Optimal. Leaf size=71 \[ -\frac {4 b (d+e x)^{11/2} (b d-a e)}{11 e^3}+\frac {2 (d+e x)^{9/2} (b d-a e)^2}{9 e^3}+\frac {2 b^2 (d+e x)^{13/2}}{13 e^3} \]
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Rubi [A] time = 0.03, antiderivative size = 71, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {27, 43} \begin {gather*} -\frac {4 b (d+e x)^{11/2} (b d-a e)}{11 e^3}+\frac {2 (d+e x)^{9/2} (b d-a e)^2}{9 e^3}+\frac {2 b^2 (d+e x)^{13/2}}{13 e^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 27
Rule 43
Rubi steps
\begin {align*} \int (d+e x)^{7/2} \left (a^2+2 a b x+b^2 x^2\right ) \, dx &=\int (a+b x)^2 (d+e x)^{7/2} \, dx\\ &=\int \left (\frac {(-b d+a e)^2 (d+e x)^{7/2}}{e^2}-\frac {2 b (b d-a e) (d+e x)^{9/2}}{e^2}+\frac {b^2 (d+e x)^{11/2}}{e^2}\right ) \, dx\\ &=\frac {2 (b d-a e)^2 (d+e x)^{9/2}}{9 e^3}-\frac {4 b (b d-a e) (d+e x)^{11/2}}{11 e^3}+\frac {2 b^2 (d+e x)^{13/2}}{13 e^3}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 61, normalized size = 0.86 \begin {gather*} \frac {2 (d+e x)^{9/2} \left (143 a^2 e^2+26 a b e (9 e x-2 d)+b^2 \left (8 d^2-36 d e x+99 e^2 x^2\right )\right )}{1287 e^3} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.05, size = 72, normalized size = 1.01 \begin {gather*} \frac {2 (d+e x)^{9/2} \left (143 a^2 e^2+234 a b e (d+e x)-286 a b d e+143 b^2 d^2+99 b^2 (d+e x)^2-234 b^2 d (d+e x)\right )}{1287 e^3} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 0.40, size = 212, normalized size = 2.99 \begin {gather*} \frac {2 \, {\left (99 \, b^{2} e^{6} x^{6} + 8 \, b^{2} d^{6} - 52 \, a b d^{5} e + 143 \, a^{2} d^{4} e^{2} + 18 \, {\left (20 \, b^{2} d e^{5} + 13 \, a b e^{6}\right )} x^{5} + {\left (458 \, b^{2} d^{2} e^{4} + 884 \, a b d e^{5} + 143 \, a^{2} e^{6}\right )} x^{4} + 4 \, {\left (53 \, b^{2} d^{3} e^{3} + 299 \, a b d^{2} e^{4} + 143 \, a^{2} d e^{5}\right )} x^{3} + 3 \, {\left (b^{2} d^{4} e^{2} + 208 \, a b d^{3} e^{3} + 286 \, a^{2} d^{2} e^{4}\right )} x^{2} - 2 \, {\left (2 \, b^{2} d^{5} e - 13 \, a b d^{4} e^{2} - 286 \, a^{2} d^{3} e^{3}\right )} x\right )} \sqrt {e x + d}}{1287 \, e^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.22, size = 840, normalized size = 11.83
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 63, normalized size = 0.89 \begin {gather*} \frac {2 \left (e x +d \right )^{\frac {9}{2}} \left (99 b^{2} e^{2} x^{2}+234 a b \,e^{2} x -36 b^{2} d e x +143 a^{2} e^{2}-52 a b d e +8 b^{2} d^{2}\right )}{1287 e^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.09, size = 68, normalized size = 0.96 \begin {gather*} \frac {2 \, {\left (99 \, {\left (e x + d\right )}^{\frac {13}{2}} b^{2} - 234 \, {\left (b^{2} d - a b e\right )} {\left (e x + d\right )}^{\frac {11}{2}} + 143 \, {\left (b^{2} d^{2} - 2 \, a b d e + a^{2} e^{2}\right )} {\left (e x + d\right )}^{\frac {9}{2}}\right )}}{1287 \, e^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.09, size = 68, normalized size = 0.96 \begin {gather*} \frac {2\,{\left (d+e\,x\right )}^{9/2}\,\left (99\,b^2\,{\left (d+e\,x\right )}^2+143\,a^2\,e^2+143\,b^2\,d^2-234\,b^2\,d\,\left (d+e\,x\right )+234\,a\,b\,e\,\left (d+e\,x\right )-286\,a\,b\,d\,e\right )}{1287\,e^3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 8.35, size = 432, normalized size = 6.08 \begin {gather*} \begin {cases} \frac {2 a^{2} d^{4} \sqrt {d + e x}}{9 e} + \frac {8 a^{2} d^{3} x \sqrt {d + e x}}{9} + \frac {4 a^{2} d^{2} e x^{2} \sqrt {d + e x}}{3} + \frac {8 a^{2} d e^{2} x^{3} \sqrt {d + e x}}{9} + \frac {2 a^{2} e^{3} x^{4} \sqrt {d + e x}}{9} - \frac {8 a b d^{5} \sqrt {d + e x}}{99 e^{2}} + \frac {4 a b d^{4} x \sqrt {d + e x}}{99 e} + \frac {32 a b d^{3} x^{2} \sqrt {d + e x}}{33} + \frac {184 a b d^{2} e x^{3} \sqrt {d + e x}}{99} + \frac {136 a b d e^{2} x^{4} \sqrt {d + e x}}{99} + \frac {4 a b e^{3} x^{5} \sqrt {d + e x}}{11} + \frac {16 b^{2} d^{6} \sqrt {d + e x}}{1287 e^{3}} - \frac {8 b^{2} d^{5} x \sqrt {d + e x}}{1287 e^{2}} + \frac {2 b^{2} d^{4} x^{2} \sqrt {d + e x}}{429 e} + \frac {424 b^{2} d^{3} x^{3} \sqrt {d + e x}}{1287} + \frac {916 b^{2} d^{2} e x^{4} \sqrt {d + e x}}{1287} + \frac {80 b^{2} d e^{2} x^{5} \sqrt {d + e x}}{143} + \frac {2 b^{2} e^{3} x^{6} \sqrt {d + e x}}{13} & \text {for}\: e \neq 0 \\d^{\frac {7}{2}} \left (a^{2} x + a b x^{2} + \frac {b^{2} x^{3}}{3}\right ) & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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