Optimal. Leaf size=137 \[ -\frac {7 e (b d-a e)^{5/2} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{b^{9/2}}+\frac {7 e \sqrt {d+e x} (b d-a e)^2}{b^4}+\frac {7 e (d+e x)^{3/2} (b d-a e)}{3 b^3}-\frac {(d+e x)^{7/2}}{b (a+b x)}+\frac {7 e (d+e x)^{5/2}}{5 b^2} \]
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Rubi [A] time = 0.08, antiderivative size = 137, 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} \[ \frac {7 e (d+e x)^{3/2} (b d-a e)}{3 b^3}+\frac {7 e \sqrt {d+e x} (b d-a e)^2}{b^4}-\frac {7 e (b d-a e)^{5/2} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{b^{9/2}}-\frac {(d+e x)^{7/2}}{b (a+b x)}+\frac {7 e (d+e x)^{5/2}}{5 b^2} \]
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}}{a^2+2 a b x+b^2 x^2} \, dx &=\int \frac {(d+e x)^{7/2}}{(a+b x)^2} \, dx\\ &=-\frac {(d+e x)^{7/2}}{b (a+b x)}+\frac {(7 e) \int \frac {(d+e x)^{5/2}}{a+b x} \, dx}{2 b}\\ &=\frac {7 e (d+e x)^{5/2}}{5 b^2}-\frac {(d+e x)^{7/2}}{b (a+b x)}+\frac {(7 e (b d-a e)) \int \frac {(d+e x)^{3/2}}{a+b x} \, dx}{2 b^2}\\ &=\frac {7 e (b d-a e) (d+e x)^{3/2}}{3 b^3}+\frac {7 e (d+e x)^{5/2}}{5 b^2}-\frac {(d+e x)^{7/2}}{b (a+b x)}+\frac {\left (7 e (b d-a e)^2\right ) \int \frac {\sqrt {d+e x}}{a+b x} \, dx}{2 b^3}\\ &=\frac {7 e (b d-a e)^2 \sqrt {d+e x}}{b^4}+\frac {7 e (b d-a e) (d+e x)^{3/2}}{3 b^3}+\frac {7 e (d+e x)^{5/2}}{5 b^2}-\frac {(d+e x)^{7/2}}{b (a+b x)}+\frac {\left (7 e (b d-a e)^3\right ) \int \frac {1}{(a+b x) \sqrt {d+e x}} \, dx}{2 b^4}\\ &=\frac {7 e (b d-a e)^2 \sqrt {d+e x}}{b^4}+\frac {7 e (b d-a e) (d+e x)^{3/2}}{3 b^3}+\frac {7 e (d+e x)^{5/2}}{5 b^2}-\frac {(d+e x)^{7/2}}{b (a+b x)}+\frac {\left (7 (b d-a 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 )}{b^4}\\ &=\frac {7 e (b d-a e)^2 \sqrt {d+e x}}{b^4}+\frac {7 e (b d-a e) (d+e x)^{3/2}}{3 b^3}+\frac {7 e (d+e x)^{5/2}}{5 b^2}-\frac {(d+e x)^{7/2}}{b (a+b x)}-\frac {7 e (b d-a e)^{5/2} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{b^{9/2}}\\ \end {align*}
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Mathematica [C] time = 0.02, size = 50, normalized size = 0.36 \[ \frac {2 e (d+e x)^{9/2} \, _2F_1\left (2,\frac {9}{2};\frac {11}{2};-\frac {b (d+e x)}{a e-b d}\right )}{9 (a e-b d)^2} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.93, size = 486, normalized size = 3.55 \[ \left [\frac {105 \, {\left (a b^{2} d^{2} e - 2 \, a^{2} b d e^{2} + a^{3} e^{3} + {\left (b^{3} d^{2} e - 2 \, a b^{2} d e^{2} + a^{2} b e^{3}\right )} x\right )} \sqrt {\frac {b d - a e}{b}} \log \left (\frac {b e x + 2 \, b d - a e - 2 \, \sqrt {e x + d} b \sqrt {\frac {b d - a e}{b}}}{b x + a}\right ) + 2 \, {\left (6 \, b^{3} e^{3} x^{3} - 15 \, b^{3} d^{3} + 161 \, a b^{2} d^{2} e - 245 \, a^{2} b d e^{2} + 105 \, a^{3} e^{3} + 2 \, {\left (16 \, b^{3} d e^{2} - 7 \, a b^{2} e^{3}\right )} x^{2} + 2 \, {\left (58 \, b^{3} d^{2} e - 84 \, a b^{2} d e^{2} + 35 \, a^{2} b e^{3}\right )} x\right )} \sqrt {e x + d}}{30 \, {\left (b^{5} x + a b^{4}\right )}}, -\frac {105 \, {\left (a b^{2} d^{2} e - 2 \, a^{2} b d e^{2} + a^{3} e^{3} + {\left (b^{3} d^{2} e - 2 \, a b^{2} d e^{2} + a^{2} b e^{3}\right )} x\right )} \sqrt {-\frac {b d - a e}{b}} \arctan \left (-\frac {\sqrt {e x + d} b \sqrt {-\frac {b d - a e}{b}}}{b d - a e}\right ) - {\left (6 \, b^{3} e^{3} x^{3} - 15 \, b^{3} d^{3} + 161 \, a b^{2} d^{2} e - 245 \, a^{2} b d e^{2} + 105 \, a^{3} e^{3} + 2 \, {\left (16 \, b^{3} d e^{2} - 7 \, a b^{2} e^{3}\right )} x^{2} + 2 \, {\left (58 \, b^{3} d^{2} e - 84 \, a b^{2} d e^{2} + 35 \, a^{2} b e^{3}\right )} x\right )} \sqrt {e x + d}}{15 \, {\left (b^{5} x + a b^{4}\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.19, size = 281, normalized size = 2.05 \[ \frac {7 \, {\left (b^{3} d^{3} e - 3 \, a b^{2} d^{2} e^{2} + 3 \, a^{2} b d e^{3} - a^{3} e^{4}\right )} \arctan \left (\frac {\sqrt {x e + d} b}{\sqrt {-b^{2} d + a b e}}\right )}{\sqrt {-b^{2} d + a b e} b^{4}} - \frac {\sqrt {x e + d} b^{3} d^{3} e - 3 \, \sqrt {x e + d} a b^{2} d^{2} e^{2} + 3 \, \sqrt {x e + d} a^{2} b d e^{3} - \sqrt {x e + d} a^{3} e^{4}}{{\left ({\left (x e + d\right )} b - b d + a e\right )} b^{4}} + \frac {2 \, {\left (3 \, {\left (x e + d\right )}^{\frac {5}{2}} b^{8} e + 10 \, {\left (x e + d\right )}^{\frac {3}{2}} b^{8} d e + 45 \, \sqrt {x e + d} b^{8} d^{2} e - 10 \, {\left (x e + d\right )}^{\frac {3}{2}} a b^{7} e^{2} - 90 \, \sqrt {x e + d} a b^{7} d e^{2} + 45 \, \sqrt {x e + d} a^{2} b^{6} e^{3}\right )}}{15 \, b^{10}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.07, size = 387, normalized size = 2.82 \[ -\frac {7 a^{3} e^{4} \arctan \left (\frac {\sqrt {e x +d}\, b}{\sqrt {\left (a e -b d \right ) b}}\right )}{\sqrt {\left (a e -b d \right ) b}\, b^{4}}+\frac {21 a^{2} d \,e^{3} \arctan \left (\frac {\sqrt {e x +d}\, b}{\sqrt {\left (a e -b d \right ) b}}\right )}{\sqrt {\left (a e -b d \right ) b}\, b^{3}}-\frac {21 a \,d^{2} e^{2} \arctan \left (\frac {\sqrt {e x +d}\, b}{\sqrt {\left (a e -b d \right ) b}}\right )}{\sqrt {\left (a e -b d \right ) b}\, b^{2}}+\frac {7 d^{3} e \arctan \left (\frac {\sqrt {e x +d}\, b}{\sqrt {\left (a e -b d \right ) b}}\right )}{\sqrt {\left (a e -b d \right ) b}\, b}+\frac {\sqrt {e x +d}\, a^{3} e^{4}}{\left (b e x +a e \right ) b^{4}}-\frac {3 \sqrt {e x +d}\, a^{2} d \,e^{3}}{\left (b e x +a e \right ) b^{3}}+\frac {3 \sqrt {e x +d}\, a \,d^{2} e^{2}}{\left (b e x +a e \right ) b^{2}}-\frac {\sqrt {e x +d}\, d^{3} e}{\left (b e x +a e \right ) b}+\frac {6 \sqrt {e x +d}\, a^{2} e^{3}}{b^{4}}-\frac {12 \sqrt {e x +d}\, a d \,e^{2}}{b^{3}}+\frac {6 \sqrt {e x +d}\, d^{2} e}{b^{2}}-\frac {4 \left (e x +d \right )^{\frac {3}{2}} a \,e^{2}}{3 b^{3}}+\frac {4 \left (e x +d \right )^{\frac {3}{2}} d e}{3 b^{2}}+\frac {2 \left (e x +d \right )^{\frac {5}{2}} e}{5 b^{2}} \]
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 \[ \text {Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.09, size = 235, normalized size = 1.72 \[ \left (\frac {2\,e\,{\left (2\,b^2\,d-2\,a\,b\,e\right )}^2}{b^6}-\frac {2\,e\,{\left (a\,e-b\,d\right )}^2}{b^4}\right )\,\sqrt {d+e\,x}+\frac {\sqrt {d+e\,x}\,\left (a^3\,e^4-3\,a^2\,b\,d\,e^3+3\,a\,b^2\,d^2\,e^2-b^3\,d^3\,e\right )}{b^5\,\left (d+e\,x\right )-b^5\,d+a\,b^4\,e}+\frac {2\,e\,{\left (d+e\,x\right )}^{5/2}}{5\,b^2}+\frac {2\,e\,\left (2\,b^2\,d-2\,a\,b\,e\right )\,{\left (d+e\,x\right )}^{3/2}}{3\,b^4}-\frac {7\,e\,\mathrm {atan}\left (\frac {\sqrt {b}\,e\,{\left (a\,e-b\,d\right )}^{5/2}\,\sqrt {d+e\,x}}{a^3\,e^4-3\,a^2\,b\,d\,e^3+3\,a\,b^2\,d^2\,e^2-b^3\,d^3\,e}\right )\,{\left (a\,e-b\,d\right )}^{5/2}}{b^{9/2}} \]
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 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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