2} e + 3 \, a^{2} b^{3} d e^{2} - a^{3} b^{2} e^{3}\right )} {\left (e x + d\right )}^{2} - 25 \, {\left (b^{5} d^{4} - 4 \, a b^{4} d^{3} e + 6 \, a^{2} b^{3} d^{2} e^{2} - 4 \, a^{3} b^{2} d e^{3} + a^{4} b e^{4}\right )} {\left (e x + d\right )}}{{\left (e x + d\right )}^{\frac {5}{2}} e^{5}}\right )}}{15 \, e} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.08, size = 255, normalized size = 1.66 \begin {gather*} \frac {2\,b^5\,{\left (d+e\,x\right )}^{5/2}}{5\,e^6}-\frac {\left (d+e\,x\right )\,\left (\frac {10\,a^4\,b\,e^4}{3}-\frac {40\,a^3\,b^2\,d\,e^3}{3}+20\,a^2\,b^3\,d^2\,e^2-\frac {40\,a\,b^4\,d^3\,e}{3}+\frac {10\,b^5\,d^4}{3}\right )-{\left (d+e\,x\right )}^2\,\left (-20\,a^3\,b^2\,e^3+60\,a^2\,b^3\,d\,e^2-60\,a\,b^4\,d^2\,e+20\,b^5\,d^3\right )+\frac {2\,a^5\,e^5}{5}-\frac {2\,b^5\,d^5}{5}-4\,a^2\,b^3\,d^3\,e^2+4\,a^3\,b^2\,d^2\,e^3+2\,a\,b^4\,d^4\,e-2\,a^4\,b\,d\,e^4}{e^6\,{\left (d+e\,x\right )}^{5/2}}-\frac {\left (10\,b^5\,d-10\,a\,b^4\,e\right )\,{\left (d+e\,x\right )}^{3/2}}{3\,e^6}+\frac {20\,b^3\,{\left (a\,e-b\,d\right )}^2\,\sqrt {d+e\,x}}{e^6} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 4.35, size = 1428, normalized size = 9.27 \begin {gather*} \begin {cases} - \frac {6 a^{5} e^{5}}{15 d^{2} e^{6} \sqrt {d + e x} + 30 d e^{7} x \sqrt {d + e x} + 15 e^{8} x^{2} \sqrt {d + e x}} - \frac {20 a^{4} b d e^{4}}{15 d^{2} e^{6} \sqrt {d + e x} + 30 d e^{7} x \sqrt {d + e x} + 15 e^{8} x^{2} \sqrt {d + e x}} - \frac {50 a^{4} b e^{5} x}{15 d^{2} e^{6} \sqrt {d + e x} + 30 d e^{7} x \sqrt {d + e x} + 15 e^{8} x^{2} \sqrt {d + e x}} - \frac {160 a^{3} b^{2} d^{2} e^{3}}{15 d^{2} e^{6} \sqrt {d + e x} + 30 d e^{7} x \sqrt {d + e x} + 15 e^{8} x^{2} \sqrt {d + e x}} - \frac {400 a^{3} b^{2} d e^{4} x}{15 d^{2} e^{6} \sqrt {d + e x} + 30 d e^{7} x \sqrt {d + e x} + 15 e^{8} x^{2} \sqrt {d + e x}} - \frac {300 a^{3} b^{2} e^{5} x^{2}}{15 d^{2} e^{6} \sqrt {d + e x} + 30 d e^{7} x \sqrt {d + e x} + 15 e^{8} x^{2} \sqrt {d + e x}} + \frac {960 a^{2} b^{3} d^{3} e^{2}}{15 d^{2} e^{6} \sqrt {d + e x} + 30 d e^{7} x \sqrt {d + e x} + 15 e^{8} x^{2} \sqrt {d + e x}} + \frac {2400 a^{2} b^{3} d^{2} e^{3} x}{15 d^{2} e^{6} \sqrt {d + e x} + 30 d e^{7} x \sqrt {d + e x} + 15 e^{8} x^{2} \sqrt {d + e x}} + \frac {1800 a^{2} b^{3} d e^{4} x^{2}}{15 d^{2} e^{6} \sqrt {d + e x} + 30 d e^{7} x \sqrt {d + e x} + 15 e^{8} x^{2} \sqrt {d + e x}} + \frac {300 a^{2} b^{3} e^{5} x^{3}}{15 d^{2} e^{6} \sqrt {d + e x} + 30 d e^{7} x \sqrt {d + e x} + 15 e^{8} x^{2} \sqrt {d + e x}} - \frac {1280 a b^{4} d^{4} e}{15 d^{2} e^{6} \sqrt {d + e x} + 30 d e^{7} x \sqrt {d + e x} + 15 e^{8} x^{2} \sqrt {d + e x}} - \frac {3200 a b^{4} d^{3} e^{2} x}{15 d^{2} e^{6} \sqrt {d + e x} + 30 d e^{7} x \sqrt {d + e x} + 15 e^{8} x^{2} \sqrt {d + e x}} - \frac {2400 a b^{4} d^{2} e^{3} x^{2}}{15 d^{2} e^{6} \sqrt {d + e x} + 30 d e^{7} x \sqrt {d + e x} + 15 e^{8} x^{2} \sqrt {d + e x}} - \frac {400 a b^{4} d e^{4} x^{3}}{15 d^{2} e^{6} \sqrt {d + e x} + 30 d e^{7} x \sqrt {d + e x} + 15 e^{8} x^{2} \sqrt {d + e x}} + \frac {50 a b^{4} e^{5} x^{4}}{15 d^{2} e^{6} \sqrt {d + e x} + 30 d e^{7} x \sqrt {d + e x} + 15 e^{8} x^{2} \sqrt {d + e x}} + \frac {512 b^{5} d^{5}}{15 d^{2} e^{6} \sqrt {d + e x} + 30 d e^{7} x \sqrt {d + e x} + 15 e^{8} x^{2} \sqrt {d + e x}} + \frac {1280 b^{5} d^{4} e x}{15 d^{2} e^{6} \sqrt {d + e x} + 30 d e^{7} x \sqrt {d + e x} + 15 e^{8} x^{2} \sqrt {d + e x}} + \frac {960 b^{5} d^{3} e^{2} x^{2}}{15 d^{2} e^{6} \sqrt {d + e x} + 30 d e^{7} x \sqrt {d + e x} + 15 e^{8} x^{2} \sqrt {d + e x}} + \frac {160 b^{5} d^{2} e^{3} x^{3}}{15 d^{2} e^{6} \sqrt {d + e x} + 30 d e^{7} x \sqrt {d + e x} + 15 e^{8} x^{2} \sqrt {d + e x}} - \frac {20 b^{5} d e^{4} x^{4}}{15 d^{2} e^{6} \sqrt {d + e x} + 30 d e^{7} x \sqrt {d + e x} + 15 e^{8} x^{2} \sqrt {d + e x}} + \frac {6 b^{5} e^{5} x^{5}}{15 d^{2} e^{6} \sqrt {d + e x} + 30 d e^{7} x \sqrt {d + e x} + 15 e^{8} x^{2} \sqrt {d + e x}} & \text {for}\: e \neq 0 \\\frac {a^{5} x + \frac {5 a^{4} b x^{2}}{2} + \frac {10 a^{3} b^{2} x^{3}}{3} + \frac {5 a^{2} b^{3} x^{4}}{2} + a b^{4} x^{5} + \frac {b^{5} x^{6}}{6}}{d^{\frac {7}{2}}} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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