Optimal. Leaf size=250 \[ -\frac {5 e^2 (-7 a B e+A b e+6 b B d) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{8 b^{9/2} \sqrt {b d-a e}}+\frac {5 e^2 \sqrt {d+e x} (-7 a B e+A b e+6 b B d)}{8 b^4 (b d-a e)}-\frac {5 e (d+e x)^{3/2} (-7 a B e+A b e+6 b B d)}{24 b^3 (a+b x) (b d-a e)}-\frac {(d+e x)^{5/2} (-7 a B e+A b e+6 b B d)}{12 b^2 (a+b x)^2 (b d-a e)}-\frac {(d+e x)^{7/2} (A b-a B)}{3 b (a+b x)^3 (b d-a e)} \]
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Rubi [A] time = 0.20, antiderivative size = 250, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {27, 78, 47, 50, 63, 208} \begin {gather*} \frac {5 e^2 \sqrt {d+e x} (-7 a B e+A b e+6 b B d)}{8 b^4 (b d-a e)}-\frac {5 e^2 (-7 a B e+A b e+6 b B d) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{8 b^{9/2} \sqrt {b d-a e}}-\frac {(d+e x)^{5/2} (-7 a B e+A b e+6 b B d)}{12 b^2 (a+b x)^2 (b d-a e)}-\frac {5 e (d+e x)^{3/2} (-7 a B e+A b e+6 b B d)}{24 b^3 (a+b x) (b d-a e)}-\frac {(d+e x)^{7/2} (A b-a B)}{3 b (a+b x)^3 (b d-a e)} \end {gather*}
Antiderivative was successfully verified.
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Rule 27
Rule 47
Rule 50
Rule 63
Rule 78
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 )^2} \, dx &=\int \frac {(A+B x) (d+e x)^{5/2}}{(a+b x)^4} \, dx\\ &=-\frac {(A b-a B) (d+e x)^{7/2}}{3 b (b d-a e) (a+b x)^3}+\frac {(6 b B d+A b e-7 a B e) \int \frac {(d+e x)^{5/2}}{(a+b x)^3} \, dx}{6 b (b d-a e)}\\ &=-\frac {(6 b B d+A b e-7 a B e) (d+e x)^{5/2}}{12 b^2 (b d-a e) (a+b x)^2}-\frac {(A b-a B) (d+e x)^{7/2}}{3 b (b d-a e) (a+b x)^3}+\frac {(5 e (6 b B d+A b e-7 a B e)) \int \frac {(d+e x)^{3/2}}{(a+b x)^2} \, dx}{24 b^2 (b d-a e)}\\ &=-\frac {5 e (6 b B d+A b e-7 a B e) (d+e x)^{3/2}}{24 b^3 (b d-a e) (a+b x)}-\frac {(6 b B d+A b e-7 a B e) (d+e x)^{5/2}}{12 b^2 (b d-a e) (a+b x)^2}-\frac {(A b-a B) (d+e x)^{7/2}}{3 b (b d-a e) (a+b x)^3}+\frac {\left (5 e^2 (6 b B d+A b e-7 a B e)\right ) \int \frac {\sqrt {d+e x}}{a+b x} \, dx}{16 b^3 (b d-a e)}\\ &=\frac {5 e^2 (6 b B d+A b e-7 a B e) \sqrt {d+e x}}{8 b^4 (b d-a e)}-\frac {5 e (6 b B d+A b e-7 a B e) (d+e x)^{3/2}}{24 b^3 (b d-a e) (a+b x)}-\frac {(6 b B d+A b e-7 a B e) (d+e x)^{5/2}}{12 b^2 (b d-a e) (a+b x)^2}-\frac {(A b-a B) (d+e x)^{7/2}}{3 b (b d-a e) (a+b x)^3}+\frac {\left (5 e^2 (6 b B d+A b e-7 a B e)\right ) \int \frac {1}{(a+b x) \sqrt {d+e x}} \, dx}{16 b^4}\\ &=\frac {5 e^2 (6 b B d+A b e-7 a B e) \sqrt {d+e x}}{8 b^4 (b d-a e)}-\frac {5 e (6 b B d+A b e-7 a B e) (d+e x)^{3/2}}{24 b^3 (b d-a e) (a+b x)}-\frac {(6 b B d+A b e-7 a B e) (d+e x)^{5/2}}{12 b^2 (b d-a e) (a+b x)^2}-\frac {(A b-a B) (d+e x)^{7/2}}{3 b (b d-a e) (a+b x)^3}+\frac {(5 e (6 b B d+A b e-7 a B e)) \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 {5 e^2 (6 b B d+A b e-7 a B e) \sqrt {d+e x}}{8 b^4 (b d-a e)}-\frac {5 e (6 b B d+A b e-7 a B e) (d+e x)^{3/2}}{24 b^3 (b d-a e) (a+b x)}-\frac {(6 b B d+A b e-7 a B e) (d+e x)^{5/2}}{12 b^2 (b d-a e) (a+b x)^2}-\frac {(A b-a B) (d+e x)^{7/2}}{3 b (b d-a e) (a+b x)^3}-\frac {5 e^2 (6 b B d+A b e-7 a B e) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{8 b^{9/2} \sqrt {b d-a e}}\\ \end {align*}
