Optimal. Leaf size=318 \[ -\frac {20 b^2 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{7/2} (b d-a e)^3}{7 e^6 (a+b x)}+\frac {2 b \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{5/2} (b d-a e)^4}{e^6 (a+b x)}-\frac {2 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{3/2} (b d-a e)^5}{3 e^6 (a+b x)}+\frac {2 b^5 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{13/2}}{13 e^6 (a+b x)}-\frac {10 b^4 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{11/2} (b d-a e)}{11 e^6 (a+b x)}+\frac {20 b^3 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{9/2} (b d-a e)^2}{9 e^6 (a+b x)} \]
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Rubi [A] time = 0.09, antiderivative size = 318, 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)^{13/2}}{13 e^6 (a+b x)}-\frac {10 b^4 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{11/2} (b d-a e)}{11 e^6 (a+b x)}+\frac {20 b^3 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{9/2} (b d-a e)^2}{9 e^6 (a+b x)}-\frac {20 b^2 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{7/2} (b d-a e)^3}{7 e^6 (a+b x)}+\frac {2 b \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{5/2} (b d-a e)^4}{e^6 (a+b x)}-\frac {2 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{3/2} (b d-a e)^5}{3 e^6 (a+b x)} \end {gather*}
Antiderivative was successfully verified.
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Rule 43
Rule 646
Rubi steps
\begin {align*} \int \sqrt {d+e x} \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 \sqrt {d+e x} \, 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 \sqrt {d+e x}}{e^5}+\frac {5 b^6 (b d-a e)^4 (d+e x)^{3/2}}{e^5}-\frac {10 b^7 (b d-a e)^3 (d+e x)^{5/2}}{e^5}+\frac {10 b^8 (b d-a e)^2 (d+e x)^{7/2}}{e^5}-\frac {5 b^9 (b d-a e) (d+e x)^{9/2}}{e^5}+\frac {b^{10} (d+e x)^{11/2}}{e^5}\right ) \, dx}{b^4 \left (a b+b^2 x\right )}\\ &=-\frac {2 (b d-a e)^5 (d+e x)^{3/2} \sqrt {a^2+2 a b x+b^2 x^2}}{3 e^6 (a+b x)}+\frac {2 b (b d-a e)^4 (d+e x)^{5/2} \sqrt {a^2+2 a b x+b^2 x^2}}{e^6 (a+b x)}-\frac {20 b^2 (b d-a e)^3 (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^3 (b d-a e)^2 (d+e x)^{9/2} \sqrt {a^2+2 a b x+b^2 x^2}}{9 e^6 (a+b x)}-\frac {10 b^4 (b d-a e) (d+e x)^{11/2} \sqrt {a^2+2 a b x+b^2 x^2}}{11 e^6 (a+b x)}+\frac {2 b^5 (d+e x)^{13/2} \sqrt {a^2+2 a b x+b^2 x^2}}{13 e^6 (a+b x)}\\ \end {align*}
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Mathematica [A] time = 0.11, size = 235, normalized size = 0.74 \begin {gather*} \frac {2 \sqrt {(a+b x)^2} (d+e x)^{3/2} \left (3003 a^5 e^5+3003 a^4 b e^4 (3 e x-2 d)+858 a^3 b^2 e^3 \left (8 d^2-12 d e x+15 e^2 x^2\right )+286 a^2 b^3 e^2 \left (-16 d^3+24 d^2 e x-30 d e^2 x^2+35 e^3 x^3\right )+13 a b^4 e \left (128 d^4-192 d^3 e x+240 d^2 e^2 x^2-280 d e^3 x^3+315 e^4 x^4\right )+b^5 \left (-256 d^5+384 d^4 e x-480 d^3 e^2 x^2+560 d^2 e^3 x^3-630 d e^4 x^4+693 e^5 x^5\right )\right )}{9009 e^6 (a+b x)} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 34.38, size = 343, normalized size = 1.08 \begin {gather*} \frac {2 (d+e x)^{3/2} \sqrt {\frac {(a e+b e x)^2}{e^2}} \left (3003 a^5 e^5+9009 a^4 b e^4 (d+e x)-15015 a^4 b d e^4+30030 a^3 b^2 d^2 e^3+12870 a^3 b^2 e^3 (d+e x)^2-36036 a^3 b^2 d e^3 (d+e x)-30030 a^2 b^3 d^3 e^2+54054 a^2 b^3 d^2 e^2 (d+e x)+10010 a^2 b^3 e^2 (d+e x)^3-38610 a^2 b^3 d e^2 (d+e x)^2+15015 a b^4 d^4 e-36036 a b^4 d^3 e (d+e x)+38610 a b^4 d^2 e (d+e x)^2+4095 a b^4 e (d+e x)^4-20020 a b^4 d e (d+e x)^3-3003 b^5 d^5+9009 b^5 d^4 (d+e x)-12870 b^5 d^3 (d+e x)^2+10010 b^5 d^2 (d+e x)^3+693 b^5 (d+e x)^5-4095 b^5 d (d+e x)^4\right )}{9009 e^5 (a e+b e x)} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.41, size = 338, normalized size = 1.06 \begin {gather*} \frac {2 \, {\left (693 \, b^{5} e^{6} x^{6} - 256 \, b^{5} d^{6} + 1664 \, a b^{4} d^{5} e - 4576 \, a^{2} b^{3} d^{4} e^{2} + 6864 \, a^{3} b^{2} d^{3} e^{3} - 6006 \, a^{4} b d^{2} e^{4} + 3003 \, a^{5} d e^{5} + 63 \, {\left (b^{5} d e^{5} + 65 \, a b^{4} e^{6}\right )} x^{5} - 35 \, {\left (2 \, b^{5} d^{2} e^{4} - 13 \, a b^{4} d e^{5} - 286 \, a^{2} b^{3} e^{6}\right )} x^{4} + 10 \, {\left (8 \, b^{5} d^{3} e^{3} - 52 \, a b^{4} d^{2} e^{4} + 143 \, a^{2} b^{3} d e^{5} + 1287 \, a^{3} b^{2} e^{6}\right )} x^{3} - 3 \, {\left (32 \, b^{5} d^{4} e^{2} - 208 \, a b^{4} d^{3} e^{3} + 572 \, a^{2} b^{3} d^{2} e^{4} - 858 \, a^{3} b^{2} d e^{5} - 3003 \, a^{4} b e^{6}\right )} x^{2} + {\left (128 \, b^{5} d^{5} e - 832 \, a b^{4} d^{4} e^{2} + 2288 \, a^{2} b^{3} d^{3} e^{3} - 3432 \, a^{3} b^{2} d^{2} e^{4} + 3003 \, a^{4} b d e^{5} + 3003 \, a^{5} e^{6}\right )} x\right )} \sqrt {e x + d}}{9009 \, e^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.37, size = 750, normalized size = 2.36
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 = 289, normalized size = 0.91 \begin {gather*} \frac {2 \left (e x +d \right )^{\frac {3}{2}} \left (693 b^{5} e^{5} x^{5}+4095 a \,b^{4} e^{5} x^{4}-630 b^{5} d \,e^{4} x^{4}+10010 a^{2} b^{3} e^{5} x^{3}-3640 a \,b^{4} d \,e^{4} x^{3}+560 b^{5} d^{2} e^{3} x^{3}+12870 a^{3} b^{2} e^{5} x^{2}-8580 a^{2} b^{3} d \,e^{4} x^{2}+3120 a \,b^{4} d^{2} e^{3} x^{2}-480 b^{5} d^{3} e^{2} x^{2}+9009 a^{4} b \,e^{5} x -10296 a^{3} b^{2} d \,e^{4} x +6864 a^{2} b^{3} d^{2} e^{3} x -2496 a \,b^{4} d^{3} e^{2} x +384 b^{5} d^{4} e x +3003 a^{5} e^{5}-6006 a^{4} b d \,e^{4}+6864 a^{3} b^{2} d^{2} e^{3}-4576 a^{2} b^{3} d^{3} e^{2}+1664 a \,b^{4} d^{4} e -256 b^{5} d^{5}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}}}{9009 \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 [A] time = 1.16, size = 338, normalized size = 1.06 \begin {gather*} \frac {2 \, {\left (693 \, b^{5} e^{6} x^{6} - 256 \, b^{5} d^{6} + 1664 \, a b^{4} d^{5} e - 4576 \, a^{2} b^{3} d^{4} e^{2} + 6864 \, a^{3} b^{2} d^{3} e^{3} - 6006 \, a^{4} b d^{2} e^{4} + 3003 \, a^{5} d e^{5} + 63 \, {\left (b^{5} d e^{5} + 65 \, a b^{4} e^{6}\right )} x^{5} - 35 \, {\left (2 \, b^{5} d^{2} e^{4} - 13 \, a b^{4} d e^{5} - 286 \, a^{2} b^{3} e^{6}\right )} x^{4} + 10 \, {\left (8 \, b^{5} d^{3} e^{3} - 52 \, a b^{4} d^{2} e^{4} + 143 \, a^{2} b^{3} d e^{5} + 1287 \, a^{3} b^{2} e^{6}\right )} x^{3} - 3 \, {\left (32 \, b^{5} d^{4} e^{2} - 208 \, a b^{4} d^{3} e^{3} + 572 \, a^{2} b^{3} d^{2} e^{4} - 858 \, a^{3} b^{2} d e^{5} - 3003 \, a^{4} b e^{6}\right )} x^{2} + {\left (128 \, b^{5} d^{5} e - 832 \, a b^{4} d^{4} e^{2} + 2288 \, a^{2} b^{3} d^{3} e^{3} - 3432 \, a^{3} b^{2} d^{2} e^{4} + 3003 \, a^{4} b d e^{5} + 3003 \, a^{5} e^{6}\right )} x\right )} \sqrt {e x + d}}{9009 \, 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 \sqrt {d+e\,x}\,{\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] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \sqrt {d + e x} \left (\left (a + b x\right )^{2}\right )^{\frac {5}{2}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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