Optimal. Leaf size=181 \[ -\frac {A b-a B}{b (a+b x) (d+e x)^{3/2} (b d-a e)}+\frac {3 a B e-5 A b e+2 b B d}{\sqrt {d+e x} (b d-a e)^3}+\frac {3 a B e-5 A b e+2 b B d}{3 b (d+e x)^{3/2} (b d-a e)^2}-\frac {\sqrt {b} (3 a B e-5 A b e+2 b B d) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{(b d-a e)^{7/2}} \]
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Rubi [A] time = 0.16, antiderivative size = 181, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {78, 51, 63, 208} \[ -\frac {A b-a B}{b (a+b x) (d+e x)^{3/2} (b d-a e)}+\frac {3 a B e-5 A b e+2 b B d}{\sqrt {d+e x} (b d-a e)^3}+\frac {3 a B e-5 A b e+2 b B d}{3 b (d+e x)^{3/2} (b d-a e)^2}-\frac {\sqrt {b} (3 a B e-5 A b e+2 b B d) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{(b d-a e)^{7/2}} \]
Antiderivative was successfully verified.
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Rule 51
Rule 63
Rule 78
Rule 208
Rubi steps
\begin {align*} \int \frac {A+B x}{(a+b x)^2 (d+e x)^{5/2}} \, dx &=-\frac {A b-a B}{b (b d-a e) (a+b x) (d+e x)^{3/2}}+\frac {(2 b B d-5 A b e+3 a B e) \int \frac {1}{(a+b x) (d+e x)^{5/2}} \, dx}{2 b (b d-a e)}\\ &=\frac {2 b B d-5 A b e+3 a B e}{3 b (b d-a e)^2 (d+e x)^{3/2}}-\frac {A b-a B}{b (b d-a e) (a+b x) (d+e x)^{3/2}}+\frac {(2 b B d-5 A b e+3 a B e) \int \frac {1}{(a+b x) (d+e x)^{3/2}} \, dx}{2 (b d-a e)^2}\\ &=\frac {2 b B d-5 A b e+3 a B e}{3 b (b d-a e)^2 (d+e x)^{3/2}}-\frac {A b-a B}{b (b d-a e) (a+b x) (d+e x)^{3/2}}+\frac {2 b B d-5 A b e+3 a B e}{(b d-a e)^3 \sqrt {d+e x}}+\frac {(b (2 b B d-5 A b e+3 a B e)) \int \frac {1}{(a+b x) \sqrt {d+e x}} \, dx}{2 (b d-a e)^3}\\ &=\frac {2 b B d-5 A b e+3 a B e}{3 b (b d-a e)^2 (d+e x)^{3/2}}-\frac {A b-a B}{b (b d-a e) (a+b x) (d+e x)^{3/2}}+\frac {2 b B d-5 A b e+3 a B e}{(b d-a e)^3 \sqrt {d+e x}}+\frac {(b (2 b B d-5 A b e+3 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 )}{e (b d-a e)^3}\\ &=\frac {2 b B d-5 A b e+3 a B e}{3 b (b d-a e)^2 (d+e x)^{3/2}}-\frac {A b-a B}{b (b d-a e) (a+b x) (d+e x)^{3/2}}+\frac {2 b B d-5 A b e+3 a B e}{(b d-a e)^3 \sqrt {d+e x}}-\frac {\sqrt {b} (2 b B d-5 A b e+3 a B e) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{(b d-a e)^{7/2}}\\ \end {align*}
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Mathematica [C] time = 0.06, size = 94, normalized size = 0.52 \[ \frac {(3 a B e-5 A b e+2 b B d) \, _2F_1\left (-\frac {3}{2},1;-\frac {1}{2};\frac {b (d+e x)}{b d-a e}\right )-\frac {3 (A b-a B) (b d-a e)}{a+b x}}{3 b (d+e x)^{3/2} (b d-a e)^2} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.78, size = 1106, normalized size = 6.11 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.33, size = 297, normalized size = 1.64 \[ \frac {{\left (2 \, B b^{2} d + 3 \, B a b e - 5 \, A