Optimal. Leaf size=145 \[ -\frac {35 e^3 \sqrt {b d-a e} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{8 b^{9/2}}-\frac {35 e^2 (d+e x)^{3/2}}{24 b^3 (a+b x)}-\frac {7 e (d+e x)^{5/2}}{12 b^2 (a+b x)^2}-\frac {(d+e x)^{7/2}}{3 b (a+b x)^3}+\frac {35 e^3 \sqrt {d+e x}}{8 b^4} \]
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Rubi [A] time = 0.07, antiderivative size = 145, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.179, Rules used = {27, 47, 50, 63, 208} \begin {gather*} -\frac {35 e^2 (d+e x)^{3/2}}{24 b^3 (a+b x)}-\frac {35 e^3 \sqrt {b d-a e} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{8 b^{9/2}}-\frac {7 e (d+e x)^{5/2}}{12 b^2 (a+b x)^2}-\frac {(d+e x)^{7/2}}{3 b (a+b x)^3}+\frac {35 e^3 \sqrt {d+e x}}{8 b^4} \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 {(d+e x)^{7/2}}{\left (a^2+2 a b x+b^2 x^2\right )^2} \, dx &=\int \frac {(d+e x)^{7/2}}{(a+b x)^4} \, dx\\ &=-\frac {(d+e x)^{7/2}}{3 b (a+b x)^3}+\frac {(7 e) \int \frac {(d+e x)^{5/2}}{(a+b x)^3} \, dx}{6 b}\\ &=-\frac {7 e (d+e x)^{5/2}}{12 b^2 (a+b x)^2}-\frac {(d+e x)^{7/2}}{3 b (a+b x)^3}+\frac {\left (35 e^2\right ) \int \frac {(d+e x)^{3/2}}{(a+b x)^2} \, dx}{24 b^2}\\ &=-\frac {35 e^2 (d+e x)^{3/2}}{24 b^3 (a+b x)}-\frac {7 e (d+e x)^{5/2}}{12 b^2 (a+b x)^2}-\frac {(d+e x)^{7/2}}{3 b (a+b x)^3}+\frac {\left (35 e^3\right ) \int \frac {\sqrt {d+e x}}{a+b x} \, dx}{16 b^3}\\ &=\frac {35 e^3 \sqrt {d+e x}}{8 b^4}-\frac {35 e^2 (d+e x)^{3/2}}{24 b^3 (a+b x)}-\frac {7 e (d+e x)^{5/2}}{12 b^2 (a+b x)^2}-\frac {(d+e x)^{7/2}}{3 b (a+b x)^3}+\frac {\left (35 e^3 (b d-a e)\right ) \int \frac {1}{(a+b x) \sqrt {d+e x}} \, dx}{16 b^4}\\ &=\frac {35 e^3 \sqrt {d+e x}}{8 b^4}-\frac {35 e^2 (d+e x)^{3/2}}{24 b^3 (a+b x)}-\frac {7 e (d+e x)^{5/2}}{12 b^2 (a+b x)^2}-\frac {(d+e x)^{7/2}}{3 b (a+b x)^3}+\frac {\left (35 e^2 (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 )}{8 b^4}\\ &=\frac {35 e^3 \sqrt {d+e x}}{8 b^4}-\frac {35 e^2 (d+e x)^{3/2}}{24 b^3 (a+b x)}-\frac {7 e (d+e x)^{5/2}}{12 b^2 (a+b x)^2}-\frac {(d+e x)^{7/2}}{3 b (a+b x)^3}-\frac {35 e^3 \sqrt {b d-a e} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{8 b^{9/2}}\\ \end {align*}
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Mathematica [C] time = 0.02, size = 52, normalized size = 0.36 \begin {gather*} \frac {2 e^3 (d+e x)^{9/2} \, _2F_1\left (4,\frac {9}{2};\frac {11}{2};-\frac {b (d+e x)}{a e-b d}\right )}{9 (a e-b d)^4} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.89, size = 215, normalized size = 1.48 \begin {gather*} \frac {e^3 \sqrt {d+e x} \left (105 a^3 e^3+280 a^2 b e^2 (d+e x)-315 a^2 b d e^2+315 a b^2 d^2 e+231 a b^2 e (d+e x)^2-560 a b^2 d e (d+e x)-105 b^3 d^3+280 b^3 d^2 (d+e x)+48 b^3 (d+e x)^3-231 b^3 d (d+e x)^2\right )}{24 b^4 (a e+b (d+e x)-b d)^3}+\frac {35 e^3 \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 )}{8 b^{9/2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 0.43, size = 498, normalized size = 3.43 \begin {gather*} \left [\frac {105 \, {\left (b^{3} e^{3} x^{3} + 3 \, a b^{2} e^{3} x^{2} + 3 \, a^{2} b e^{3} x + a^{3} e^{3}\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 (48 \, b^{3} e^{3} x^{3} - 8 \, b^{3} d^{3} - 14 \, a b^{2} d^{2} e - 35 \, a^{2} b d e^{2} + 105 \, a^{3} e^{3} - 3 \, {\left (29 \, b^{3} d e^{2} - 77 \, a b^{2} e^{3}\right )} x^{2} - 2 \, {\left (19 \, b^{3} d^{2} e + 49 \, a b^{2} d e^{2} - 140 \, a^{2} b e^{3}\right )} x\right )} \sqrt {e x + d}}{48 \, {\left (b^{7} x^{3} + 3 \, a b^{6} x^{2} + 3 \, a^{2} b^{5} x + a^{3} b^{4}\right )}}, -\frac {105 \, {\left (b^{3} e^{3} x^{3} + 3 \, a b^{2} e^{3} x^{2} + 3 \, a^{2} b e^{3} x + a^{3} e^{3}\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 (48 \, b^{3} e^{3} x^{3} - 8 \, b^{3} d^{3} - 14 \, a b^{2} d^{2} e - 35 \, a^{2} b d e^{2} + 105 \, a^{3} e^{3} - 3 \, {\left (29 \, b^{3} d e^{2} - 77 \, a b^{2} e^{3}\right )} x^{2} - 2 \, {\left (19 \, b^{3} d^{2} e + 49 \, a b^{2} d e^{2} - 140 \, a^{2} b e^{3}\right )} x\right )} \sqrt {e x + d}}{24 \, {\left (b^{7} x^{3} + 3 \, a b^{6} x^{2} + 3 \, a^{2} b^{5} x + a^{3} b^{4}\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.22, size = 248, normalized size = 1.71 \begin {gather*} \frac {35 \, {\left (b d e^{3} - a e^{4}\right )} \arctan \left (\frac {\sqrt {x e + d} b}{\sqrt {-b^{2} d + a b e}}\right )}{8 \, \sqrt {-b^{2} d + a b e} b^{4}} + \frac {2 \, \sqrt {x e + d} e^{3}}{b^{4}} - \frac {87 \, {\left (x e + d\right )}^{\frac {5}{2}} b^{3} d e^{3} - 136 \, {\left (x e + d\right )}^{\frac {3}{2}} b^{3} d^{2} e^{3} + 57 \, \sqrt {x e + d} b^{3} d^{3} e^{3} - 87 \, {\left (x e + d\right )}^{\frac {5}{2}} a b^{2} e^{4} + 272 \, {\left (x e + d\right )}^{\frac {3}{2}} a b^{2} d e^{4} - 171 \, \sqrt {x e + d} a b^{2} d^{2} e^{4} - 136 \, {\left (x e + d\right )}^{\frac {3}{2}} a^{2} b e^{5} + 171 \, \sqrt {x e + d} a^{2} b d e^{5} - 57 \, \sqrt {x e + d} a^{3} e^{6}}{24 \, {\left ({\left (x e + d\right )} b - b d + a e\right )}^{3} b^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.07, size = 352, normalized size = 2.43 \begin {gather*} \frac {19 \sqrt {e x +d}\, a^{3} e^{6}}{8 \left (b e x +a e \right )^{3} b^{4}}-\frac {57 \sqrt {e x +d}\, a^{2} d \,e^{5}}{8 \left (b e x +a e \right )^{3} b^{3}}+\frac {57 \sqrt {e x +d}\, a \,d^{2} e^{4}}{8 \left (b e x +a e \right )^{3} b^{2}}-\frac {19 \sqrt {e x +d}\, d^{3} e^{3}}{8 \left (b e x +a e \right )^{3} b}+\frac {17 \left (e x +d \right )^{\frac {3}{2}} a^{2} e^{5}}{3 \left (b e x +a e \right )^{3} b^{3}}-\frac {34 \left (e x +d \right )^{\frac {3}{2}} a d \,e^{4}}{3 \left (b e x +a e \right )^{3} b^{2}}+\frac {17 \left (e x +d \right )^{\frac {3}{2}} d^{2} e^{3}}{3 \left (b e x +a e \right )^{3} b}+\frac {29 \left (e x +d \right )^{\frac {5}{2}} a \,e^{4}}{8 \left (b e x +a e \right )^{3} b^{2}}-\frac {29 \left (e x +d \right )^{\frac {5}{2}} d \,e^{3}}{8 \left (b e x +a e \right )^{3} b}-\frac {35 a \,e^{4} \arctan \left (\frac {\sqrt {e x +d}\, b}{\sqrt {\left (a e -b d \right ) b}}\right )}{8 \sqrt {\left (a e -b d \right ) b}\, b^{4}}+\frac {35 d \,e^{3} \arctan \left (\frac {\sqrt {e x +d}\, b}{\sqrt {\left (a e -b d \right ) b}}\right )}{8 \sqrt {\left (a e -b d \right ) b}\, b^{3}}+\frac {2 \sqrt {e x +d}\, e^{3}}{b^{4}} \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 = 0.69, size = 303, normalized size = 2.09 \begin {gather*} \frac {\sqrt {d+e\,x}\,\left (\frac {19\,a^3\,e^6}{8}-\frac {57\,a^2\,b\,d\,e^5}{8}+\frac {57\,a\,b^2\,d^2\,e^4}{8}-\frac {19\,b^3\,d^3\,e^3}{8}\right )+\left (\frac {29\,a\,b^2\,e^4}{8}-\frac {29\,b^3\,d\,e^3}{8}\right )\,{\left (d+e\,x\right )}^{5/2}+{\left (d+e\,x\right )}^{3/2}\,\left (\frac {17\,a^2\,b\,e^5}{3}-\frac {34\,a\,b^2\,d\,e^4}{3}+\frac {17\,b^3\,d^2\,e^3}{3}\right )}{b^7\,{\left (d+e\,x\right )}^3-\left (3\,b^7\,d-3\,a\,b^6\,e\right )\,{\left (d+e\,x\right )}^2+\left (d+e\,x\right )\,\left (3\,a^2\,b^5\,e^2-6\,a\,b^6\,d\,e+3\,b^7\,d^2\right )-b^7\,d^3+a^3\,b^4\,e^3-3\,a^2\,b^5\,d\,e^2+3\,a\,b^6\,d^2\,e}+\frac {2\,e^3\,\sqrt {d+e\,x}}{b^4}-\frac {35\,e^3\,\mathrm {atan}\left (\frac {\sqrt {b}\,e^3\,\sqrt {a\,e-b\,d}\,\sqrt {d+e\,x}}{a\,e^4-b\,d\,e^3}\right )\,\sqrt {a\,e-b\,d}}{8\,b^{9/2}} \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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