Optimal. Leaf size=291 \[ \frac {35 e^2 (a B e-3 A b e+2 b B d)}{8 \sqrt {d+e x} (b d-a e)^5}+\frac {35 e^2 (a B e-3 A b e+2 b B d)}{24 b (d+e x)^{3/2} (b d-a e)^4}-\frac {35 \sqrt {b} e^2 (a B e-3 A b e+2 b B d) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{8 (b d-a e)^{11/2}}+\frac {7 e (a B e-3 A b e+2 b B d)}{8 b (a+b x) (d+e x)^{3/2} (b d-a e)^3}-\frac {a B e-3 A b e+2 b B d}{4 b (a+b x)^2 (d+e x)^{3/2} (b d-a e)^2}-\frac {A b-a B}{3 b (a+b x)^3 (d+e x)^{3/2} (b d-a e)} \]
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Rubi [A] time = 0.30, antiderivative size = 291, 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, 78, 51, 63, 208} \begin {gather*} \frac {35 e^2 (a B e-3 A b e+2 b B d)}{8 \sqrt {d+e x} (b d-a e)^5}+\frac {35 e^2 (a B e-3 A b e+2 b B d)}{24 b (d+e x)^{3/2} (b d-a e)^4}-\frac {35 \sqrt {b} e^2 (a B e-3 A b e+2 b B d) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{8 (b d-a e)^{11/2}}+\frac {7 e (a B e-3 A b e+2 b B d)}{8 b (a+b x) (d+e x)^{3/2} (b d-a e)^3}-\frac {a B e-3 A b e+2 b B d}{4 b (a+b x)^2 (d+e x)^{3/2} (b d-a e)^2}-\frac {A b-a B}{3 b (a+b x)^3 (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 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}{(a+b x)^4 (d+e x)^{5/2}} \, dx\\ &=-\frac {A b-a B}{3 b (b d-a e) (a+b x)^3 (d+e x)^{3/2}}+\frac {(2 b B d-3 A b e+a B e) \int \frac {1}{(a+b x)^3 (d+e x)^{5/2}} \, dx}{2 b (b d-a e)}\\ &=-\frac {A b-a B}{3 b (b d-a e) (a+b x)^3 (d+e x)^{3/2}}-\frac {2 b B d-3 A b e+a B e}{4 b (b d-a e)^2 (a+b x)^2 (d+e x)^{3/2}}-\frac {(7 e (2 b B d-3 A b e+a B e)) \int \frac {1}{(a+b x)^2 (d+e x)^{5/2}} \, dx}{8 b (b d-a e)^2}\\ &=-\frac {A b-a B}{3 b (b d-a e) (a+b x)^3 (d+e x)^{3/2}}-\frac {2 b B d-3 A b e+a B e}{4 b (b d-a e)^2 (a+b x)^2 (d+e x)^{3/2}}+\frac {7 e (2 b B d-3 A b e+a B e)}{8 b (b d-a e)^3 (a+b x) (d+e x)^{3/2}}+\frac {\left (35 e^2 (2 b B d-3 A b e+a B e)\right ) \int \frac {1}{(a+b x) (d+e x)^{5/2}} \, dx}{16 b (b d-a e)^3}\\ &=\frac {35 e^2 (2 b B d-3 A b e+a B e)}{24 b (b d-a e)^4 (d+e x)^{3/2}}-\frac {A b-a B}{3 b (b d-a e) (a+b x)^3 (d+e x)^{3/2}}-\frac {2 b B d-3 A b e+a B e}{4 b (b d-a e)^2 (a+b x)^2 (d+e x)^{3/2}}+\frac {7 e (2 b B d-3 A b e+a B e)}{8 b (b d-a e)^3 (a+b x) (d+e x)^{3/2}}+\frac {\left (35 e^2 (2 b B d-3 A b e+a B e)\right ) \int \frac {1}{(a+b x) (d+e x)^{3/2}} \, dx}{16 (b d-a e)^4}\\ &=\frac {35 e^2 (2 b B d-3 