b^{2} e\right )} \arctan \left (\frac {\sqrt {x e + d} b}{\sqrt {-b^{2} d + a b e}}\right )}{{\left (b^{3} d^{3} - 3 \, a b^{2} d^{2} e + 3 \, a^{2} b d e^{2} - a^{3} e^{3}\right )} \sqrt {-b^{2} d + a b e}} + \frac {\sqrt {x e + d} B a b e - \sqrt {x e + d} A b^{2} e}{{\left (b^{3} d^{3} - 3 \, a b^{2} d^{2} e + 3 \, a^{2} b d e^{2} - a^{3} e^{3}\right )} {\left ({\left (x e + d\right )} b - b d + a e\right )}} + \frac {2 \, {\left (3 \, {\left (x e + d\right )} B b d + B b d^{2} + 3 \, {\left (x e + d\right )} B a e - 6 \, {\left (x e + d\right )} A b e - B a d e - A b d e + A a e^{2}\right )}}{3 \, {\left (b^{3} d^{3} - 3 \, a b^{2} d^{2} e + 3 \, a^{2} b d e^{2} - a^{3} e^{3}\right )} {\left (x e + d\right )}^{\frac {3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.02, size = 328, normalized size = 1.81 \[ \frac {5 A \,b^{2} e \arctan \left (\frac {\sqrt {e x +d}\, b}{\sqrt {\left (a e -b d \right ) b}}\right )}{\left (a e -b d \right )^{3} \sqrt {\left (a e -b d \right ) b}}-\frac {3 B a b e \arctan \left (\frac {\sqrt {e x +d}\, b}{\sqrt {\left (a e -b d \right ) b}}\right )}{\left (a e -b d \right )^{3} \sqrt {\left (a e -b d \right ) b}}-\frac {2 B \,b^{2} d \arctan \left (\frac {\sqrt {e x +d}\, b}{\sqrt {\left (a e -b d \right ) b}}\right )}{\left (a e -b d \right )^{3} \sqrt {\left (a e -b d \right ) b}}+\frac {\sqrt {e x +d}\, A \,b^{2} e}{\left (a e -b d \right )^{3} \left (b x e +a e \right )}-\frac {\sqrt {e x +d}\, B a b e}{\left (a e -b d \right )^{3} \left (b x e +a e \right )}+\frac {4 A b e}{\left (a e -b d \right )^{3} \sqrt {e x +d}}-\frac {2 B a e}{\left (a e -b d \right )^{3} \sqrt {e x +d}}-\frac {2 B b d}{\left (a e -b d \right )^{3} \sqrt {e x +d}}-\frac {2 A e}{3 \left (a e -b d \right )^{2} \left (e x +d \right )^{\frac {3}{2}}}+\frac {2 B d}{3 \left (a e -b d \right )^{2} \left (e x +d \right )^{\frac {3}{2}}} \]
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 = 1.35, size = 210, normalized size = 1.16 \[ -\frac {\frac {2\,\left (A\,e-B\,d\right )}{3\,\left (a\,e-b\,d\right )}+\frac {2\,\left (d+e\,x\right )\,\left (3\,B\,a\,e-5\,A\,b\,e+2\,B\,b\,d\right )}{3\,{\left (a\,e-b\,d\right )}^2}+\frac {b\,{\left (d+e\,x\right )}^2\,\left (3\,B\,a\,e-5\,A\,b\,e+2\,B\,b\,d\right )}{{\left (a\,e-b\,d\right )}^3}}{b\,{\left (d+e\,x\right )}^{5/2}+\left (a\,e-b\,d\right )\,{\left (d+e\,x\right )}^{3/2}}-\frac {\sqrt {b}\,\mathrm {atan}\left (\frac {\sqrt {b}\,\sqrt {d+e\,x}\,\left (a^3\,e^3-3\,a^2\,b\,d\,e^2+3\,a\,b^2\,d^2\,e-b^3\,d^3\right )}{{\left (a\,e-b\,d\right )}^{7/2}}\right )\,\left (3\,B\,a\,e-5\,A\,b\,e+2\,B\,b\,d\right )}{{\left (a\,e-b\,d\right )}^{7/2}} \]
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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