Optimal. Leaf size=100 \[ -\frac {6 b^2 (d+e x)^{11/2} (b d-a e)}{11 e^4}+\frac {2 b (d+e x)^{9/2} (b d-a e)^2}{3 e^4}-\frac {2 (d+e x)^{7/2} (b d-a e)^3}{7 e^4}+\frac {2 b^3 (d+e x)^{13/2}}{13 e^4} \]
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Rubi [A] time = 0.04, antiderivative size = 100, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.065, Rules used = {27, 43} \[ -\frac {6 b^2 (d+e x)^{11/2} (b d-a e)}{11 e^4}+\frac {2 b (d+e x)^{9/2} (b d-a e)^2}{3 e^4}-\frac {2 (d+e x)^{7/2} (b d-a e)^3}{7 e^4}+\frac {2 b^3 (d+e x)^{13/2}}{13 e^4} \]
Antiderivative was successfully verified.
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Rule 27
Rule 43
Rubi steps
\begin {align*} \int (a+b x) (d+e x)^{5/2} \left (a^2+2 a b x+b^2 x^2\right ) \, dx &=\int (a+b x)^3 (d+e x)^{5/2} \, dx\\ &=\int \left (\frac {(-b d+a e)^3 (d+e x)^{5/2}}{e^3}+\frac {3 b (b d-a e)^2 (d+e x)^{7/2}}{e^3}-\frac {3 b^2 (b d-a e) (d+e x)^{9/2}}{e^3}+\frac {b^3 (d+e x)^{11/2}}{e^3}\right ) \, dx\\ &=-\frac {2 (b d-a e)^3 (d+e x)^{7/2}}{7 e^4}+\frac {2 b (b d-a e)^2 (d+e x)^{9/2}}{3 e^4}-\frac {6 b^2 (b d-a e) (d+e x)^{11/2}}{11 e^4}+\frac {2 b^3 (d+e x)^{13/2}}{13 e^4}\\ \end {align*}
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Mathematica [A] time = 0.06, size = 79, normalized size = 0.79 \[ \frac {2 (d+e x)^{7/2} \left (-819 b^2 (d+e x)^2 (b d-a e)+1001 b (d+e x) (b d-a e)^2-429 (b d-a e)^3+231 b^3 (d+e x)^3\right )}{3003 e^4} \]
Antiderivative was successfully verified.
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fricas [B] time = 1.01, size = 268, normalized size = 2.68 \[ \frac {2 \, {\left (231 \, b^{3} e^{6} x^{6} - 16 \, b^{3} d^{6} + 104 \, a b^{2} d^{5} e - 286 \, a^{2} b d^{4} e^{2} + 429 \, a^{3} d^{3} e^{3} + 63 \, {\left (9 \, b^{3} d e^{5} + 13 \, a b^{2} e^{6}\right )} x^{5} + 7 \, {\left (53 \, b^{3} d^{2} e^{4} + 299 \, a b^{2} d e^{5} + 143 \, a^{2} b e^{6}\right )} x^{4} + {\left (5 \, b^{3} d^{3} e^{3} + 1469 \, a b^{2} d^{2} e^{4} + 2717 \, a^{2} b d e^{5} + 429 \, a^{3} e^{6}\right )} x^{3} - 3 \, {\left (2 \, b^{3} d^{4} e^{2} - 13 \, a b^{2} d^{3} e^{3} - 715 \, a^{2} b d^{2} e^{4} - 429 \, a^{3} d e^{5}\right )} x^{2} + {\left (8 \, b^{3} d^{5} e - 52 \, a b^{2} d^{4} e^{2} + 143 \, a^{2} b d^{3} e^{3} + 1287 \, a^{3} d^{2} e^{4}\right )} x\right )} \sqrt {e x + d}}{3003 \, e^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.22, size = 908, normalized size = 9.08 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 116, normalized size = 1.16 \[ \frac {2 \left (e x +d \right )^{\frac {7}{2}} \left (231 b^{3} e^{3} x^{3}+819 a \,b^{2} e^{3} x^{2}-126 b^{3} d \,e^{2} x^{2}+1001 a^{2} b \,e^{3} x -364 a \,b^{2} d \,e^{2} x +56 b^{3} d^{2} e x +429 a^{3} e^{3}-286 a^{2} b d \,e^{2}+104 a \,b^{2} d^{2} e -16 b^{3} d^{3}\right )}{3003 e^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.57, size = 118, normalized size = 1.18 \[ \frac {2 \, {\left (231 \, {\left (e x + d\right )}^{\frac {13}{2}} b^{3} - 819 \, {\left (b^{3} d - a b^{2} e\right )} {\left (e x + d\right )}^{\frac {11}{2}} + 1001 \, {\left (b^{3} d^{2} - 2 \, a b^{2} d e + a^{2} b e^{2}\right )} {\left (e x + d\right )}^{\frac {9}{2}} - 429 \, {\left (b^{3} d^{3} - 3 \, a b^{2} d^{2} e + 3 \, a^{2} b d e^{2} - a^{3} e^{3}\right )} {\left (e x + d\right )}^{\frac {7}{2}}\right )}}{3003 \, e^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.07, size = 87, normalized size = 0.87 \[ \frac {2\,b^3\,{\left (d+e\,x\right )}^{13/2}}{13\,e^4}-\frac {\left (6\,b^3\,d-6\,a\,b^2\,e\right )\,{\left (d+e\,x\right )}^{11/2}}{11\,e^4}+\frac {2\,{\left (a\,e-b\,d\right )}^3\,{\left (d+e\,x\right )}^{7/2}}{7\,e^4}+\frac {2\,b\,{\left (a\,e-b\,d\right )}^2\,{\left (d+e\,x\right )}^{9/2}}{3\,e^4} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 4.59, size = 549, normalized size = 5.49 \[ \begin {cases} \frac {2 a^{3} d^{3} \sqrt {d + e x}}{7 e} + \frac {6 a^{3} d^{2} x \sqrt {d + e x}}{7} + \frac {6 a^{3} d e x^{2} \sqrt {d + e x}}{7} + \frac {2 a^{3} e^{2} x^{3} \sqrt {d + e x}}{7} - \frac {4 a^{2} b d^{4} \sqrt {d + e x}}{21 e^{2}} + \frac {2 a^{2} b d^{3} x \sqrt {d + e x}}{21 e} + \frac {10 a^{2} b d^{2} x^{2} \sqrt {d + e x}}{7} + \frac {38 a^{2} b d e x^{3} \sqrt {d + e x}}{21} + \frac {2 a^{2} b e^{2} x^{4} \sqrt {d + e x}}{3} + \frac {16 a b^{2} d^{5} \sqrt {d + e x}}{231 e^{3}} - \frac {8 a b^{2} d^{4} x \sqrt {d + e x}}{231 e^{2}} + \frac {2 a b^{2} d^{3} x^{2} \sqrt {d + e x}}{77 e} + \frac {226 a b^{2} d^{2} x^{3} \sqrt {d + e x}}{231} + \frac {46 a b^{2} d e x^{4} \sqrt {d + e x}}{33} + \frac {6 a b^{2} e^{2} x^{5} \sqrt {d + e x}}{11} - \frac {32 b^{3} d^{6} \sqrt {d + e x}}{3003 e^{4}} + \frac {16 b^{3} d^{5} x \sqrt {d + e x}}{3003 e^{3}} - \frac {4 b^{3} d^{4} x^{2} \sqrt {d + e x}}{1001 e^{2}} + \frac {10 b^{3} d^{3} x^{3} \sqrt {d + e x}}{3003 e} + \frac {106 b^{3} d^{2} x^{4} \sqrt {d + e x}}{429} + \frac {54 b^{3} d e x^{5} \sqrt {d + e x}}{143} + \frac {2 b^{3} e^{2} x^{6} \sqrt {d + e x}}{13} & \text {for}\: e \neq 0 \\d^{\frac {5}{2}} \left (a^{3} x + \frac {3 a^{2} b x^{2}}{2} + a b^{2} x^{3} + \frac {b^{3} x^{4}}{4}\right ) & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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