A b e+a B e)}{24 b (b d-a e)^4 (d+e x)^{3/2}}-\frac {A b-a B}{3 b (b d-a e) (a+b x)^3 (d+e x)^{3/2}}-\frac {2 b B d-3 A b e+a B e}{4 b (b d-a e)^2 (a+b x)^2 (d+e x)^{3/2}}+\frac {7 e (2 b B d-3 A b e+a B e)}{8 b (b d-a e)^3 (a+b x) (d+e x)^{3/2}}+\frac {35 e^2 (2 b B d-3 A b e+a B e)}{8 (b d-a e)^5 \sqrt {d+e x}}+\frac {\left (35 b e^2 (2 b B d-3 A b e+a B e)\right ) \int \frac {1}{(a+b x) \sqrt {d+e x}} \, dx}{16 (b d-a e)^5}\\ &=\frac {35 e^2 (2 b B d-3 A b e+a B e)}{24 b (b d-a e)^4 (d+e x)^{3/2}}-\frac {A b-a B}{3 b (b d-a e) (a+b x)^3 (d+e x)^{3/2}}-\frac {2 b B d-3 A b e+a B e}{4 b (b d-a e)^2 (a+b x)^2 (d+e x)^{3/2}}+\frac {7 e (2 b B d-3 A b e+a B e)}{8 b (b d-a e)^3 (a+b x) (d+e x)^{3/2}}+\frac {35 e^2 (2 b B d-3 A b e+a B e)}{8 (b d-a e)^5 \sqrt {d+e x}}+\frac {(35 b e (2 b B d-3 A b e+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 d-a e)^5}\\ &=\frac {35 e^2 (2 b B d-3 A b e+a B e)}{24 b (b d-a e)^4 (d+e x)^{3/2}}-\frac {A b-a B}{3 b (b d-a e) (a+b x)^3 (d+e x)^{3/2}}-\frac {2 b B d-3 A b e+a B e}{4 b (b d-a e)^2 (a+b x)^2 (d+e x)^{3/2}}+\frac {7 e (2 b B d-3 A b e+a B e)}{8 b (b d-a e)^3 (a+b x) (d+e x)^{3/2}}+\frac {35 e^2 (2 b B d-3 A b e+a B e)}{8 (b d-a e)^5 \sqrt {d+e x}}-\frac {35 \sqrt {b} e^2 (2 b B d-3 A b e+a B e) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{8 (b d-a e)^{11/2}}\\ \end {align*}
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Mathematica [C] time = 0.06, size = 100, normalized size = 0.34 \begin {gather*} \frac {\frac {3 a B-3 A b}{(a+b x)^3}-\frac {3 e^2 (-a B e+3 A b e-2 b B d) \, _2F_1\left (-\frac {3}{2},3;-\frac {1}{2};\frac {b (d+e x)}{b d-a e}\right )}{(b d-a e)^3}}{9 b (d+e x)^{3/2} (b d-a e)} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [B] time = 2.06, size = 638, normalized size = 2.19 \begin {gather*} -\frac {e^2 \left (16 a^4 A e^5+48 a^4 B e^4 (d+e x)-16 a^4 B d e^4-144 a^3 A b e^4 (d+e x)-64 a^3 A b d e^4+64 a^3 b B d^2 e^3-48 a^3 b B d e^3 (d+e x)+231 a^3 b B e^3 (d+e x)^2+96 a^2 A b^2 d^2 e^3+432 a^2 A b^2 d e^3 (d+e x)-693 a^2 A b^2 e^3 (d+e x)^2-96 a^2 b^2 B d^3 e^2-144 a^2 b^2 B d^2 e^2 (d+e x)+280 a^2 b^2 B e^2 (d+e x)^3-64 a A b^3 d^3 e^2-432 a A b^3 d^2 e^2 (d+e x)+1386 a A b^3 d e^2 (d+e x)^2-840 a A b^3 e^2 (d+e x)^3+64 a b^3 B d^4 e+240 a b^3 B d^3 e (d+e x)-693 a b^3 B d^2 e (d+e x)^2+280 a b^3 B d e (d+e x)^3+105 a b^3 B e (d+e x)^4+16 A b^4 d^4 e+144 