Optimal. Leaf size=233 \[ -\frac {1155 b^{3/2} e^4 \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{64 (b d-a e)^{13/2}}+\frac {1155 b e^4}{64 \sqrt {d+e x} (b d-a e)^6}+\frac {385 e^4}{64 (d+e x)^{3/2} (b d-a e)^5}+\frac {231 e^3}{64 (a+b x) (d+e x)^{3/2} (b d-a e)^4}-\frac {33 e^2}{32 (a+b x)^2 (d+e x)^{3/2} (b d-a e)^3}+\frac {11 e}{24 (a+b x)^3 (d+e x)^{3/2} (b d-a e)^2}-\frac {1}{4 (a+b x)^4 (d+e x)^{3/2} (b d-a e)} \]
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Rubi [A] time = 0.20, antiderivative size = 233, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 4, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.121, Rules used = {27, 51, 63, 208} \begin {gather*} -\frac {1155 b^{3/2} e^4 \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{64 (b d-a e)^{13/2}}+\frac {1155 b e^4}{64 \sqrt {d+e x} (b d-a e)^6}+\frac {385 e^4}{64 (d+e x)^{3/2} (b d-a e)^5}+\frac {231 e^3}{64 (a+b x) (d+e x)^{3/2} (b d-a e)^4}-\frac {33 e^2}{32 (a+b x)^2 (d+e x)^{3/2} (b d-a e)^3}+\frac {11 e}{24 (a+b x)^3 (d+e x)^{3/2} (b d-a e)^2}-\frac {1}{4 (a+b x)^4 (d+e x)^{3/2} (b d-a e)} \end {gather*}
Antiderivative was successfully verified.
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Rule 27
Rule 51
Rule 63
Rule 208
Rubi steps
\begin {align*} \int \frac {a+b x}{(d+e x)^{5/2} \left (a^2+2 a b x+b^2 x^2\right )^3} \, dx &=\int \frac {1}{(a+b x)^5 (d+e x)^{5/2}} \, dx\\ &=-\frac {1}{4 (b d-a e) (a+b x)^4 (d+e x)^{3/2}}-\frac {(11 e) \int \frac {1}{(a+b x)^4 (d+e x)^{5/2}} \, dx}{8 (b d-a e)}\\ &=-\frac {1}{4 (b d-a e) (a+b x)^4 (d+e x)^{3/2}}+\frac {11 e}{24 (b d-a e)^2 (a+b x)^3 (d+e x)^{3/2}}+\frac {\left (33 e^2\right ) \int \frac {1}{(a+b x)^3 (d+e x)^{5/2}} \, dx}{16 (b d-a e)^2}\\ &=-\frac {1}{4 (b d-a e) (a+b x)^4 (d+e x)^{3/2}}+\frac {11 e}{24 (b d-a e)^2 (a+b x)^3 (d+e x)^{3/2}}-\frac {33 e^2}{32 (b d-a e)^3 (a+b x)^2 (d+e x)^{3/2}}-\frac {\left (231 e^3\right ) \int \frac {1}{(a+b x)^2 (d+e x)^{5/2}} \, dx}{64 (b d-a e)^3}\\ &=-\frac {1}{4 (b d-a e) (a+b x)^4 (d+e x)^{3/2}}+\frac {11 e}{24 (b d-a e)^2 (a+b x)^3 (d+e x)^{3/2}}-\frac {33 e^2}{32 (b d-a e)^3 (a+b x)^2 (d+e x)^{3/2}}+\frac {231 e^3}{64 (b d-a e)^4 (a+b x) (d+e x)^{3/2}}+\frac {\left (1155 e^4\right ) \int \frac {1}{(a+b x) (d+e x)^{5/2}} \, dx}{128 (b d-a e)^4}\\ &=\frac {385 e^4}{64 (b d-a e)^5 (d+e x)^{3/2}}-\frac {1}{4 (b d-a e) (a+b x)^4 (d+e x)^{3/2}}+\frac {11 e}{24 (b d-a e)^2 (a+b x)^3 (d+e