Optimal. Leaf size=320 \[ -\frac {20 b^2 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{11/2} (b d-a e)^3}{11 e^6 (a+b x)}+\frac {10 b \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{9/2} (b d-a e)^4}{9 e^6 (a+b x)}-\frac {2 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{7/2} (b d-a e)^5}{7 e^6 (a+b x)}+\frac {2 b^5 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{17/2}}{17 e^6 (a+b x)}-\frac {2 b^4 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{15/2} (b d-a e)}{3 e^6 (a+b x)}+\frac {20 b^3 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{13/2} (b d-a e)^2}{13 e^6 (a+b x)} \]
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Rubi [A] time = 0.12, 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} \begin {gather*} \frac {2 b^5 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{17/2}}{17 e^6 (a+b x)}-\frac {2 b^4 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{15/2} (b d-a e)}{3 e^6 (a+b x)}+\frac {20 b^3 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{13/2} (b d-a e)^2}{13 e^6 (a+b x)}-\frac {20 b^2 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{11/2} (b d-a e)^3}{11 e^6 (a+b x)}+\frac {10 b \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{9/2} (b d-a e)^4}{9 e^6 (a+b x)}-\frac {2 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{7/2} (b d-a e)^5}{7 e^6 (a+b x)} \end {gather*}
Antiderivative was successfully verified.
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Rule 43
Rule 646
Rubi steps
\begin {align*} \int (d+e x)^{5/2} \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)^{5/2} \, 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)^{5/2}}{e^5}+\frac {5 b^6 (b d-a e)^4 (d+e x)^{7/2}}{e^5}-\frac {10 b^7 (b d-a e)^3 (d+e x)^{9/2}}{e^5}+\frac {10 b^8 (b d-a e)^2 (d+e x)^{11/2}}{e^5}-\frac {5 b^9 (b d-a e) (d+e x)^{13/2}}{e^5}+\frac {b^{10} (d+e x)^{15/2}}{e^5}\right ) \, dx}{b^4 \left (a b+b^2 x\right )}\\ &=-\frac {2 (b d-a e)^5 (d+e x)^{7/2} \sqrt {a^2+2 a b x+b^2 x^2}}{7 e^6 (a+b x)}+\frac {10 b (b d-a e)^4 (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^2 (b d-a e)^3 (d+e x)^{11/2} \sqrt {a^2+2 a b x+b^2 x^2}}{11 e^6 (a+b x)}+\frac {20 b^3 (b d-a e)^2 (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^4 (b d-a e) (d+e x)^{15/2} \sqrt {a^2+2 a b x+b^2 x^2}}{3 e^6 (a+b x)}+\frac {2 b^5 (d+e x)^{17/2} \sqrt {a^2+2 a b x+b^2 x^2}}{17 e^6 (a+b x)}\\ \end {align*}
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Mathematica [A] time = 0.17, size = 141, normalized size = 0.44 \begin {gather*} \frac {2 \left ((a+b x)^2\right )^{5/2} (d+e x)^{7/2} \left (-51051 b^4 (d+e x)^4 (b d-a e)+117810 b^3 (d+e x)^3 (b d-a e)^2-139230 b^2 (d+e x)^2 (b d-a e)^3+85085 b (d+e x) (b d-a e)^4-21879 (b d-a e)^5+9009 b^5 (d+e x)^5\right )}{153153 e^6 (a+b x)^5} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 50.68, size = 343, normalized size = 1.07 \begin {gather*} \frac {2 (d+e x)^{7/2} \sqrt {\frac {(a e+b e x)^2}{e^2}} \left (21879 a^5 e^5+85085 a^4 b e^4 (d+e x)-109395 a^4 b d e^4+218790 a^3 b^2 d^2 e^3+139230 a^3 b^2 e^3 (d+e x)^2-340340 a^3 b^2 d e^3 (d+e x)-218790 a^2 b^3 d^3 e^2+510510 a^2 b^3 d^2 e^2 (d+e x)+117810 a^2 b^3 e^2 (d+e x)^3-417690 a^2 b^3 d e^2 (d+e x)^2+109395 a b^4 d^4 e-340340 a b^4 d^3 e (d+e x)+417690 a b^4 d^2 e (d+e x)^2+51051 a b^4 e (d+e x)^4-235620 a b^4 d e (d+e x)^3-21879 b^5 d^5+85085 b^5 d^4 (d+e x)-139230 b^5 d^3 (d+e x)^2+117810 b^5 d^2 (d+e x)^3+9009 b^5 (d+e x)^5-51051 b^5 d (d+e x)^4\right )}{153153 e^5 (a e+b e x)} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 0.41, size = 497, normalized size = 1.55 \begin {gather*} \frac {2 \, {\left (9009 \, b^{5} e^{8} x^{8} - 256 \, b^{5} d^{8} + 2176 \, a b^{4} d^{7} e - 8160 \, a^{2} b^{3} d^{6} e^{2} + 17680 \, a^{3} b^{2} d^{5} e^{3} - 24310 \, a^{4} b d^{4} e^{4} + 21879 \, a^{5} d^{3} e^{5} + 