Optimal. Leaf size=320 \[ -\frac {20 b^2 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{9/2} (b d-a e)^3}{9 e^6 (a+b x)}+\frac {10 b \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{7/2} (b d-a e)^4}{7 e^6 (a+b x)}-\frac {2 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{5/2} (b d-a e)^5}{5 e^6 (a+b x)}+\frac {2 b^5 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{15/2}}{15 e^6 (a+b x)}-\frac {10 b^4 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{13/2} (b d-a e)}{13 e^6 (a+b x)}+\frac {20 b^3 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{11/2} (b d-a e)^2}{11 e^6 (a+b x)} \]
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Rubi [A] time = 0.10, antiderivative size = 320, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {646, 43} \[ \frac {2 b^5 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{15/2}}{15 e^6 (a+b x)}-\frac {10 b^4 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{13/2} (b d-a e)}{13 e^6 (a+b x)}+\frac {20 b^3 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{11/2} (b d-a e)^2}{11 e^6 (a+b x)}-\frac {20 b^2 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{9/2} (b d-a e)^3}{9 e^6 (a+b x)}+\frac {10 b \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{7/2} (b d-a e)^4}{7 e^6 (a+b x)}-\frac {2 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{5/2} (b d-a e)^5}{5 e^6 (a+b x)} \]
Antiderivative was successfully verified.
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Rule 43
Rule 646
Rubi steps
\begin {align*} \int (d+e x)^{3/2} \left (a^2+2 a b x+b^2 x^2\right )^{5/2} \, dx &=\frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \left (a b+b^2 x\right )^5 (d+e x)^{3/2} \, dx}{b^4 \left (a b+b^2 x\right )}\\ &=\frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \left (-\frac {b^5 (b d-a e)^5 (d+e x)^{3/2}}{e^5}+\frac {5 b^6 (b d-a e)^4 (d+e x)^{5/2}}{e^5}-\frac {10 b^7 (b d-a e)^3 (d+e x)^{7/2}}{e^5}+\frac {10 b^8 (b d-a e)^2 (d+e x)^{9/2}}{e^5}-\frac {5 b^9 (b d-a e) (d+e x)^{11/2}}{e^5}+\frac {b^{10} (d+e x)^{13/2}}{e^5}\right ) \, dx}{b^4 \left (a b+b^2 x\right )}\\ &=-\frac {2 (b d-a e)^5 (d+e x)^{5/2} \sqrt {a^2+2 a b x+b^2 x^2}}{5 e^6 (a+b x)}+\frac {10 b (b d-a e)^4 (d+e x)^{7/2} \sqrt {a^2+2 a b x+b^2 x^2}}{7 e^6 (a+b x)}-\frac {20 b^2 (b d-a e)^3 (d+e x)^{9/2} \sqrt {a^2+2 a b x+b^2 x^2}}{9 e^6 (a+b x)}+\frac {20 b^3 (b d-a e)^2 (d+e x)^{11/2} \sqrt {a^2+2 a b x+b^2 x^2}}{11 e^6 (a+b x)}-\frac {10 b^4 (b d-a e) (d+e x)^{13/2} \sqrt {a^2+2 a b x+b^2 x^2}}{13 e^6 (a+b x)}+\frac {2 b^5 (d+e x)^{15/2} \sqrt {a^2+2 a b x+b^2 x^2}}{15 e^6 (a+b x)}\\ \end {align*}
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Mathematica [A] time = 0.13, size = 141, normalized size = 0.44 \[ \frac {2 \left ((a+b x)^2\right )^{5/2} (d+e x)^{5/2} \left (-17325 b^4 (d+e x)^4 (b d-a e)+40950 b^3 (d+e x)^3 (b d-a e)^2-50050 b^2 (d+e x)^2 (b d-a e)^3+32175 b (d+e x) (b d-a e)^4-9009 (b d-a e)^5+3003 b^5 (d+e x)^5\right )}{45045 e^6 (a+b x)^5} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.78, size = 418, normalized size = 1.31 \[ \frac {2 \, {\left (3003 \, b^{5} e^{7} x^{7} - 256 \, b^{5} d^{7} + 1920 \, a b^{4} d^{6} e - 6240 \, a^{2} b^{3} d^{5} e^{2} + 11440 \, a^{3} b^{2} d^{4} e^{3} - 12870 \, a^{4} b d^{3} e^{4} + 9009 \, a^{5} d^{2} e^{5} + 231 \, {\left (16 \, b^{5} d