A b^4 d^3 e (d+e x)-693 A b^4 d^2 e (d+e x)^2+840 A b^4 d e (d+e x)^3-315 A b^4 e (d+e x)^4-16 b^4 B d^5-96 b^4 B d^4 (d+e x)+462 b^4 B d^3 (d+e x)^2-560 b^4 B d^2 (d+e x)^3+210 b^4 B d (d+e x)^4\right )}{24 (d+e x)^{3/2} (b d-a e)^5 (-a e-b (d+e x)+b d)^3}-\frac {35 \left (a \sqrt {b} B e^3-3 A b^{3/2} e^3+2 b^{3/2} 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 d-a e)^5 \sqrt {a e-b d}} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 0.50, size = 2648, normalized size = 9.10
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.29, size = 750, normalized size = 2.58 \begin {gather*} \frac {35 \, {\left (2 \, B b^{2} d e^{2} + B a b e^{3} - 3 \, A b^{2} e^{3}\right )} \arctan \left (\frac {\sqrt {x e + d} b}{\sqrt {-b^{2} d + a b e}}\right )}{8 \, {\left (b^{5} d^{5} - 5 \, a b^{4} d^{4} e + 10 \, a^{2} b^{3} d^{3} e^{2} - 10 \, a^{3} b^{2} d^{2} e^{3} + 5 \, a^{4} b d e^{4} - a^{5} e^{5}\right )} \sqrt {-b^{2} d + a b e}} + \frac {210 \, {\left (x e + d\right )}^{4} B b^{4} d e^{2} - 560 \, {\left (x e + d\right )}^{3} B b^{4} d^{2} e^{2} + 462 \, {\left (x e + d\right )}^{2} B b^{4} d^{3} e^{2} - 96 \, {\left (x e + d\right )} B b^{4} d^{4} e^{2} - 16 \, B b^{4} d^{5} e^{2} + 105 \, {\left (x e + d\right )}^{4} B a b^{3} e^{3} - 315 \, {\left (x e + d\right )}^{4} A b^{4} e^{3} + 280 \, {\left (x e + d\right )}^{3} B a b^{3} d e^{3} + 840 \, {\left (x e + d\right )}^{3} A b^{4} d e^{3} - 693 \, {\left (x e + d\right )}^{2} B a b^{3} d^{2} e^{3} - 693 \, {\left (x e + d\right )}^{2} A b^{4} d^{2} e^{3} + 240 \, {\left (x e + d\right )} B a b^{3} d^{3} e^{3} + 144 \, {\left (x e + d\right )} A b^{4} d^{3} e^{3} + 64 \, B a b^{3} d^{4} e^{3} + 16 \, A b^{4} d^{4} e^{3} + 280 \, {\left (x e + d\right )}^{3} B a^{2} b^{2} e^{4} - 840 \, {\left (x e + d\right )}^{3} A a b^{3} e^{4} + 1386 \, {\left (x e + d\right )}^{2} A a b^{3} d e^{4} - 144 \, {\left (x e + d\right )} B a^{2} b^{2} d^{2} e^{4} - 432 \, {\left (x e + d\right )} A a b^{3} d^{2} e^{4} - 96 \, B a^{2} b^{2} d^{3} e^{4} - 64 \, A a b^{3} d^{3} e^{4} + 231 \, {\left (x e + d\right )}^{2} B a^{3} b e^{5} - 693 \, {\left (x e + d\right )}^{2} A a^{2} b^{2} e^{5} - 48 \, {\left (x e + d\right )} B a^{3} b d e^{5} + 432 \, {\left (x e + d\right )} A a^{2} b^{2} d e^{5} + 64 \, B a^{3} b d^{2} e^{5} + 96 \, A a^{2} b^{2} d^{2} e^{5} + 48 \, {\left (x e + d\right )} B a^{4} e^{6} - 