3003 \, {\left (7 \, b^{5} d e^{7} + 17 \, a b^{4} e^{8}\right )} x^{7} + 231 \, {\left (55 \, b^{5} d^{2} e^{6} + 527 \, a b^{4} d e^{7} + 510 \, a^{2} b^{3} e^{8}\right )} x^{6} + 63 \, {\left (b^{5} d^{3} e^{5} + 1207 \, a b^{4} d^{2} e^{6} + 4590 \, a^{2} b^{3} d e^{7} + 2210 \, a^{3} b^{2} e^{8}\right )} x^{5} - 35 \, {\left (2 \, b^{5} d^{4} e^{4} - 17 \, a b^{4} d^{3} e^{5} - 5406 \, a^{2} b^{3} d^{2} e^{6} - 10166 \, a^{3} b^{2} d e^{7} - 2431 \, a^{4} b e^{8}\right )} x^{4} + {\left (80 \, b^{5} d^{5} e^{3} - 680 \, a b^{4} d^{4} e^{4} + 2550 \, a^{2} b^{3} d^{3} e^{5} + 249730 \, a^{3} b^{2} d^{2} e^{6} + 230945 \, a^{4} b d e^{7} + 21879 \, a^{5} e^{8}\right )} x^{3} - 3 \, {\left (32 \, b^{5} d^{6} e^{2} - 272 \, a b^{4} d^{5} e^{3} + 1020 \, a^{2} b^{3} d^{4} e^{4} - 2210 \, a^{3} b^{2} d^{3} e^{5} - 60775 \, a^{4} b d^{2} e^{6} - 21879 \, a^{5} d e^{7}\right )} x^{2} + {\left (128 \, b^{5} d^{7} e - 1088 \, a b^{4} d^{6} e^{2} + 4080 \, a^{2} b^{3} d^{5} e^{3} - 8840 \, a^{3} b^{2} d^{4} e^{4} + 12155 \, a^{4} b d^{3} e^{5} + 65637 \, a^{5} d^{2} e^{6}\right )} x\right )} \sqrt {e x + d}}{153153 \, e^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.38, size = 1842, normalized size = 5.76
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 \begin {gather*} \frac {2 \left (e x +d \right )^{\frac {7}{2}} \left (9009 b^{5} e^{5} x^{5}+51051 a \,b^{4} e^{5} x^{4}-6006 b^{5} d \,e^{4} x^{4}+117810 a^{2} b^{3} e^{5} x^{3}-31416 a \,b^{4} d \,e^{4} x^{3}+3696 b^{5} d^{2} e^{3} x^{3}+139230 a^{3} b^{2} e^{5} x^{2}-64260 a^{2} b^{3} d \,e^{4} x^{2}+17136 a \,b^{4} d^{2} e^{3} x^{2}-2016 b^{5} d^{3} e^{2} x^{2}+85085 a^{4} b \,e^{5} x -61880 a^{3} b^{2} d \,e^{4} x +28560 a^{2} b^{3} d^{2} e^{3} x -7616 a \,b^{4} d^{3} e^{2} x +896 b^{5} d^{4} e x +21879 a^{5} e^{5}-24310 a^{4} b d \,e^{4}+17680 a^{3} b^{2} d^{2} e^{3}-8160 a^{2} b^{3} d^{3} e^{2}+2176 a \,b^{4} d^{4} e -256 b^{5} d^{5}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}}}{153153 \left (b x +a \right )^{5} e^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.19, size = 497, normalized size = 1.55 \begin {gather*} \frac {2 \, {\left (9009 \, b^{5} e^{8} x^{8} - 256 \, b^{5} d^{8} + 2176 \, a b^{4} d^{7} e - 8160 \, a^{2} b^{3} d^{6} e^{2} + 17680 \, a^{3} b^{2} d^{5} e^{3} - 24310 \, a^{4} b d^{4} e^{4} + 21879 \, a^{5} d^{3} e^{5} + 3003 \, {\left (7 \, b^{5} d e^{7} + 17 \, a b^{4} e^{8}\right )} x^{7} + 231 \, {\left (55 \, b^{5} d^{2} e^{6} + 527 \, a b^{4} d e^{7} + 510 \, a^{2} b^{3} e^{8}\right )} x^{6} + 63 \, {\left (b^{5} d^{3} e^{5} + 1207 \, a b^{4} d^{2} e^{6} + 4590 \, a^{2} b^{3} d e^{7} + 2210 \, a^{3} b^{2} e^{8}\right )} x^{5} - 35 \, {\left (2 \, b^{5} d^{4} e^{4} - 17 \, a b^{4} d^{3} e^{5} - 5406 \, a^{2} b^{3} d^{2} e^{6} - 10166 \, a^{3} b^{2} d e^{7} - 2431 \, a^{4} b e^{8}\right )} x^{4} + {\left (80 \, b^{5} d^{5} e^{3} - 680 \, a b^{4} d^{4} e^{4} + 2550 \, a^{2} b^{3} d^{3} e^{5} + 249730 \, a^{3} b^{2} d^{2} e^{6} + 230945 \, a^{4} b d e^{7} + 21879 \, a^{5} e^{8}\right )} x^{3} - 3 \, {\left (32 \, b^{5} d^{6} e^{2} - 272 \, a b^{4} d^{5} e^{3} + 1020 \, a^{2} b^{3} d^{4} e^{4} - 2210 \, a^{3} b^{2} d^{3} e^{5} - 60775 \, a^{4} b d^{2} e^{6} - 21879 \, a^{5} d e^{7}\right )} x^{2} + {\left (128 \, b^{5} d^{7} e - 1088 \, a b^{4} d^{6} e^{2} + 4080 \, a^{2} b^{3} d^{5} e^{3} - 8840 \, a^{3} b^{2} d^{4} e^{4} + 12155 \, a^{4} b d^{3} e^{5} + 65637 \, a^{5} d^{2} e^{6}\right )} x\right )} \sqrt {e x + d}}{153153 \, e^{6}} \end {gather*}
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 \begin {gather*} \int {\left (d+e\,x\right )}^{5/2}\,{\left (a^2+2\,a\,b\,x+b^2\,x^2\right )}^{5/2} \,d x \end {gather*}
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 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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