e^{6} + 75 \, a b^{4} e^{7}\right )} x^{6} + 63 \, {\left (b^{5} d^{2} e^{5} + 350 \, a b^{4} d e^{6} + 650 \, a^{2} b^{3} e^{7}\right )} x^{5} - 35 \, {\left (2 \, b^{5} d^{3} e^{4} - 15 \, a b^{4} d^{2} e^{5} - 1560 \, a^{2} b^{3} d e^{6} - 1430 \, a^{3} b^{2} e^{7}\right )} x^{4} + 5 \, {\left (16 \, b^{5} d^{4} e^{3} - 120 \, a b^{4} d^{3} e^{4} + 390 \, a^{2} b^{3} d^{2} e^{5} + 14300 \, a^{3} b^{2} d e^{6} + 6435 \, a^{4} b e^{7}\right )} x^{3} - 3 \, {\left (32 \, b^{5} d^{5} e^{2} - 240 \, a b^{4} d^{4} e^{3} + 780 \, a^{2} b^{3} d^{3} e^{4} - 1430 \, a^{3} b^{2} d^{2} e^{5} - 17160 \, a^{4} b d e^{6} - 3003 \, a^{5} e^{7}\right )} x^{2} + {\left (128 \, b^{5} d^{6} e - 960 \, a b^{4} d^{5} e^{2} + 3120 \, a^{2} b^{3} d^{4} e^{3} - 5720 \, a^{3} b^{2} d^{3} e^{4} + 6435 \, a^{4} b d^{2} e^{5} + 18018 \, a^{5} d e^{6}\right )} x\right )} \sqrt {e x + d}}{45045 \, e^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.32, size = 1257, normalized size = 3.93 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 289, normalized size = 0.90 \[ \frac {2 \left (e x +d \right )^{\frac {5}{2}} \left (3003 b^{5} e^{5} x^{5}+17325 a \,b^{4} e^{5} x^{4}-2310 b^{5} d \,e^{4} x^{4}+40950 a^{2} b^{3} e^{5} x^{3}-12600 a \,b^{4} d \,e^{4} x^{3}+1680 b^{5} d^{2} e^{3} x^{3}+50050 a^{3} b^{2} e^{5} x^{2}-27300 a^{2} b^{3} d \,e^{4} x^{2}+8400 a \,b^{4} d^{2} e^{3} x^{2}-1120 b^{5} d^{3} e^{2} x^{2}+32175 a^{4} b \,e^{5} x -28600 a^{3} b^{2} d \,e^{4} x +15600 a^{2} b^{3} d^{2} e^{3} x -4800 a \,b^{4} d^{3} e^{2} x +640 b^{5} d^{4} e x +9009 a^{5} e^{5}-12870 a^{4} b d \,e^{4}+11440 a^{3} b^{2} d^{2} e^{3}-6240 a^{2} b^{3} d^{3} e^{2}+1920 a \,b^{4} d^{4} e -256 b^{5} d^{5}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}}}{45045 \left (b x +a \right )^{5} e^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.16, size = 418, normalized size = 1.31 \[ \frac {2 \, {\left (3003 \, b^{5} e^{7} x^{7} - 256 \, b^{5} d^{7} + 1920 \, a b^{4} d^{6} e - 6240 \, a^{2} b^{3} d^{5} e^{2} + 11440 \, a^{3} b^{2} d^{4} e^{3} - 12870 \, a^{4} b d^{3} e^{4} + 9009 \, a^{5} d^{2} e^{5} + 231 \, {\left (16 \, b^{5} d e^{6} + 75 \, a b^{4} e^{7}\right )} x^{6} + 63 \, {\left (b^{5} d^{2} e^{5} + 350 \, a b^{4} d e^{6} + 650 \, a^{2} b^{3} e^{7}\right )} x^{5} - 35 \, {\left (2 \, b^{5} d^{3} e^{4} - 15 \, a b^{4} d^{2} e^{5} - 1560 \, a^{2} b^{3} d e^{6} - 1430 \, a^{3} b^{2} e^{7}\right )} x^{4} + 5 \, {\left (16 \, b^{5} d^{4} e^{3} - 120 \, a b^{4} d^{3} e^{4} + 390 \, a^{2} b^{3} d^{2} e^{5} + 14300 \, a^{3} b^{2} d e^{6} + 6435 \, a^{4} b e^{7}\right )} x^{3} - 3 \, {\left (32 \, b^{5} d^{5} e^{2} - 240 \, a b^{4} d^{4} e^{3} + 780 \, a^{2} b^{3} d^{3} e^{4} - 1430 \, a^{3} b^{2} d^{2} e^{5} - 17160 \, a^{4} b d e^{6} - 3003 \, a^{5} e^{7}\right )} x^{2} + {\left (128 \, b^{5} d^{6} e - 960 \, a b^{4} d^{5} e^{2} + 3120 \, a^{2} b^{3} d^{4} e^{3} - 5720 \, a^{3} b^{2} d^{3} e^{4} + 6435 \, a^{4} b d^{2} e^{5} + 18018 \, a^{5} d e^{6}\right )} x\right )} \sqrt {e x + d}}{45045 \, e^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int {\left (d+e\,x\right )}^{3/2}\,{\left (a^2+2\,a\,b\,x+b^2\,x^2\right )}^{5/2} \,d x \]
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 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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