Optimal. Leaf size=179 \[ -\frac {12 b^5 (d+e x)^{5/2} (b d-a e)}{5 e^7}+\frac {10 b^4 (d+e x)^{3/2} (b d-a e)^2}{e^7}-\frac {40 b^3 \sqrt {d+e x} (b d-a e)^3}{e^7}-\frac {30 b^2 (b d-a e)^4}{e^7 \sqrt {d+e x}}+\frac {4 b (b d-a e)^5}{e^7 (d+e x)^{3/2}}-\frac {2 (b d-a e)^6}{5 e^7 (d+e x)^{5/2}}+\frac {2 b^6 (d+e x)^{7/2}}{7 e^7} \]
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Rubi [A] time = 0.06, antiderivative size = 179, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {27, 43} \begin {gather*} -\frac {12 b^5 (d+e x)^{5/2} (b d-a e)}{5 e^7}+\frac {10 b^4 (d+e x)^{3/2} (b d-a e)^2}{e^7}-\frac {40 b^3 \sqrt {d+e x} (b d-a e)^3}{e^7}-\frac {30 b^2 (b d-a e)^4}{e^7 \sqrt {d+e x}}+\frac {4 b (b d-a e)^5}{e^7 (d+e x)^{3/2}}-\frac {2 (b d-a e)^6}{5 e^7 (d+e x)^{5/2}}+\frac {2 b^6 (d+e x)^{7/2}}{7 e^7} \end {gather*}
Antiderivative was successfully verified.
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Rule 27
Rule 43
Rubi steps
\begin {align*} \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^3}{(d+e x)^{7/2}} \, dx &=\int \frac {(a+b x)^6}{(d+e x)^{7/2}} \, dx\\ &=\int \left (\frac {(-b d+a e)^6}{e^6 (d+e x)^{7/2}}-\frac {6 b (b d-a e)^5}{e^6 (d+e x)^{5/2}}+\frac {15 b^2 (b d-a e)^4}{e^6 (d+e x)^{3/2}}-\frac {20 b^3 (b d-a e)^3}{e^6 \sqrt {d+e x}}+\frac {15 b^4 (b d-a e)^2 \sqrt {d+e x}}{e^6}-\frac {6 b^5 (b d-a e) (d+e x)^{3/2}}{e^6}+\frac {b^6 (d+e x)^{5/2}}{e^6}\right ) \, dx\\ &=-\frac {2 (b d-a e)^6}{5 e^7 (d+e x)^{5/2}}+\frac {4 b (b d-a e)^5}{e^7 (d+e x)^{3/2}}-\frac {30 b^2 (b d-a e)^4}{e^7 \sqrt {d+e x}}-\frac {40 b^3 (b d-a e)^3 \sqrt {d+e x}}{e^7}+\frac {10 b^4 (b d-a e)^2 (d+e x)^{3/2}}{e^7}-\frac {12 b^5 (b d-a e) (d+e x)^{5/2}}{5 e^7}+\frac {2 b^6 (d+e x)^{7/2}}{7 e^7}\\ \end {align*}
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Mathematica [A] time = 0.08, size = 145, normalized size = 0.81 \begin {gather*} \frac {2 \left (-42 b^5 (d+e x)^5 (b d-a e)+175 b^4 (d+e x)^4 (b d-a e)^2-700 b^3 (d+e x)^3 (b d-a e)^3-525 b^2 (d+e x)^2 (b d-a e)^4+70 b (d+e x) (b d-a e)^5-7 (b d-a e)^6+5 b^6 (d+e x)^6\right )}{35 e^7 (d+e x)^{5/2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [B] time = 0.10, size = 438, normalized size = 2.45 \begin {gather*} \frac {2 \left (-7 a^6 e^6-70 a^5 b e^5 (d+e x)+42 a^5 b d e^5-105 a^4 b^2 d^2 e^4-525 a^4 b^2 e^4 (d+e x)^2+350 a^4 b^2 d e^4 (d+e x)+140 a^3 