Optimal. Leaf size=221 \[ -\frac {b^{3/2} (5 a B e-7 A b e+2 b B d) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{(b d-a e)^{9/2}}+\frac {b (5 a B e-7 A b e+2 b B d)}{\sqrt {d+e x} (b d-a e)^4}+\frac {5 a B e-7 A b e+2 b B d}{3 (d+e x)^{3/2} (b d-a e)^3}+\frac {5 a B e-7 A b e+2 b B d}{5 b (d+e x)^{5/2} (b d-a e)^2}-\frac {A b-a B}{b (a+b x) (d+e x)^{5/2} (b d-a e)} \]
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Rubi [A] time = 0.23, antiderivative size = 221, 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} \[ -\frac {b^{3/2} (5 a B e-7 A b e+2 b B d) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{(b d-a e)^{9/2}}+\frac {b (5 a B e-7 A b e+2 b B d)}{\sqrt {d+e x} (b d-a e)^4}+\frac {5 a B e-7 A b e+2 b B d}{3 (d+e x)^{3/2} (b d-a e)^3}+\frac {5 a B e-7 A b e+2 b B d}{5 b (d+e x)^{5/2} (b d-a e)^2}-\frac {A b-a B}{b (a+b x) (d+e x)^{5/2} (b d-a e)} \]
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)^{7/2} \left (a^2+2 a b x+b^2 x^2\right )} \, dx &=\int \frac {A+B x}{(a+b x)^2 (d+e x)^{7/2}} \, dx\\ &=-\frac {A b-a B}{b (b d-a e) (a+b x) (d+e x)^{5/2}}+\frac {(2 b B d-7 A b e+5 a B e) \int \frac {1}{(a+b x) (d+e x)^{7/2}} \, dx}{2 b (b d-a e)}\\ &=\frac {2 b B d-7 A b e+5 a B e}{5 b (b d-a e)^2 (d+e x)^{5/2}}-\frac {A b-a B}{b (b d-a e) (a+b x) (d+e x)^{5/2}}+\frac {(2 b B d-7 A b e+5 a B e) \int \frac {1}{(a+b x) (d+e x)^{5/2}} \, dx}{2 (b d-a e)^2}\\ &=\frac {2 b B d-7 A b e+5 a B e}{5 b (b d-a e)^2 (d+e x)^{5/2}}-\frac {A b-a B}{b (b d-a e) (a+b x) (d+e x)^{5/2}}+\frac {2 b B d-7 A b e+5 a B e}{3 (b d-a e)^3 (d+e x)^{3/2}}+\frac {(b (2 b B d-7 A b e+5 a B e)) \int \frac {1}{(a+b x) (d+e x)^{3/2}} \, dx}{2 (b d-a e)^3}\\ &=\frac {2 b B d-7 A b e+5 a B e}{5 b (b d-a e)^2 (d+e x)^{5/2}}-\frac {A b-a B}{b (b d-a e) (a+b x) (d+e x)^{5/2}}+\frac {2 b B d-7 A b e+5 a B e}{3 (b d-a e)^3 (d+e x)^{3/2}}+\frac {b (2 b B d-7 A b e+5 a B e)}{(b d-a e)^4 \sqrt {d+e x}}+\frac {\left (b^2 (2 b B d-7 A b e+5 a B e)\right ) \int \frac {1}{(a+b x) \sqrt {d+e x}} \, dx}{2 (b d-a e)^4}\\ &=\frac {2 b B d-7 A b e+5 a B e}{5 b (b d-a e)^2 (d+e x)^{5/2}}-\frac {A b-a B}{b (b d-a e) (a+b x) (d+e x)^{5/2}}+\frac {2 b B d-7 A b e+5 a B e}{3 (b d-a e)^3 (d+e x)^{3/2}}+\frac {b (2 b B d-7 A b e+5 a B e)}{(b d-a e)^4 \sqrt {d+e x}}+\frac {\left (b^2 (2 b B d-7 A b e+5 a B 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 )}{e (b d-a e)^4}\\ &=\frac {2 b B d-7 A b e+5 a B e}{5 b (b d-a e)^2 (d+e x)^{5/2}}-\frac {A b-a B}{b (b d-a e) (a+b x) (d+e x)^{5/2}}+\frac {2 b B d-7 A b e+5 a B e}{3 (b d-a e)^3 (d+e x)^{3/2}}+\frac {b (2 b B d-7 A b e+5 a B e)}{(b d-a e)^4 \sqrt {d+e x}}-\frac {b^{3/2} (2 b B d-7 A b e+5 a B e) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{(b d-a e)^{9/2}}\\ \end {align*}
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Mathematica [C] time = 0.04, size = 94, normalized size = 0.43 \[ \frac {(5 a B e-7 A b e+2 b B d) \, _2F_1\left (-\frac {5}{2},1;-\frac {3}{2};\frac {b (d+e x)}{b d-a e}\right )-\frac {5 (A b-a B) (b d-a e)}{a+b x}}{5 b (d+e x)^{5/2} (b d-a e)^2} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.85, size = 1749, normalized size = 7.91 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.23, size = 435, normalized size = 1.97 \[ \frac {{\left (2 \, B b^{3} d + 5 \, B a b^{2} e - 7 \, A b^{3} e\right )} \arctan \left (\frac {\sqrt {x e + d} b}{\sqrt {-b^{2} d + a b e}}\right )}{{\left (b^{4} d^{4} - 4 \, a b^{3} d^{3} e + 6 \, a^{2} b^{2} d^{2} e^{2} - 4 \, a^{3} b d e^{3} + a^{4} e^{4}\right )} \sqrt {-b^{2} d + a b e}} + \frac {\sqrt {x e + d} B a b^{2} e - \sqrt {x e + d} A b^{3} e}{{\left (b^{4} d^{4} - 4 \, a b^{3} d^{3} e + 6 \, a^{2} b^{2} d^{2} e^{2} - 4 \, a^{3} b d e^{3} + a^{4} e^{4}\right )} {\left ({\left (x e + d\right )} b - b d + a e\right )}} + \frac {2 \, {\left (15 \, {\left (x e + d\right )}^{2} B b^{2} d + 5 \, {\left (x e + d\right )} B b^{2} d^{2} + 3 \, B b^{2} d^{3} + 30 \, {\left (x e + d\right )}^{2} B a b e - 45 \, {\left (x e + d\right )}^{2} A b^{2} e - 10 \, {\left (x e + d\right )} A b^{2} d e - 6 \, B a b d^{2} e - 3 \, A b^{2} d^{2} e - 5 \, {\left (x e + d\right )} B a^{2} e^{2} + 10 \, {\left (x e + d\right )} A a b e^{2} + 3 \, B a^{2} d e^{2} + 6 \, A a b d e^{2} - 3 \, A a^{2} e^{3}\right )}}{15 \, {\left (b^{4} d^{4} - 4 \, a b^{3} d^{3} e + 6 \, a^{2} b^{2} d^{2} e^{2} - 4 \, a^{3} b d e^{3} + a^{4} e^{4}\right )} {\left (x e + d\right )}^{\frac {5}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.07, size = 403, normalized size = 1.82 \[ -\frac {7 A \,b^{3} e \arctan \left (\frac {\sqrt {e x +d}\, b}{\sqrt {\left (a e -b d \right ) b}}\right )}{\left (a e -b d \right )^{4} \sqrt {\left (a e -b d \right ) b}}+\frac {5 B a \,b^{2} e \arctan \left (\frac {\sqrt {e x +d}\, b}{\sqrt {\left (a e -b d \right ) b}}\right )}{\left (a e -b d \right )^{4} \sqrt {\left (a e -b d \right ) b}}+\frac {2 B \,b^{3} d \arctan \left (\frac {\sqrt {e x +d}\, b}{\sqrt {\left (a e -b d \right ) b}}\right )}{\left (a e -b d \right )^{4} \sqrt {\left (a e -b d \right ) b}}-\frac {\sqrt {e x +d}\, A \,b^{3} e}{\left (a e -b d \right )^{4} \left (b e x +a e \right )}+\frac {\sqrt {e x +d}\, B a \,b^{2} e}{\left (a e -b d \right )^{4} \left (b e x +a e \right )}-\frac {6 A \,b^{2} e}{\left (a e -b d \right )^{4} \sqrt {e x +d}}+\frac {4 B a b e}{\left (a e -b d \right )^{4} \sqrt {e x +d}}+\frac {2 B \,b^{2} d}{\left (a e -b d \right )^{4} \sqrt {e x +d}}+\frac {4 A b e}{3 \left (a e -b d \right )^{3} \left (e x +d \right )^{\frac {3}{2}}}-\frac {2 B a e}{3 \left (a e -b d \right )^{3} \left (e x +d \right )^{\frac {3}{2}}}-\frac {2 B b d}{3 \left (a e -b d \right )^{3} \left (e x +d \right )^{\frac {3}{2}}}-\frac {2 A e}{5 \left (a e -b d \right )^{2} \left (e x +d \right )^{\frac {5}{2}}}+\frac {2 B d}{5 \left (a e -b d \right )^{2} \left (e x +d \right )^{\frac {5}{2}}} \]
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 = 2.16, size = 261, normalized size = 1.18 \[ \frac {b^{3/2}\,\mathrm {atan}\left (\frac {\sqrt {b}\,\sqrt {d+e\,x}\,\left (a^4\,e^4-4\,a^3\,b\,d\,e^3+6\,a^2\,b^2\,d^2\,e^2-4\,a\,b^3\,d^3\,e+b^4\,d^4\right )}{{\left (a\,e-b\,d\right )}^{9/2}}\right )\,\left (5\,B\,a\,e-7\,A\,b\,e+2\,B\,b\,d\right )}{{\left (a\,e-b\,d\right )}^{9/2}}-\frac {\frac {2\,\left (A\,e-B\,d\right )}{5\,\left (a\,e-b\,d\right )}+\frac {2\,\left (d+e\,x\right )\,\left (5\,B\,a\,e-7\,A\,b\,e+2\,B\,b\,d\right )}{15\,{\left (a\,e-b\,d\right )}^2}-\frac {b^2\,{\left (d+e\,x\right )}^3\,\left (5\,B\,a\,e-7\,A\,b\,e+2\,B\,b\,d\right )}{{\left (a\,e-b\,d\right )}^4}-\frac {2\,b\,{\left (d+e\,x\right )}^2\,\left (5\,B\,a\,e-7\,A\,b\,e+2\,B\,b\,d\right )}{3\,{\left (a\,e-b\,d\right )}^3}}{b\,{\left (d+e\,x\right )}^{7/2}+\left (a\,e-b\,d\right )\,{\left (d+e\,x\right )}^{5/2}} \]
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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