Optimal. Leaf size=171 \[ -\frac {315 e^4 \sqrt {b d-a e} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{64 b^{11/2}}-\frac {105 e^3 (d+e x)^{3/2}}{64 b^4 (a+b x)}-\frac {21 e^2 (d+e x)^{5/2}}{32 b^3 (a+b x)^2}-\frac {3 e (d+e x)^{7/2}}{8 b^2 (a+b x)^3}-\frac {(d+e x)^{9/2}}{4 b (a+b x)^4}+\frac {315 e^4 \sqrt {d+e x}}{64 b^5} \]
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Rubi [A] time = 0.08, antiderivative size = 171, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 5, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.152, Rules used = {27, 47, 50, 63, 208} \begin {gather*} -\frac {21 e^2 (d+e x)^{5/2}}{32 b^3 (a+b x)^2}-\frac {105 e^3 (d+e x)^{3/2}}{64 b^4 (a+b x)}-\frac {315 e^4 \sqrt {b d-a e} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{64 b^{11/2}}-\frac {3 e (d+e x)^{7/2}}{8 b^2 (a+b x)^3}-\frac {(d+e x)^{9/2}}{4 b (a+b x)^4}+\frac {315 e^4 \sqrt {d+e x}}{64 b^5} \end {gather*}
Antiderivative was successfully verified.
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Rule 27
Rule 47
Rule 50
Rule 63
Rule 208
Rubi steps
\begin {align*} \int \frac {(a+b x) (d+e x)^{9/2}}{\left (a^2+2 a b x+b^2 x^2\right )^3} \, dx &=\int \frac {(d+e x)^{9/2}}{(a+b x)^5} \, dx\\ &=-\frac {(d+e x)^{9/2}}{4 b (a+b x)^4}+\frac {(9 e) \int \frac {(d+e x)^{7/2}}{(a+b x)^4} \, dx}{8 b}\\ &=-\frac {3 e (d+e x)^{7/2}}{8 b^2 (a+b x)^3}-\frac {(d+e x)^{9/2}}{4 b (a+b x)^4}+\frac {\left (21 e^2\right ) \int \frac {(d+e x)^{5/2}}{(a+b x)^3} \, dx}{16 b^2}\\ &=-\frac {21 e^2 (d+e x)^{5/2}}{32 b^3 (a+b x)^2}-\frac {3 e (d+e x)^{7/2}}{8 b^2 (a+b x)^3}-\frac {(d+e x)^{9/2}}{4 b (a+b x)^4}+\frac {\left (105 e^3\right ) \int \frac {(d+e x)^{3/2}}{(a+b x)^2} \, dx}{64 b^3}\\ &=-\frac {105 e^3 (d+e x)^{3/2}}{64 b^4 (a+b x)}-\frac {21 e^2 (d+e x)^{5/2}}{32 b^3 (a+b x)^2}-\frac {3 e (d+e x)^{7/2}}{8 b^2 (a+b x)^3}-\frac {(d+e x)^{9/2}}{4 b (a+b x)^4}+\frac {\left (315 e^4\right ) \int \frac {\sqrt {d+e x}}{a+b x} \, dx}{128 b^4}\\ &=\frac {315 e^4 \sqrt {d+e x}}{64 b^5}-\frac {105 e^3 (d+e x)^{3/2}}{64 b^4 (a+b x)}-\frac {21 e^2 (d+e x)^{5/2}}{32 b^3 (a+b x)^2}-\frac {3 e (d+e x)^{7/2}}{8 b^2 (a+b x)^3}-\frac {(d+e x)^{9/2}}{4 b (a+b x)^4}+\frac {\left (315 e^4 (b d-a e)\right ) \int \frac {1}{(a+b x) \sqrt {d+e x}} \, dx}{128 b^5}\\ &=\frac {315 e^4 \sqrt {d+e x}}{64 b^5}-\frac {105 e^3 (d+e x)^{3/2}}{64 b^4 (a+b x)}-\frac {21 e^2 (d+e x)^{5/2}}{32 b^3 (a+b x)^2}-\frac {3 e (d+e x)^{7/2}}{8 b^2 (a+b x)^3}-\frac {(d+e x)^{9/2}}{4 b (a+b x)^4}+\frac {\left (315 e^3 (b d-a e)\right ) \operatorname {Subst}\left (\int \frac {1}{a-\frac {b d}{e}+\frac {b x^2}{e}} \, dx,x,\sqrt {d+e x}\right )}{64 b^5}\\ &=\frac {315 e^4 \sqrt {d+e x}}{64 b^5}-\frac {105 e^3 (d+e x)^{3/2}}{64 b^4 (a+b x)}-\frac {21 e^2 (d+e x)^{5/2}}{32 b^3 (a+b x)^2}-\frac {3 e (d+e x)^{7/2}}{8 b^2 (a+b x)^3}-\frac {(d+e x)^{9/2}}{4 b (a+b x)^4}-\frac {315 e^4 \sqrt {b d-a e} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{64 b^{11/2}}\\ \end {align*}
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Mathematica [C] time = 0.02, size = 52, normalized size = 0.30 \begin {gather*} \frac {2 e^4 (d+e x)^{11/2} \, _2F_1\left (5,\frac {11}{2};\frac {13}{2};-\frac {b (d+e x)}{a e-b d}\right )}{11 (a e-b d)^5} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 1.34, size = 296, normalized size = 1.73 \begin {gather*} \frac {e^4 \sqrt {d+e x} \left (315 a^4 e^4+1155 a^3 b e^3 (d+e x)-1260 a^3 b d e^3+1890 a^2 b^2 d^2 e^2+1533 a^2 b^2 e^2 (d+e x)^2-3465 a^2 b^2 d e^2 (d+e x)-1260 a b^3 d^3 e+3465 a b^3 d^2 e (d+e x)+837 a b^3 e (d+e x)^3-3066 a b^3 d e (d+e x)^2+315 b^4 d^4-1155 b^4 d^3 (d+e x)+1533 b^4 d^2 (d+e x)^2+128 b^4 (d+e x)^4-837 b^4 d (d+e x)^3\right )}{64 b^5 (a e+b (d+e x)-b d)^4}+\frac {315 e^4 \sqrt {a e-b d} \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x} \sqrt {a e-b d}}{b d-a e}\right )}{64 b^{11/2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 0.44, size = 680, normalized size = 3.98 \begin {gather*} \left [\frac {315 \, {\left (b^{4} e^{4} x^{4} + 4 \, a b^{3} e^{4} x^{3} + 6 \, a^{2} b^{2} e^{4} x^{2} + 4 \, a^{3} b e^{4} x + a^{4} e^{4}\right )} \sqrt {\frac {b d - a e}{b}} \log \left (\frac {b e x + 2 \, b d - a e - 2 \, \sqrt {e x + d} b \sqrt {\frac {b d - a e}{b}}}{b x + a}\right ) + 2 \, {\left (128 \, b^{4} e^{4} x^{4} - 16 \, b^{4} d^{4} - 24 \, a b^{3} d^{3} e - 42 \, a^{2} b^{2} d^{2} e^{2} - 105 \, a^{3} b d e^{3} + 315 \, a^{4} e^{4} - {\left (325 \, b^{4} d e^{3} - 837 \, a b^{3} e^{4}\right )} x^{3} - 3 \, {\left (70 \, b^{4} d^{2} e^{2} + 185 \, a b^{3} d e^{3} - 511 \, a^{2} b^{2} e^{4}\right )} x^{2} - {\left (88 \, b^{4} d^{3} e + 156 \, a b^{3} d^{2} e^{2} + 399 \, a^{2} b^{2} d e^{3} - 1155 \, a^{3} b e^{4}\right )} x\right )} \sqrt {e x + d}}{128 \, {\left (b^{9} x^{4} + 4 \, a b^{8} x^{3} + 6 \, a^{2} b^{7} x^{2} + 4 \, a^{3} b^{6} x + a^{4} b^{5}\right )}}, -\frac {315 \, {\left (b^{4} e^{4} x^{4} + 4 \, a b^{3} e^{4} x^{3} + 6 \, a^{2} b^{2} e^{4} x^{2} + 4 \, a^{3} b e^{4} x + a^{4} e^{4}\right )} \sqrt {-\frac {b d - a e}{b}} \arctan \left (-\frac {\sqrt {e x + d} b \sqrt {-\frac {b d - a e}{b}}}{b d - a e}\right ) - {\left (128 \, b^{4} e^{4} x^{4} - 16 \, b^{4} d^{4} - 24 \, a b^{3} d^{3} e - 42 \, a^{2} b^{2} d^{2} e^{2} - 105 \, a^{3} b d e^{3} + 315 \, a^{4} e^{4} - {\left (325 \, b^{4} d e^{3} - 837 \, a b^{3} e^{4}\right )} x^{3} - 3 \, {\left (70 \, b^{4} d^{2} e^{2} + 185 \, a b^{3} d e^{3} - 511 \, a^{2} b^{2} e^{4}\right )} x^{2} - {\left (88 \, b^{4} d^{3} e + 156 \, a b^{3} d^{2} e^{2} + 399 \, a^{2} b^{2} d e^{3} - 1155 \, a^{3} b e^{4}\right )} x\right )} \sqrt {e x + d}}{64 \, {\left (b^{9} x^{4} + 4 \, a b^{8} x^{3} + 6 \, a^{2} b^{7} x^{2} + 4 \, a^{3} b^{6} x + a^{4} b^{5}\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.23, size = 343, normalized size = 2.01 \begin {gather*} \frac {315 \, {\left (b d e^{4} - a e^{5}\right )} \arctan \left (\frac {\sqrt {x e + d} b}{\sqrt {-b^{2} d + a b e}}\right )}{64 \, \sqrt {-b^{2} d + a b e} b^{5}} + \frac {2 \, \sqrt {x e + d} e^{4}}{b^{5}} - \frac {325 \, {\left (x e + d\right )}^{\frac {7}{2}} b^{4} d e^{4} - 765 \, {\left (x e + d\right )}^{\frac {5}{2}} b^{4} d^{2} e^{4} + 643 \, {\left (x e + d\right )}^{\frac {3}{2}} b^{4} d^{3} e^{4} - 187 \, \sqrt {x e + d} b^{4} d^{4} e^{4} - 325 \, {\left (x e + d\right )}^{\frac {7}{2}} a b^{3} e^{5} + 1530 \, {\left (x e + d\right )}^{\frac {5}{2}} a b^{3} d e^{5} - 1929 \, {\left (x e + d\right )}^{\frac {3}{2}} a b^{3} d^{2} e^{5} + 748 \, \sqrt {x e + d} a b^{3} d^{3} e^{5} - 765 \, {\left (x e + d\right )}^{\frac {5}{2}} a^{2} b^{2} e^{6} + 1929 \, {\left (x e + d\right )}^{\frac {3}{2}} a^{2} b^{2} d e^{6} - 1122 \, \sqrt {x e + d} a^{2} b^{2} d^{2} e^{6} - 643 \, {\left (x e + d\right )}^{\frac {3}{2}} a^{3} b e^{7} + 748 \, \sqrt {x e + d} a^{3} b d e^{7} - 187 \, \sqrt {x e + d} a^{4} e^{8}}{64 \, {\left ({\left (x e + d\right )} b - b d + a e\right )}^{4} b^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.07, size = 497, normalized size = 2.91 \begin {gather*} \frac {187 \sqrt {e x +d}\, a^{4} e^{8}}{64 \left (b e x +a e \right )^{4} b^{5}}-\frac {187 \sqrt {e x +d}\, a^{3} d \,e^{7}}{16 \left (b e x +a e \right )^{4} b^{4}}+\frac {561 \sqrt {e x +d}\, a^{2} d^{2} e^{6}}{32 \left (b e x +a e \right )^{4} b^{3}}-\frac {187 \sqrt {e x +d}\, a \,d^{3} e^{5}}{16 \left (b e x +a e \right )^{4} b^{2}}+\frac {187 \sqrt {e x +d}\, d^{4} e^{4}}{64 \left (b e x +a e \right )^{4} b}+\frac {643 \left (e x +d \right )^{\frac {3}{2}} a^{3} e^{7}}{64 \left (b e x +a e \right )^{4} b^{4}}-\frac {1929 \left (e x +d \right )^{\frac {3}{2}} a^{2} d \,e^{6}}{64 \left (b e x +a e \right )^{4} b^{3}}+\frac {1929 \left (e x +d \right )^{\frac {3}{2}} a \,d^{2} e^{5}}{64 \left (b e x +a e \right )^{4} b^{2}}-\frac {643 \left (e x +d \right )^{\frac {3}{2}} d^{3} e^{4}}{64 \left (b e x +a e \right )^{4} b}+\frac {765 \left (e x +d \right )^{\frac {5}{2}} a^{2} e^{6}}{64 \left (b e x +a e \right )^{4} b^{3}}-\frac {765 \left (e x +d \right )^{\frac {5}{2}} a d \,e^{5}}{32 \left (b e x +a e \right )^{4} b^{2}}+\frac {765 \left (e x +d \right )^{\frac {5}{2}} d^{2} e^{4}}{64 \left (b e x +a e \right )^{4} b}+\frac {325 \left (e x +d \right )^{\frac {7}{2}} a \,e^{5}}{64 \left (b e x +a e \right )^{4} b^{2}}-\frac {325 \left (e x +d \right )^{\frac {7}{2}} d \,e^{4}}{64 \left (b e x +a e \right )^{4} b}-\frac {315 a \,e^{5} \arctan \left (\frac {\sqrt {e x +d}\, b}{\sqrt {\left (a e -b d \right ) b}}\right )}{64 \sqrt {\left (a e -b d \right ) b}\, b^{5}}+\frac {315 d \,e^{4} \arctan \left (\frac {\sqrt {e x +d}\, b}{\sqrt {\left (a e -b d \right ) b}}\right )}{64 \sqrt {\left (a e -b d \right ) b}\, b^{4}}+\frac {2 \sqrt {e x +d}\, e^{4}}{b^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.25, size = 436, normalized size = 2.55 \begin {gather*} \frac {{\left (d+e\,x\right )}^{5/2}\,\left (\frac {765\,a^2\,b^2\,e^6}{64}-\frac {765\,a\,b^3\,d\,e^5}{32}+\frac {765\,b^4\,d^2\,e^4}{64}\right )+{\left (d+e\,x\right )}^{3/2}\,\left (\frac {643\,a^3\,b\,e^7}{64}-\frac {1929\,a^2\,b^2\,d\,e^6}{64}+\frac {1929\,a\,b^3\,d^2\,e^5}{64}-\frac {643\,b^4\,d^3\,e^4}{64}\right )+\left (\frac {325\,a\,b^3\,e^5}{64}-\frac {325\,b^4\,d\,e^4}{64}\right )\,{\left (d+e\,x\right )}^{7/2}+\sqrt {d+e\,x}\,\left (\frac {187\,a^4\,e^8}{64}-\frac {187\,a^3\,b\,d\,e^7}{16}+\frac {561\,a^2\,b^2\,d^2\,e^6}{32}-\frac {187\,a\,b^3\,d^3\,e^5}{16}+\frac {187\,b^4\,d^4\,e^4}{64}\right )}{b^9\,{\left (d+e\,x\right )}^4-\left (4\,b^9\,d-4\,a\,b^8\,e\right )\,{\left (d+e\,x\right )}^3+b^9\,d^4+{\left (d+e\,x\right )}^2\,\left (6\,a^2\,b^7\,e^2-12\,a\,b^8\,d\,e+6\,b^9\,d^2\right )-\left (d+e\,x\right )\,\left (-4\,a^3\,b^6\,e^3+12\,a^2\,b^7\,d\,e^2-12\,a\,b^8\,d^2\,e+4\,b^9\,d^3\right )+a^4\,b^5\,e^4-4\,a^3\,b^6\,d\,e^3+6\,a^2\,b^7\,d^2\,e^2-4\,a\,b^8\,d^3\,e}+\frac {2\,e^4\,\sqrt {d+e\,x}}{b^5}-\frac {315\,e^4\,\mathrm {atan}\left (\frac {\sqrt {b}\,e^4\,\sqrt {a\,e-b\,d}\,\sqrt {d+e\,x}}{a\,e^5-b\,d\,e^4}\right )\,\sqrt {a\,e-b\,d}}{64\,b^{11/2}} \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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