144 \, {\left (x e + d\right )} A a^{3} b e^{6} - 16 \, B a^{4} d e^{6} - 64 \, A a^{3} b d e^{6} + 16 \, A a^{4} e^{7}}{24 \, {\left (b^{5} d^{5} - 5 \, a b^{4} d^{4} e + 10 \, a^{2} b^{3} d^{3} e^{2} - 10 \, a^{3} b^{2} d^{2} e^{3} + 5 \, a^{4} b d e^{4} - a^{5} e^{5}\right )} {\left ({\left (x e + d\right )}^{\frac {3}{2}} b - \sqrt {x e + d} b d + \sqrt {x e + d} a e\right )}^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.08, size = 853, normalized size = 2.93 \begin {gather*} \frac {55 \sqrt {e x +d}\, A \,a^{2} b^{2} e^{5}}{8 \left (a e -b d \right )^{5} \left (b e x +a e \right )^{3}}-\frac {55 \sqrt {e x +d}\, A a \,b^{3} d \,e^{4}}{4 \left (a e -b d \right )^{5} \left (b e x +a e \right )^{3}}+\frac {55 \sqrt {e x +d}\, A \,b^{4} d^{2} e^{3}}{8 \left (a e -b d \right )^{5} \left (b e x +a e \right )^{3}}-\frac {29 \sqrt {e x +d}\, B \,a^{3} b \,e^{5}}{8 \left (a e -b d \right )^{5} \left (b e x +a e \right )^{3}}+\frac {4 \sqrt {e x +d}\, B \,a^{2} b^{2} d \,e^{4}}{\left (a e -b d \right )^{5} \left (b e x +a e \right )^{3}}+\frac {23 \sqrt {e x +d}\, B a \,b^{3} d^{2} e^{3}}{8 \left (a e -b d \right )^{5} \left (b e x +a e \right )^{3}}-\frac {13 \sqrt {e x +d}\, B \,b^{4} d^{3} e^{2}}{4 \left (a e -b d \right )^{5} \left (b e x +a e \right )^{3}}+\frac {35 \left (e x +d \right )^{\frac {3}{2}} A a \,b^{3} e^{4}}{3 \left (a e -b d \right )^{5} \left (b e x +a e \right )^{3}}-\frac {35 \left (e x +d \right )^{\frac {3}{2}} A \,b^{4} d \,e^{3}}{3 \left (a e -b d \right )^{5} \left (b e x +a e \right )^{3}}-\frac {17 \left (e x +d \right )^{\frac {3}{2}} B \,a^{2} b^{2} e^{4}}{3 \left (a e -b d \right )^{5} \left (b e x +a e \right )^{3}}-\frac {\left (e x +d \right )^{\frac {3}{2}} B a \,b^{3} d \,e^{3}}{3 \left (a e -b d \right )^{5} \left (b e x +a e \right )^{3}}+\frac {6 \left (e x +d \right )^{\frac {3}{2}} B \,b^{4} d^{2} e^{2}}{\left (a e -b d \right )^{5} \left (b e x +a e \right )^{3}}+\frac {41 \left (e x +d \right )^{\frac {5}{2}} A \,b^{4} e^{3}}{8 \left (a e -b d \right )^{5} \left (b e x +a e \right )^{3}}-\frac {19 \left (e x +d \right )^{\frac {5}{2}} B a \,b^{3} e^{3}}{8 \left (a e -b d \right )^{5} \left (b e x +a e \right )^{3}}-\frac {11 \left (e x +d \right )^{\frac {5}{2}} B \,b^{4} d \,e^{2}}{4 \left (a e -b d \right )^{5} \left (b e x +a e \right )^{3}}+\frac {105 A \,b^{2} e^{3} \arctan \left (\frac {\sqrt {e x +d}\, b}{\sqrt {\left (a e -b d \right ) b}}\right )}{8 \left (a e -b d \right )^{5} \sqrt {\left (a e -b d \right ) b}}-\frac {35 B a b \,e^{3} \arctan \left (\frac {\sqrt {e x +d}\, b}{\sqrt {\left (a e -b d \right ) b}}\right )}{8 \left (a e -b d \right )^{5} \sqrt {\left (a e -b d \right ) b}}-\frac {35 B \,b^{2} d \,e^{2} \arctan \left (\frac {\sqrt {e x +d}\, b}{\sqrt {\left (a e -b d \right ) b}}\right )}{4 \left (a e -b d \right )^{5} \sqrt {\left (a e -b d \right ) b}}+\frac {8 A b \,e^{3}}{\left (a e -b d \right )^{5} \sqrt {e x +d}}-\frac {2 B a \,e^{3}}{\left (a e -b d \right )^{5} \sqrt {e x +d}}-\frac {6 B b d \,e^{2}}{\left (a e -b d \right )^{5} \sqrt {e x +d}}-\frac {2 A \,e^{3}}{3 \left (a e -b d \right )^{4} \left (e x +d \right )^{\frac {3}{2}}}+\frac {2 B d \,e^{2}}{3 \left (a e -b d \right )^{4} \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 = 2.51, size = 483, normalized size = 1.66 \begin {gather*} -\frac {\frac {2\,\left (A\,e^3-B\,d\,e^2\right )}{3\,\left (a\,e-b\,d\right )}+\frac {77\,{\left (d+e\,x\right )}^2\,\left (-3\,A\,b^2\,e^3+2\,B\,d\,b^2\,e^2+B\,a\,b\,e^3\right )}{8\,{\left (a\,e-b\,d\right )}^3}+\frac {35\,{\left (d+e\,x\right )}^3\,\left (-3\,A\,b^3\,e^3+2\,B\,d\,b^3\,e^2+B\,a\,b^2\,e^3\right )}{3\,{\left (a\,e-b\,d\right )}^4}+\frac {2\,\left (d+e\,x\right )\,\left (B\,a\,e^3-3\,A\,b\,e^3+2\,B\,b\,d\,e^2\right )}{{\left (a\,e-b\,d\right )}^2}+\frac {35\,b^3\,{\left (d+e\,x\right )}^4\,\left (B\,a\,e^3-3\,A\,b\,e^3+2\,B\,b\,d\,e^2\right )}{8\,{\left (a\,e-b\,d\right )}^5}}{{\left (d+e\,x\right )}^{3/2}\,\left (a^3\,e^3-3\,a^2\,b\,d\,e^2+3\,a\,b^2\,d^2\,e-b^3\,d^3\right )+b^3\,{\left (d+e\,x\right )}^{9/2}-\left (3\,b^3\,d-3\,a\,b^2\,e\right )\,{\left (d+e\,x\right )}^{7/2}+{\left (d+e\,x\right )}^{5/2}\,\left (3\,a^2\,b\,e^2-6\,a\,b^2\,d\,e+3\,b^3\,d^2\right )}-\frac {35\,\sqrt {b}\,e^2\,\mathrm {atan}\left (\frac {\sqrt {b}\,e^2\,\sqrt {d+e\,x}\,\left (B\,a\,e-3\,A\,b\,e+2\,B\,b\,d\right )\,\left (a^5\,e^5-5\,a^4\,b\,d\,e^4+10\,a^3\,b^2\,d^2\,e^3-10\,a^2\,b^3\,d^3\,e^2+5\,a\,b^4\,d^4\,e-b^5\,d^5\right )}{{\left (a\,e-b\,d\right )}^{11/2}\,\left (B\,a\,e^3-3\,A\,b\,e^3+2\,B\,b\,d\,e^2\right )}\right )\,\left (B\,a\,e-3\,A\,b\,e+2\,B\,b\,d\right )}{8\,{\left (a\,e-b\,d\right )}^{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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