Optimal. Leaf size=250 \[ \frac {5 e^2 (a B e-7 A b e+6 b B d)}{8 b \sqrt {d+e x} (b d-a e)^4}-\frac {5 e^2 (a B e-7 A b e+6 b B d) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{8 \sqrt {b} (b d-a e)^{9/2}}+\frac {5 e (a B e-7 A b e+6 b B d)}{24 b (a+b x) \sqrt {d+e x} (b d-a e)^3}-\frac {a B e-7 A b e+6 b B d}{12 b (a+b x)^2 \sqrt {d+e x} (b d-a e)^2}-\frac {A b-a B}{3 b (a+b x)^3 \sqrt {d+e x} (b d-a e)} \]
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Rubi [A] time = 0.24, antiderivative size = 250, normalized size of antiderivative = 1.00, number of steps used = 7, 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 {5 e^2 (a B e-7 A b e+6 b B d)}{8 b \sqrt {d+e x} (b d-a e)^4}-\frac {5 e^2 (a B e-7 A b e+6 b B d) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{8 \sqrt {b} (b d-a e)^{9/2}}+\frac {5 e (a B e-7 A b e+6 b B d)}{24 b (a+b x) \sqrt {d+e x} (b d-a e)^3}-\frac {a B e-7 A b e+6 b B d}{12 b (a+b x)^2 \sqrt {d+e x} (b d-a e)^2}-\frac {A b-a B}{3 b (a+b x)^3 \sqrt {d+e x} (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)^{3/2} \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)^{3/2}} \, dx\\ &=-\frac {A b-a B}{3 b (b d-a e) (a+b x)^3 \sqrt {d+e x}}+\frac {(6 b B d-7 A b e+a B e) \int \frac {1}{(a+b x)^3 (d+e x)^{3/2}} \, dx}{6 b (b d-a e)}\\ &=-\frac {A b-a B}{3 b (b d-a e) (a+b x)^3 \sqrt {d+e x}}-\frac {6 b B d-7 A b e+a B e}{12 b (b d-a e)^2 (a+b x)^2 \sqrt {d+e x}}-\frac {(5 e (6 b B d-7 A b e+a B e)) \int \frac {1}{(a+b x)^2 (d+e x)^{3/2}} \, dx}{24 b (b d-a e)^2}\\ &=-\frac {A b-a B}{3 b (b d-a e) (a+b x)^3 \sqrt {d+e x}}-\frac {6 b B d-7 A b e+a B e}{12 b (b d-a e)^2 (a+b x)^2 \sqrt {d+e x}}+\frac {5 e (6 b B d-7 A b e+a B e)}{24 b (b d-a e)^3 (a+b x) \sqrt {d+e x}}+\frac {\left (5 e^2 (6 b B d-7 A b e+a B e)\right ) \int \frac {1}{(a+b x) (d+e x)^{3/2}} \, dx}{16 b (b d-a e)^3}\\ &=\frac {5 e^2 (6 b B d-7 A b e+a B e)}{8 b (b d-a e)^4 \sqrt {d+e x}}-\frac {A b-a B}{3 b (b d-a e) (a+b x)^3 \sqrt {d+e x}}-\frac {6 b B d-7 A b e+a B e}{12 b (b d-a e)^2 (a+b x)^2 \sqrt {d+e x}}+\frac {5 e (6 b B d-7 A b e+a B e)}{24 b (b d-a e)^3 (a+b x) \sqrt {d+e x}}+\frac {\left (5 e^2 (6 b B d-7 A b e+a B e)\right ) \int \frac {1}{(a+b x) \sqrt {d+e x}} \, dx}{16 (b d-a e)^4}\\ &=\frac {5 e^2 (6 b B d-7 A b e+a B e)}{8 b (b d-a e)^4 \sqrt {d+e x}}-\frac {A b-a B}{3 b (b d-a e) (a+b x)^3 \sqrt {d+e x}}-\frac {6 b B d-7 A b e+a B e}{12 b (b d-a e)^2 (a+b x)^2 \sqrt {d+e x}}+\frac {5 e (6 b B d-7 A b e+a B e)}{24 b (b d-a e)^3 (a+b x) \sqrt {d+e x}}+\frac {(5 e (6 b B d-7 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)^4}\\ &=\frac {5 e^2 (6 b B d-7 A b e+a B e)}{8 b (b d-a e)^4 \sqrt {d+e x}}-\frac {A b-a B}{3 b (b d-a e) (a+b x)^3 \sqrt {d+e x}}-\frac {6 b B d-7 A b e+a B e}{12 b (b d-a e)^2 (a+b x)^2 \sqrt {d+e x}}+\frac {5 e (6 b B d-7 A b e+a B e)}{24 b (b d-a e)^3 (a+b x) \sqrt {d+e x}}-\frac {5 e^2 (6 b B d-7 A b e+a B e) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{8 \sqrt {b} (b d-a e)^{9/2}}\\ \end {align*}