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Mathematica [C] time = 0.08, size = 99, normalized size = 0.40 \begin {gather*} \frac {(d+e x)^{7/2} \left (\frac {7 a B-7 A b}{(a+b x)^3}-\frac {e^2 (-7 a B e+A b e+6 b B d) \, _2F_1\left (3,\frac {7}{2};\frac {9}{2};\frac {b (d+e x)}{b d-a e}\right )}{(b d-a e)^3}\right )}{21 b (b d-a e)} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 1.76, size = 317, normalized size = 1.27 \begin {gather*} -\frac {e^2 \sqrt {d+e x} \left (-105 a^3 B e^3+15 a^2 A b e^3-280 a^2 b B e^2 (d+e x)+300 a^2 b B d e^2+40 a A b^2 e^2 (d+e x)-30 a A b^2 d e^2-285 a b^2 B d^2 e-231 a b^2 B e (d+e x)^2+520 a b^2 B d e (d+e x)+15 A b^3 d^2 e+33 A b^3 e (d+e x)^2-40 A b^3 d e (d+e x)+90 b^3 B d^3-240 b^3 B d^2 (d+e x)-48 b^3 B (d+e x)^3+198 b^3 B d (d+e x)^2\right )}{24 b^4 (a e+b (d+e x)-b d)^3}-\frac {5 \left (-7 a B e^3+A b e^3+6 b B d e^2\right ) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x} \sqrt {a e-b d}}{b d-a e}\right )}{8 b^{9/2} \sqrt {a e-b d}} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 0.46, size = 1075, normalized size = 4.30 \begin {gather*} \left [\frac {15 \, {\left (6 \, B a^{3} b d e^{2} - {\left (7 \, B a^{4} - A a^{3} b\right )} e^{3} + {\left (6 \, B b^{4} d e^{2} - {\left (7 \, B a b^{3} - A b^{4}\right )} e^{3}\right )} x^{3} + 3 \, {\left (6 \, B a b^{3} d e^{2} - {\left (7 \, B a^{2} b^{2} - A a b^{3}\right )} e^{3}\right )} x^{2} + 3 \, {\left (6 \, B a^{2} b^{2} d e^{2} - {\left (7 \, B a^{3} b - A a^{2} b^{2}\right )} e^{3}\right )} x\right )} \sqrt {b^{2} d - a b e} \log \left (\frac {b e x + 2 \, b d - a e - 2 \, \sqrt {b^{2} d - a b e} \sqrt {e x + d}}{b x + a}\right ) - 2 \, {\left (4 \, {\left (B a b^{4} + 2 \, A b^{5}\right )} d^{3} + 2 \, {\left (8 \, B a^{2} b^{3} + A a b^{4}\right )} d^{2} e - 5 \, {\left (25 \, B a^{3} b^{2} - A a^{2} b^{3}\right )} d e^{2} + 15 \, {\left (7 \, B a^{4} b - A a^{3} b^{2}\right )} e^{3} - 48 \, {\left (B b^{5} d e^{2} - B a b^{4} e^{3}\right )} x^{3} + 3 \, {\left (18 \, B b^{5} d^{2} e - {\left (95 \, B a b^{4} - 11 \, A b^{5}\right )} d e^{2} + 11 \, {\left (7 \, B a^{2} b^{3} - A a b^{4}\right )} e^{3}\right )} x^{2} + 2 \, {\left (6 \, B b^{5} d^{3} + {\left (23 \, B a b^{4} + 13 \, A b^{5}\right )} d^{2} e - {\left (169 \, B a^{2} b^{3} - 7 \, A a b^{4}\right )} d e^{2} + 20 \, {\left (7 \, B a^{3} b^{2} - A a^{2} b^{3}\right )} e^{3}\right )} x\right )} \sqrt {e x + d}}{48 \, {\left (a^{3} b^{6} d - a^{4} b^{5} e + {\left (b^{9} d - a b^{8} e\right )} x^{3} + 3 \, {\left (a b^{8} d - a^{2} b^{7} e\right )} x^{2} + 3 \, {\left (a^{2} b^{7} d - a^{3} b^{6} e\right )} x\right )}}, \frac {15 \, {\left (6 \, B a^{3} b d e^{2} - {\left (7 \, B a^{4} - A a^{3} b\right )} e^{3} + {\left (6 \, B b^{4} d e^{2} - {\left (7 \, B a b^{3} - A b^{4}\right )} e^{3}\right )} x^{3} + 3 \, {\left (6 \, B a b^{3} d e^{2} - {\left (7 \, B a^{2} b^{2} - A a b^{3}\right )} e^{3}\right )} x^{2} + 3 \, {\left (6 \, B a^{2} b^{2} d e^{2} - {\left (7 \, B a^{3} b - A a^{2} b^{2}\right )} e^{3}\right )} x\right )} \sqrt {-b^{2} d + a b e} \arctan \left (\frac {\sqrt {-b^{2} d + a b e} \sqrt {e x + d}}{b e x + b d}\right ) - {\left (4 \, {\left (B a b^{4} + 2 \, A b^{5}\right )} d^{3} + 2 \, {\left (8 \, B a^{2} b^{3} + A a b^{4}\right )} d^{2} e - 5 \, {\left (25 \, B a^{3} b^{2} - A a^{2} b^{3}\right )} d e^{2} + 15 \, {\left (7 \, B a^{4} b - A a^{3} b^{2}\right )} e^{3} - 48 \, {\left (B b^{5} d e^{2} - B a b^{4} e^{3}\right )} x^{3} + 3 \, {\left (18 \, B b^{5} d^{2} e - {\left (95 \, B a b^{4} - 11 \, A b^{5}\right )} d e^{2} + 11 \, {\left (7 \, B a^{2} b^{3} - A a b^{4}\right )} e^{3}\right )} x^{2} + 2 \, {\left (6 \, B b^{5} d^{3} + {\left (23 \, B a b^{4} + 13 \, A b^{5}\right )} d^{2} e - {\left (169 \, B a^{2} b^{3} - 7 \, A a b^{4}\right )} d e^{2} + 20 \, {\left (7 \, B a^{3} b^{2} - A a^{2} b^{3}\right )} e^{3}\right )} x\right )} \sqrt {e x + d}}{24 \, {\left (a^{3} b^{6} d - a^{4} b^{5} e + {\left (b^{9} d - a b^{8} e\right )} x^{3} + 3 \, {\left (a b^{8} d - a^{2} b^{7} e\right )} x^{2} + 3 \, {\left (a^{2} b^{7} d - a^{3} b^{6} e\right )} x\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.27, size = 370, normalized size = 1.48 \begin {gather*} \frac {2 \, \sqrt {x e + d} B e^{2}}{b^{4}} + \frac {5 \, {\left (6 \, B b d e^{2} - 7 \, B a e^{3} + A b e^{3}\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 {54 \, {\left (x e + d\right )}^{\frac {5}{2}} B b^{3} d e^{2} - 96 \, {\left (x e + d\right )}^{\frac {3}{2}} B b^{3} d^{2} e^{2} + 42 \, \sqrt {x e + d} B b^{3} d^{3} e^{2} - 87 \, {\left (x e + d\right )}^{\frac {5}{2}} B a b^{2} e^{3} + 33 \, {\left (x e + d\right )}^{\frac {5}{2}} A b^{3} e^{3} + 232 \, {\left (x e + d\right )}^{\frac {3}{2}} B a b^{2} d e^{3} - 40 \, {\left (x e + d\right )}^{\frac {3}{2}} A b^{3} d e^{3} - 141 \, \sqrt {x e + d} B a b^{2} d^{2} e^{3} + 15 \, \sqrt {x e + d} A b^{3} d^{2} e^{3} - 136 \, {\left (x e + d\right )}^{\frac {3}{2}} B a^{2} b e^{4} + 40 \, {\left (x e + d\right )}^{\frac {3}{2}} A a b^{2} e^{4} + 156 \, \sqrt {x e + d} B a^{2} b d e^{4} - 30 \, \sqrt {x e + d} A a b^{2} d e^{4} - 57 \, \sqrt {x e + d} B a^{3} e^{5} + 15 \, \sqrt {x e + d} A a^{2} b e^{5}}{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 = 0.08, size = 573, normalized size = 2.29 \begin {gather*} -\frac {5 \sqrt {e