x)^{3/2}}-\frac {33 e^2}{32 (b d-a e)^3 (a+b x)^2 (d+e x)^{3/2}}+\frac {231 e^3}{64 (b d-a e)^4 (a+b x) (d+e x)^{3/2}}+\frac {\left (1155 b e^4\right ) \int \frac {1}{(a+b x) (d+e x)^{3/2}} \, dx}{128 (b d-a e)^5}\\ &=\frac {385 e^4}{64 (b d-a e)^5 (d+e x)^{3/2}}-\frac {1}{4 (b d-a e) (a+b x)^4 (d+e x)^{3/2}}+\frac {11 e}{24 (b d-a e)^2 (a+b x)^3 (d+e x)^{3/2}}-\frac {33 e^2}{32 (b d-a e)^3 (a+b x)^2 (d+e x)^{3/2}}+\frac {231 e^3}{64 (b d-a e)^4 (a+b x) (d+e x)^{3/2}}+\frac {1155 b e^4}{64 (b d-a e)^6 \sqrt {d+e x}}+\frac {\left (1155 b^2 e^4\right ) \int \frac {1}{(a+b x) \sqrt {d+e x}} \, dx}{128 (b d-a e)^6}\\ &=\frac {385 e^4}{64 (b d-a e)^5 (d+e x)^{3/2}}-\frac {1}{4 (b d-a e) (a+b x)^4 (d+e x)^{3/2}}+\frac {11 e}{24 (b d-a e)^2 (a+b x)^3 (d+e x)^{3/2}}-\frac {33 e^2}{32 (b d-a e)^3 (a+b x)^2 (d+e x)^{3/2}}+\frac {231 e^3}{64 (b d-a e)^4 (a+b x) (d+e x)^{3/2}}+\frac {1155 b e^4}{64 (b d-a e)^6 \sqrt {d+e x}}+\frac {\left (1155 b^2 e^3\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 d-a e)^6}\\ &=\frac {385 e^4}{64 (b d-a e)^5 (d+e x)^{3/2}}-\frac {1}{4 (b d-a e) (a+b x)^4 (d+e x)^{3/2}}+\frac {11 e}{24 (b d-a e)^2 (a+b x)^3 (d+e x)^{3/2}}-\frac {33 e^2}{32 (b d-a e)^3 (a+b x)^2 (d+e x)^{3/2}}+\frac {231 e^3}{64 (b d-a e)^4 (a+b x) (d+e x)^{3/2}}+\frac {1155 b e^4}{64 (b d-a e)^6 \sqrt {d+e x}}-\frac {1155 b^{3/2} e^4 \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{64 (b d-a e)^{13/2}}\\ \end {align*}
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Mathematica [C] time = 0.02, size = 52, normalized size = 0.22 \begin {gather*} -\frac {2 e^4 \, _2F_1\left (-\frac {3}{2},5;-\frac {1}{2};-\frac {b (d+e x)}{a e-b d}\right )}{3 (d+e x)^{3/2} (a e-b d)^5} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 1.58, size = 406, normalized size = 1.74 \begin {gather*} -\frac {e^4 \left (128 a^5 e^5-1408 a^4 b e^4 (d+e x)-640 a^4 b d e^4+1280 a^3 b^2 d^2 e^3-9207 a^3 b^2 e^3 (d+e x)^2+5632 a^3 b^2 d e^3 (d+e x)-1280 a^2 b^3 d^3 e^2-8448 a^2 b^3 d^2 e^2 (d+e x)-16863 a^2 b^3 e^2 (d+e x)^3+27621 a^2 b^3 d e^2 (d+e x)^2+640 a b^4 d^4 e+5632 a b^4 d^3 e (d+e x)-27621 a b^4 d^2 e (d+e x)^2-12705 a b^4 e (d+e x)^4+33726 a b^4 d e (d+e x)^3-128 b^5 d^5-1408 b^5 d^4 (d+e x)+9207 b^5 d^3 (d+e x)^2-16863 b^5 d^2 (d+e x)^3-3465 b^5 (d+e x)^5+12705 b^5 d (d+e x)^4\right )}{192 (d+e x)^{3/2} (b d-a e)^6 (-a e-b (d+e x)+b d)^4}-\frac {1155 b^{3/2} e^4 \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x} \sqrt {a e-b d}}{b d-a e}\right )}{64 (a e-b d)^{13/2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 0.49, size = 2494, normalized size = 10.70