b^3 d^3 e^3-700 a^3 b^3 d^2 e^3 (d+e x)+700 a^3 b^3 e^3 (d+e x)^3+2100 a^3 b^3 d e^3 (d+e x)^2-105 a^2 b^4 d^4 e^2+700 a^2 b^4 d^3 e^2 (d+e x)-3150 a^2 b^4 d^2 e^2 (d+e x)^2+175 a^2 b^4 e^2 (d+e x)^4-2100 a^2 b^4 d e^2 (d+e x)^3+42 a b^5 d^5 e-350 a b^5 d^4 e (d+e x)+2100 a b^5 d^3 e (d+e x)^2+2100 a b^5 d^2 e (d+e x)^3+42 a b^5 e (d+e x)^5-350 a b^5 d e (d+e x)^4-7 b^6 d^6+70 b^6 d^5 (d+e x)-525 b^6 d^4 (d+e x)^2-700 b^6 d^3 (d+e x)^3+175 b^6 d^2 (d+e x)^4+5 b^6 (d+e x)^6-42 b^6 d (d+e x)^5\right )}{35 e^7 (d+e x)^{5/2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 0.40, size = 388, normalized size = 2.17 \begin {gather*} \frac {2 \, {\left (5 \, b^{6} e^{6} x^{6} - 1024 \, b^{6} d^{6} + 3584 \, a b^{5} d^{5} e - 4480 \, a^{2} b^{4} d^{4} e^{2} + 2240 \, a^{3} b^{3} d^{3} e^{3} - 280 \, a^{4} b^{2} d^{2} e^{4} - 28 \, a^{5} b d e^{5} - 7 \, a^{6} e^{6} - 6 \, {\left (2 \, b^{6} d e^{5} - 7 \, a b^{5} e^{6}\right )} x^{5} + 5 \, {\left (8 \, b^{6} d^{2} e^{4} - 28 \, a b^{5} d e^{5} + 35 \, a^{2} b^{4} e^{6}\right )} x^{4} - 20 \, {\left (16 \, b^{6} d^{3} e^{3} - 56 \, a b^{5} d^{2} e^{4} + 70 \, a^{2} b^{4} d e^{5} - 35 \, a^{3} b^{3} e^{6}\right )} x^{3} - 15 \, {\left (128 \, b^{6} d^{4} e^{2} - 448 \, a b^{5} d^{3} e^{3} + 560 \, a^{2} b^{4} d^{2} e^{4} - 280 \, a^{3} b^{3} d e^{5} + 35 \, a^{4} b^{2} e^{6}\right )} x^{2} - 10 \, {\left (256 \, b^{6} d^{5} e - 896 \, a b^{5} d^{4} e^{2} + 1120 \, a^{2} b^{4} d^{3} e^{3} - 560 \, a^{3} b^{3} d^{2} e^{4} + 70 \, a^{4} b^{2} d e^{5} + 7 \, a^{5} b e^{6}\right )} x\right )} \sqrt {e x + d}}{35 \, {\left (e^{10} x^{3} + 3 \, d e^{9} x^{2} + 3 \, d^{2} e^{8} x + d^{3} e^{7}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.22, size = 458, normalized size = 2.56 \begin {gather*} \frac {2}{35} \, {\left (5 \, {\left (x e + d\right )}^{\frac {7}{2}} b^{6} e^{42} - 42 \, {\left (x e + d\right )}^{\frac {5}{2}} b^{6} d e^{42} + 175 \, {\left (x e + d\right )}^{\frac {3}{2}} b^{6} d^{2} e^{42} - 700 \, \sqrt {x e + d} b^{6} d^{3} e^{42} + 42 \, {\left (x e + d\right )}^{\frac {5}{2}} a b^{5} e^{43} - 350 \, {\left (x e + d\right )}^{\frac {3}{2}} a b^{5} d e^{43} + 2100 \, \sqrt {x e + d} a b^{5} d^{2} e^{43} + 175 \, {\left (x e + d\right )}^{\frac {3}{2}} a^{2} b^{4} e^{44} - 2100 \, \sqrt {x e + d} a^{2} b^{4} d e^{44} + 700 \, \sqrt {x e + d} a^{3} b^{3} e^{45}\right )} e^{\left (-49\right )} - \frac {2 \, {\left (75 \, {\left (x e + d\right )}^{2} b^{6} d^{4} - 10 \, {\left (x e + d\right )} b^{6} d^{5} + b^{6} d^{6} - 300 \, {\left (x e + d\right )}^{2} a b^{5} d^{3} e + 50 \, {\left (x e + d\right )} a b^{5} d^{4} e - 6 \, a b^{5} d^{5} e + 450 \, {\left (x e + d\right )}^{2} a^{2} b^{4} d^{2} e^{2} - 100 \, {\left (x e + d\right )} a^{2} b^{4} d^{3} e^{2} + 15 \, a^{2} b^{4} d^{4} e^{2} - 300 \, {\left (x e + d\right )}^{2} a^{3} b^{3} d e^{3} + 100 \, {\left (x e + d\right )} a^{3} b^{3} d^{2} e^{3} - 20 \, a^{3} b^{3} d^{3} e^{3} + 75 \, {\left (x e + d\right )}^{2} a^{4} b^{2} e^{4} - 50 \, {\left (x e + d\right )} a^{4} b^{2} d e^{4} + 15 \, a^{4} b^{2} d^{2} e^{4} + 10 \, {\left (x e + d\right )} a^{5} b e^{5} - 6 \, a^{5} b d e^{5} + a^{6} e^{6}\right )} e^{\left (-7\right )}}{5 \, {\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 = 0.05, size = 377, normalized size = 2.11 \begin {gather*} -\frac {2 \left (-5 b^{6} e^{6} x^{6}-42 a \,b^{5} e^{6} x^{5}+12 b^{6} d \,e^{5} x^{5}-175 a^{2} b^{4} e^{6} x^{4}+140 a \,b^{5} d \,e^{5} x^{4}-40 b^{6} d^{2} e^{4} x^{4}-700 a^{3} b^{3} e^{6} x^{3}+1400 a^{2} b^{4} d \,e^{5} x^{3}-1120 a \,b^{5} d^{2} e^{4} x^{3}+320 b^{6} d^{3} e^{3} x^{3}+525 a^{4} b^{2} e^{6} x^{2}-4200 a^{3} b^{3} d \,e^{5} x^{2}+8400 a^{2} b^{4} d^{2} e^{4} x^{2}-6720 a \,b^{5} d^{3} e^{3} x^{2}+1920 b^{6} d^{4} e^{2} x^{2}+70 a^{5} b \,e^{6} x +700 a^{4} b^{2} d \,e^{5} x -5600 a^{3} b^{3} d^{2} e^{4} x +11200 a^{2} b^{4} d^{3} e^{3} x -8960 a \,b^{5} d^{4} e^{2} x +2560 b^{6} d^{5} e x +7 a^{6} e^{6}+28 a^{5} b d \,e^{5}+280 a^{4} b^{2} d^{2} e^{4}-2240 a^{3} b^{3} d^{3} e^{3}+4480 a^{2} b^{4} d^{4} e^{2}-3584 a \,b^{5} d^{5} e +1024 b^{6} d^{6}\right )}{35 \left (e x +d \right )^{\frac {5}{2}} e^{7}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.05, size = 356, normalized size = 1.99 \begin {gather*} \frac {2 \, {\left (\frac {5 \, {\left (e x + d\right )}^{\frac {7}{2}} b^{6} - 42 \, {\left (b^{6} d - a b^{5} e\right )} {\left (e x + d\right )}^{\frac {5}{2}} + 175 \, {\left (b^{6} d^{2} - 2 \, a b^{5} d e + a^{2} b^{4} e^{2}\right )} {\left (e x + d\right )}^{\frac {3}{2}} - 700 \, {\left (b^{6} d^{3} - 3 \, a b^{5} d^{2} e + 3 \, a^{2} b^{4} d e^{2} - a^{3} b^{3} e^{3}\right )} \sqrt {e x + d}}{e^{6}} - \frac {7 \, {\left (b^{6} d^{6} - 6 \, a