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Mathematica [C] time = 0.06, size = 99, normalized size = 0.40 \begin {gather*} \frac {\frac {a B-A b}{(a+b x)^3}-\frac {e^2 (-a B e+7 A b e-6 b B d) \, _2F_1\left (-\frac {1}{2},3;\frac {1}{2};\frac {b (d+e x)}{b d-a e}\right )}{(b d-a e)^3}}{3 b \sqrt {d+e x} (b d-a e)} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 1.69, size = 453, normalized size = 1.81 \begin {gather*} \frac {e^2 \left (48 a^3 A e^4-33 a^3 B e^3 (d+e x)-48 a^3 B d e^3+231 a^2 A b e^3 (d+e x)-144 a^2 A b d e^3+144 a^2 b B d^2 e^2-132 a^2 b B d e^2 (d+e x)-40 a^2 b B e^2 (d+e x)^2+144 a A b^2 d^2 e^2-462 a A b^2 d e^2 (d+e x)+280 a A b^2 e^2 (d+e x)^2-144 a b^2 B d^3 e+363 a b^2 B d^2 e (d+e x)-200 a b^2 B d e (d+e x)^2-15 a b^2 B e (d+e x)^3-48 A b^3 d^3 e+231 A b^3 d^2 e (d+e x)-280 A b^3 d e (d+e x)^2+105 A b^3 e (d+e x)^3+48 b^3 B d^4-198 b^3 B d^3 (d+e x)+240 b^3 B d^2 (d+e x)^2-90 b^3 B d (d+e x)^3\right )}{24 \sqrt {d+e x} (b d-a e)^4 (-a e-b (d+e x)+b d)^3}-\frac {5 \left (a B e^3-7 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 \sqrt {b} (b d-a e)^4 \sqrt {a e-b d}} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 0.49, size = 2114, normalized size = 8.46
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.26, size = 516, normalized size = 2.06 \begin {gather*} \frac {5 \, {\left (6 \, B b d e^{2} + B a e^{3} - 7 \, A b e^{3}\right )} \arctan \left (\frac {\sqrt {x e + d} b}{\sqrt {-b^{2} d + a b e}}\right )}{8 \, {\left (b^{4} d^{4} - 4 \, a b^{3} d^{3} e + 6 \, a^{2} b^{2} d^{2} e^{2} - 4 \, a^{3} b d e^{3} + a^{4} e^{4}\right )} \sqrt {-b^{2} d + a b e}} + \frac {2 \, {\left (B d e^{2} - A e^{3}\right )}}{{\left (b^{4} d^{4} - 4 \, a b^{3} d^{3} e + 6 \, a^{2} b^{2} d^{2} e^{2} - 4 \, a^{3} b d e^{3} + a^{4} e^{4}\right )} \sqrt {x e + d}} + \frac {42 \, {\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} + 54 \, \sqrt {x e + d} B b^{3} d^{3} e^{2} + 15 \, {\left (x e + d\right )}^{\frac {5}{2}} B a b^{2} e^{3} - 57 \, {\left (x e + d\right )}^{\frac {5}{2}} A b^{3} e^{3} + 56 \, {\left (x e + d\right )}^{\frac {3}{2}} B a b^{2} d e^{3} + 136 \, {\left (x e + d\right )}^{\frac {3}{2}} A b^{3} d e^{3} - 75 \, \sqrt {x e + d} B a b^{2} d^{2} e^{3} - 87 \, \sqrt {x e + d} A b^{3} d^{2} e^{3} + 40 \, {\left (x e + d\right )}^{\frac {3}{2}} B a^{2} b e^{4} - 136 \, {\left (x e + d\right )}^{\frac {3}{2}} A a b^{2} e^{4} - 12 \, \sqrt {x e + d} B a^{2} b d e^{4} + 174 \, \sqrt {x e + d} A a b^{2} d e^{4} + 33 \, \sqrt {x e + d} B a^{3} e^{5} - 87 \, \sqrt {x e + d} A a^{2} b e^{5}}{24 \, {\left (b^{4} d^{4} - 4 \, a b^{3} d^{3} e + 6 \, a^{2} b^{2} d^{2} e^{2} - 4 \, a^{3} b d e^{3} + a^{4} e^{4}\right )} {\left ({\left (x e + d\right )} b - b 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 = 768, normalized size = 3.07 \begin {gather*} -\frac {29 \sqrt {e x +d}\, A \,a^{2} b \,e^{5}}{8 \left (a e -b d \right )^{4} \left (b e x +a e \right )^{3}}+\frac {29 \sqrt {e x +d}\, A a \,b^{2} d \,e^{4}}{4 \left (a e -b d \right )^{4} \left (b e x +a e \right )^{3}}-\frac {29 \sqrt {e x +d}\, A \,b^{3} d^{2} e^{3}}{8 \left (a e -b d \right )^{4} \left (b e x +a e \right )^{3}}+\frac {11 \sqrt {e x +d}\, B \,a^{3} e^{5}}{8 \left (a e -b d \right )^{4} \left (b e x +a e \right )^{3}}-\frac {\sqrt {e x +d}\, B \,a^{2} b d \,e^{4}}{2 \left (a e -b d \right )^{4} \left (b e x +a e \right )^{3}}-\frac {25 \sqrt {e x +d}\, B a \,b^{2} d^{2} e^{3}}{8 \left (a e -b d \right )^{4} \left (b e x +a e \right )^{3}}+\frac {9 \sqrt {e x +d}\, B \,b^{3} d^{3} e^{2}}{4 \left (a e -b d \right )^{4} \left (b e x +a e \right )^{3}}-\frac {17 \left (e x +d \right )^{\frac {3}{2}} A a \,b^{2} e^{4}}{3 \left (a e -b d \right )^{4} \left (b e x +a e \right )^{3}}+\frac {17 \left (e x +d \right )^{\frac {3}{2}} A \,b^{3} d \,e^{3}}{3 \left (a e -b d \right )^{4} \left (b e x +a e \right )^{3}}+\frac {5 \left (e x +d \right )^{\frac {3}{2}} B \,a^{2} b \,e^{4}}{3 \left (a e -b d \right )^{4} \left (b e x +a e \right )^{3}}+\frac {7 \left (e x +d \right )^{\frac {3}{2}} B a \,b^{2} d \,e^{3}}{3 \left (a e -b d \right )^{4} \left (b e x +a e \right )^{3}}-\frac {4 \left (e x +d \right )^{\frac {3}{2}} B \,b^{3} d^{2} e^{2}}{\left (a e -b d \right )^{4} \left (b e x +a e \right )^{3}}-\frac {19 \left (e x +d \right )^{\frac {5}{2}} A \,b^{3} e^{3}}{8 \left (a e -b d \right )^{4} \left (b e x +a e \right )^{3}}+\frac {5 \left (e x +d \right )^{\frac {5}{2}} B a \,b^{2} e^{3}}{8 \left (a e -b d \right )^{4} \left (b e x +a e \right )^{3}}+\frac {7 \left (e x +d \right )^{\frac {5}{2}} B \,b^{3} d \,e^{2}}{4 \left (a e -b d \right )^{4} \left (b e x +a e \right )^{3}}-\frac {35 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 )^{4} \sqrt {\left (a e -b d \right ) b}}+\frac {5 B a \,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 )^{4} \sqrt {\left (a e -b d \right ) b}}+\frac {15 B b 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 )^{4} \sqrt {\left (a e -b d \right ) b}}-\frac {2 A \,e^{3}}{\left (a e -b d \right )^{4} \sqrt {e x +d}}+\frac {2 B d \,e^{2}}{\left (a e -b d \right )^{4} \sqrt {e x +d}} \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 = 2.34, size = 420, normalized size = 1.68 \begin {gather*} \frac {\frac {5\,{\left (d+e\,x\right )}^2\,\left (-7\,A\,b^2\,e^3+6\,B\,d\,b^2\,e^2+B\,a\,b\,e^3\right )}{3\,{\left (a\,e-b\,d\right )}^3}-\frac {2\,\left (A\,e^3-B\,d\,e^2\right )}{a\,e-b\,d}+\frac {11\,\left (d+e\,x\right )\,\left (B\,a\,e^3-7\,A\,b\,e^3+6\,B\,b\,d\,e^2\right )}{8\,{\left (a\,e-b\,d\right )}^2}+\frac {5\,b^2\,{\left (d+e\,x\right )}^3\,\left (B\,a\,e^3-7\,A\,b\,e^3+6\,B\,b\,d\,e^2\right )}{8\,{\left (a\,e-b\,d\right )}^4}}{\sqrt {d+e\,x}\,\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 )}^{7/2}-\left (3\,b^3\,d-3\,a\,b^2\,e\right )\,{\left (d+e\,x\right )}^{5/2}+{\left (d+e\,x\right )}^{3/2}\,\left (3\,a^2\,b\,e^2-6\,a\,b^2\,d\,e+3\,b^3\,d^2\right )}+\frac {5\,e^2\,\mathrm {atan}\left (\frac {5\,\sqrt {b}\,e^2\,\sqrt {d+e\,x}\,\left (B\,a\,e-7\,A\,b\,e+6\,B\,b\,d\right )\,\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 (a\,e-b\,d\right )}^{9/2}\,\left (5\,B\,a\,e^3-35\,A\,b\,e^3+30\,B\,b\,d\,e^2\right )}\right )\,\left (B\,a\,e-7\,A\,b\,e+6\,B\,b\,d\right )}{8\,\sqrt {b}\,{\left (a\,e-b\,d\right )}^{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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