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.27, size = 500, normalized size = 2.15 \begin {gather*} \frac {1155 \, b^{2} \arctan \left (\frac {\sqrt {x e + d} b}{\sqrt {-b^{2} d + a b e}}\right ) e^{4}}{64 \, {\left (b^{6} d^{6} - 6 \, a b^{5} d^{5} e + 15 \, a^{2} b^{4} d^{4} e^{2} - 20 \, a^{3} b^{3} d^{3} e^{3} + 15 \, a^{4} b^{2} d^{2} e^{4} - 6 \, a^{5} b d e^{5} + a^{6} e^{6}\right )} \sqrt {-b^{2} d + a b e}} + \frac {2 \, {\left (15 \, {\left (x e + d\right )} b e^{4} + b d e^{4} - a e^{5}\right )}}{3 \, {\left (b^{6} d^{6} - 6 \, a b^{5} d^{5} e + 15 \, a^{2} b^{4} d^{4} e^{2} - 20 \, a^{3} b^{3} d^{3} e^{3} + 15 \, a^{4} b^{2} d^{2} e^{4} - 6 \, a^{5} b d e^{5} + a^{6} e^{6}\right )} {\left (x e + d\right )}^{\frac {3}{2}}} + \frac {1545 \, {\left (x e + d\right )}^{\frac {7}{2}} b^{5} e^{4} - 5153 \, {\left (x e + d\right )}^{\frac {5}{2}} b^{5} d e^{4} + 5855 \, {\left (x e + d\right )}^{\frac {3}{2}} b^{5} d^{2} e^{4} - 2295 \, \sqrt {x e + d} b^{5} d^{3} e^{4} + 5153 \, {\left (x e + d\right )}^{\frac {5}{2}} a b^{4} e^{5} - 11710 \, {\left (x e + d\right )}^{\frac {3}{2}} a b^{4} d e^{5} + 6885 \, \sqrt {x e + d} a b^{4} d^{2} e^{5} + 5855 \, {\left (x e + d\right )}^{\frac {3}{2}} a^{2} b^{3} e^{6} - 6885 \, \sqrt {x e + d} a^{2} b^{3} d e^{6} + 2295 \, \sqrt {x e + d} a^{3} b^{2} e^{7}}{192 \, {\left (b^{6} d^{6} - 6 \, a b^{5} d^{5} e + 15 \, a^{2} b^{4} d^{4} e^{2} - 20 \, a^{3} b^{3} d^{3} e^{3} + 15 \, a^{4} b^{2} d^{2} e^{4} - 6 \, a^{5} b d e^{5} + a^{6} e^{6}\right )} {\left ({\left (x e + d\right )} b - b d + a e\right )}^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.08, size = 473, normalized size = 2.03 \begin {gather*} \frac {765 \sqrt {e x +d}\, a^{3} b^{2} e^{7}}{64 \left (a e -b d \right )^{6} \left (b e x +a e \right )^{4}}-\frac {2295 \sqrt {e x +d}\, a^{2} b^{3} d \,e^{6}}{64 \left (a e -b d \right )^{6} \left (b e x +a e \right )^{4}}+\frac {2295 \sqrt {e x +d}\, a \,b^{4} d^{2} e^{5}}{64 \left (a e -b d \right )^{6} \left (b e x +a e \right )^{4}}-\frac {765 \sqrt {e x +d}\, b^{5} d^{3} e^{4}}{64 \left (a e -b d \right )^{6} \left (b e x +a e \right )^{4}}+\frac {5855 \left (e x +d \right )^{\frac {3}{2}} a^{2} b^{3} e^{6}}{192 \left (a e -b d \right )^{6} \left (b e x +a