b^{5} d^{5} e + 15 \, a^{2} b^{4} d^{4} e^{2} - 20 \, a^{3} b^{3} d^{3} e^{3} + 15 \, a^{4} b^{2} d^{2} e^{4} - 6 \, a^{5} b d e^{5} + a^{6} e^{6} + 75 \, {\left (b^{6} d^{4} - 4 \, a b^{5} d^{3} e + 6 \, a^{2} b^{4} d^{2} e^{2} - 4 \, a^{3} b^{3} d e^{3} + a^{4} b^{2} e^{4}\right )} {\left (e x + d\right )}^{2} - 10 \, {\left (b^{6} d^{5} - 5 \, a b^{5} d^{4} e + 10 \, a^{2} b^{4} d^{3} e^{2} - 10 \, a^{3} b^{3} d^{2} e^{3} + 5 \, a^{4} b^{2} d e^{4} - a^{5} b e^{5}\right )} {\left (e x + d\right )}\right )}}{{\left (e x + d\right )}^{\frac {5}{2}} e^{6}}\right )}}{35 \, e} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.55, size = 322, normalized size = 1.80 \begin {gather*} \frac {2\,b^6\,{\left (d+e\,x\right )}^{7/2}}{7\,e^7}-\frac {\left (12\,b^6\,d-12\,a\,b^5\,e\right )\,{\left (d+e\,x\right )}^{5/2}}{5\,e^7}-\frac {{\left (d+e\,x\right )}^2\,\left (30\,a^4\,b^2\,e^4-120\,a^3\,b^3\,d\,e^3+180\,a^2\,b^4\,d^2\,e^2-120\,a\,b^5\,d^3\,e+30\,b^6\,d^4\right )-\left (d+e\,x\right )\,\left (-4\,a^5\,b\,e^5+20\,a^4\,b^2\,d\,e^4-40\,a^3\,b^3\,d^2\,e^3+40\,a^2\,b^4\,d^3\,e^2-20\,a\,b^5\,d^4\,e+4\,b^6\,d^5\right )+\frac {2\,a^6\,e^6}{5}+\frac {2\,b^6\,d^6}{5}+6\,a^2\,b^4\,d^4\,e^2-8\,a^3\,b^3\,d^3\,e^3+6\,a^4\,b^2\,d^2\,e^4-\frac {12\,a\,b^5\,d^5\,e}{5}-\frac {12\,a^5\,b\,d\,e^5}{5}}{e^7\,{\left (d+e\,x\right )}^{5/2}}+\frac {40\,b^3\,{\left (a\,e-b\,d\right )}^3\,\sqrt {d+e\,x}}{e^7}+\frac {10\,b^4\,{\left (a\,e-b\,d\right )}^2\,{\left (d+e\,x\right )}^{3/2}}{e^7} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 149.33, size = 221, normalized size = 1.23 \begin {gather*} \frac {2 b^{6} \left (d + e x\right )^{\frac {7}{2}}}{7 e^{7}} - \frac {30 b^{2} \left (a e - b d\right )^{4}}{e^{7} \sqrt {d + e x}} - \frac {4 b \left (a e - b d\right )^{5}}{e^{7} \left (d + e x\right )^{\frac {3}{2}}} + \frac {\left (d + e x\right )^{\frac {5}{2}} \left (12 a b^{5} e - 12 b^{6} d\right )}{5 e^{7}} + \frac {\left (d + e x\right )^{\frac {3}{2}} \left (30 a^{2} b^{4} e^{2} - 60 a b^{5} d e + 30 b^{6} d^{2}\right )}{3 e^{7}} + \frac {\sqrt {d + e x} \left (40 a^{3} b^{3} e^{3} - 120 a^{2} b^{4} d e^{2} + 120 a b^{5} d^{2} e - 40 b^{6} d^{3}\right )}{e^{7}} - \frac {2 \left (a e - b d\right )^{6}}{5 e^{7} \left (d + e x\right )^{\frac {5}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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