x +d}\, A \,a^{2} e^{5}}{8 \left (b e x +a e \right )^{3} b^{3}}+\frac {5 \sqrt {e x +d}\, A a d \,e^{4}}{4 \left (b e x +a e \right )^{3} b^{2}}-\frac {5 \sqrt {e x +d}\, A \,d^{2} e^{3}}{8 \left (b e x +a e \right )^{3} b}+\frac {19 \sqrt {e x +d}\, B \,a^{3} e^{5}}{8 \left (b e x +a e \right )^{3} b^{4}}-\frac {13 \sqrt {e x +d}\, B \,a^{2} d \,e^{4}}{2 \left (b e x +a e \right )^{3} b^{3}}+\frac {47 \sqrt {e x +d}\, B a \,d^{2} e^{3}}{8 \left (b e x +a e \right )^{3} b^{2}}-\frac {7 \sqrt {e x +d}\, B \,d^{3} e^{2}}{4 \left (b e x +a e \right )^{3} b}-\frac {5 \left (e x +d \right )^{\frac {3}{2}} A a \,e^{4}}{3 \left (b e x +a e \right )^{3} b^{2}}+\frac {5 \left (e x +d \right )^{\frac {3}{2}} A d \,e^{3}}{3 \left (b e x +a e \right )^{3} b}+\frac {17 \left (e x +d \right )^{\frac {3}{2}} B \,a^{2} e^{4}}{3 \left (b e x +a e \right )^{3} b^{3}}-\frac {29 \left (e x +d \right )^{\frac {3}{2}} B a d \,e^{3}}{3 \left (b e x +a e \right )^{3} b^{2}}+\frac {4 \left (e x +d \right )^{\frac {3}{2}} B \,d^{2} e^{2}}{\left (b e x +a e \right )^{3} b}-\frac {11 \left (e x +d \right )^{\frac {5}{2}} A \,e^{3}}{8 \left (b e x +a e \right )^{3} b}+\frac {29 \left (e x +d \right )^{\frac {5}{2}} B a \,e^{3}}{8 \left (b e x +a e \right )^{3} b^{2}}-\frac {9 \left (e x +d \right )^{\frac {5}{2}} B d \,e^{2}}{4 \left (b e x +a e \right )^{3} b}+\frac {5 A \,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 {35 B a \,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^{4}}+\frac {15 B d \,e^{2} \arctan \left (\frac {\sqrt {e x +d}\, b}{\sqrt {\left (a e -b d \right ) b}}\right )}{4 \sqrt {\left (a e -b d \right ) b}\, b^{3}}+\frac {2 \sqrt {e x +d}\, B \,e^{2}}{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 = 0.23, size = 416, normalized size = 1.66 \begin {gather*} \frac {2\,B\,e^2\,\sqrt {d+e\,x}}{b^4}-\frac {{\left (d+e\,x\right )}^{5/2}\,\left (\frac {11\,A\,b^3\,e^3}{8}+\frac {9\,B\,d\,b^3\,e^2}{4}-\frac {29\,B\,a\,b^2\,e^3}{8}\right )-{\left (d+e\,x\right )}^{3/2}\,\left (\frac {17\,B\,a^2\,b\,e^4}{3}-\frac {29\,B\,a\,b^2\,d\,e^3}{3}-\frac {5\,A\,a\,b^2\,e^4}{3}+4\,B\,b^3\,d^2\,e^2+\frac {5\,A\,b^3\,d\,e^3}{3}\right )+\sqrt {d+e\,x}\,\left (-\frac {19\,B\,a^3\,e^5}{8}+\frac {13\,B\,a^2\,b\,d\,e^4}{2}+\frac {5\,A\,a^2\,b\,e^5}{8}-\frac {47\,B\,a\,b^2\,d^2\,e^3}{8}-\frac {5\,A\,a\,b^2\,d\,e^4}{4}+\frac {7\,B\,b^3\,d^3\,e^2}{4}+\frac {5\,A\,b^3\,d^2\,e^3}{8}\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 {5\,e^2\,\mathrm {atan}\left (\frac {\sqrt {b}\,e^2\,\sqrt {d+e\,x}\,\left (A\,b\,e-7\,B\,a\,e+6\,B\,b\,d\right )}{\sqrt {a\,e-b\,d}\,\left (A\,b\,e^3-7\,B\,a\,e^3+6\,B\,b\,d\,e^2\right )}\right )\,\left (A\,b\,e-7\,B\,a\,e+6\,B\,b\,d\right )}{8\,b^{9/2}\,\sqrt {a\,e-b\,d}} \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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