e \right )^{4}}-\frac {5855 \left (e x +d \right )^{\frac {3}{2}} a \,b^{4} d \,e^{5}}{96 \left (a e -b d \right )^{6} \left (b e x +a e \right )^{4}}+\frac {5855 \left (e x +d \right )^{\frac {3}{2}} b^{5} d^{2} e^{4}}{192 \left (a e -b d \right )^{6} \left (b e x +a e \right )^{4}}+\frac {5153 \left (e x +d \right )^{\frac {5}{2}} a \,b^{4} e^{5}}{192 \left (a e -b d \right )^{6} \left (b e x +a e \right )^{4}}-\frac {5153 \left (e x +d \right )^{\frac {5}{2}} b^{5} d \,e^{4}}{192 \left (a e -b d \right )^{6} \left (b e x +a e \right )^{4}}+\frac {515 \left (e x +d \right )^{\frac {7}{2}} b^{5} e^{4}}{64 \left (a e -b d \right )^{6} \left (b e x +a e \right )^{4}}+\frac {1155 b^{2} e^{4} \arctan \left (\frac {\sqrt {e x +d}\, b}{\sqrt {\left (a e -b d \right ) b}}\right )}{64 \left (a e -b d \right )^{6} \sqrt {\left (a e -b d \right ) b}}+\frac {10 b \,e^{4}}{\left (a e -b d \right )^{6} \sqrt {e x +d}}-\frac {2 e^{4}}{3 \left (a e -b d \right )^{5} \left (e x +d \right )^{\frac {3}{2}}} \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 = 2.64, size = 436, normalized size = 1.87 \begin {gather*} \frac {\frac {3069\,b^2\,e^4\,{\left (d+e\,x\right )}^2}{64\,{\left (a\,e-b\,d\right )}^3}-\frac {2\,e^4}{3\,\left (a\,e-b\,d\right )}+\frac {5621\,b^3\,e^4\,{\left (d+e\,x\right )}^3}{64\,{\left (a\,e-b\,d\right )}^4}+\frac {4235\,b^4\,e^4\,{\left (d+e\,x\right )}^4}{64\,{\left (a\,e-b\,d\right )}^5}+\frac {1155\,b^5\,e^4\,{\left (d+e\,x\right )}^5}{64\,{\left (a\,e-b\,d\right )}^6}+\frac {22\,b\,e^4\,\left (d+e\,x\right )}{3\,{\left (a\,e-b\,d\right )}^2}}{b^4\,{\left (d+e\,x\right )}^{11/2}-\left (4\,b^4\,d-4\,a\,b^3\,e\right )\,{\left (d+e\,x\right )}^{9/2}+{\left (d+e\,x\right )}^{3/2}\,\left (a^4\,e^4-4\,a^3\,b\,d\,e^3+6\,a^2\,b^2\,d^2\,e^2-4\,a\,b^3\,d^3\,e+b^4\,d^4\right )+{\left (d+e\,x\right )}^{7/2}\,\left (6\,a^2\,b^2\,e^2-12\,a\,b^3\,d\,e+6\,b^4\,d^2\right )-{\left (d+e\,x\right )}^{5/2}\,\left (-4\,a^3\,b\,e^3+12\,a^2\,b^2\,d\,e^2-12\,a\,b^3\,d^2\,e+4\,b^4\,d^3\right )}+\frac {1155\,b^{3/2}\,e^4\,\mathrm {atan}\left (\frac {\sqrt {b}\,\sqrt {d+e\,x}\,\left (a^6\,e^6-6\,a^5\,b\,d\,e^5+15\,a^4\,b^2\,d^2\,e^4-20\,a^3\,b^3\,d^3\,e^3+15\,a^2\,b^4\,d^4\,e^2-6\,a\,b^5\,d^5\,e+b^6\,d^6\right )}{{\left (a\,e-b\,d\right )}^{13/2}}\right )}{64\,{\left (a\,e-b\,d\right )}^{13/2}} \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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