Optimal. Leaf size=151 \[ -\frac {2 b^{3/2} (A b-a B) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{(b d-a e)^{7/2}}+\frac {2 b (A b-a B)}{\sqrt {d+e x} (b d-a e)^3}+\frac {2 (A b-a B)}{3 (d+e x)^{3/2} (b d-a e)^2}-\frac {2 (B d-A e)}{5 e (d+e x)^{5/2} (b d-a e)} \]
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Rubi [A] time = 0.09, antiderivative size = 151, 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} \begin {gather*} -\frac {2 b^{3/2} (A b-a B) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{(b d-a e)^{7/2}}+\frac {2 b (A b-a B)}{\sqrt {d+e x} (b d-a e)^3}+\frac {2 (A b-a B)}{3 (d+e x)^{3/2} (b d-a e)^2}-\frac {2 (B d-A e)}{5 e (d+e x)^{5/2} (b d-a e)} \end {gather*}
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) (d+e x)^{7/2}} \, dx &=-\frac {2 (B d-A e)}{5 e (b d-a e) (d+e x)^{5/2}}+\frac {(A b-a B) \int \frac {1}{(a+b x) (d+e x)^{5/2}} \, dx}{b d-a e}\\ &=-\frac {2 (B d-A e)}{5 e (b d-a e) (d+e x)^{5/2}}+\frac {2 (A b-a B)}{3 (b d-a e)^2 (d+e x)^{3/2}}+\frac {(b (A b-a B)) \int \frac {1}{(a+b x) (d+e x)^{3/2}} \, dx}{(b d-a e)^2}\\ &=-\frac {2 (B d-A e)}{5 e (b d-a e) (d+e x)^{5/2}}+\frac {2 (A b-a B)}{3 (b d-a e)^2 (d+e x)^{3/2}}+\frac {2 b (A b-a B)}{(b d-a e)^3 \sqrt {d+e x}}+\frac {\left (b^2 (A b-a B)\right ) \int \frac {1}{(a+b x) \sqrt {d+e x}} \, dx}{(b d-a e)^3}\\ &=-\frac {2 (B d-A e)}{5 e (b d-a e) (d+e x)^{5/2}}+\frac {2 (A b-a B)}{3 (b d-a e)^2 (d+e x)^{3/2}}+\frac {2 b (A b-a B)}{(b d-a e)^3 \sqrt {d+e x}}+\frac {\left (2 b^2 (A b-a B)\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)^3}\\ &=-\frac {2 (B d-A e)}{5 e (b d-a e) (d+e x)^{5/2}}+\frac {2 (A b-a B)}{3 (b d-a e)^2 (d+e x)^{3/2}}+\frac {2 b (A b-a B)}{(b d-a e)^3 \sqrt {d+e x}}-\frac {2 b^{3/2} (A b-a B) \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 = 86, normalized size = 0.57 \begin {gather*} \frac {10 e (d+e x) (A b-a B) \, _2F_1\left (-\frac {3}{2},1;-\frac {1}{2};\frac {b (d+e x)}{b d-a e}\right )-6 (b d-a e) (B d-A e)}{15 e (d+e x)^{5/2} (b d-a e)^2} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.25, size = 232, normalized size = 1.54 \begin {gather*} \frac {2 \left (A b^{5/2}-a b^{3/2} B\right ) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x} \sqrt {a e-b d}}{b d-a e}\right )}{(a e-b d)^{7/2}}-\frac {2 \left (3 a^2 A e^3+5 a^2 B e^2 (d+e x)-3 a^2 B d e^2-5 a A b e^2 (d+e x)-6 a A b d e^2+6 a b B d^2 e-5 a b B d e (d+e x)-15 a b B e (d+e x)^2+3 A b^2 d^2 e+5 A b^2 d e (d+e x)+15 A b^2 e (d+e x)^2-3 b^2 B d^3\right )}{15 e (d+e x)^{5/2} (a e-b d)^3} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 0.94, size = 902, normalized size = 5.97 \begin {gather*} \left [\frac {15 \, {\left ({\left (B a b - A b^{2}\right )} e^{4} x^{3} + 3 \, {\left (B a b - A b^{2}\right )} d e^{3} x^{2} + 3 \, {\left (B a b - A b^{2}\right )} d^{2} e^{2} x + {\left (B a b - A b^{2}\right )} d^{3} e\right )} \sqrt {\frac {b}{b d - a e}} \log \left (\frac {b e x + 2 \, b d - a e + 2 \, {\left (b d - a e\right )} \sqrt {e x + d} \sqrt {\frac {b}{b d - a e}}}{b x + a}\right ) - 2 \, {\left (3 \, B b^{2} d^{3} - 3 \, A a^{2} e^{3} + 15 \, {\left (B a b - A b^{2}\right )} e^{3} x^{2} + {\left (14 \, B a b - 23 \, A b^{2}\right )} d^{2} e - {\left (2 \, B a^{2} - 11 \, A a b\right )} d e^{2} + 5 \, {\left (7 \, {\left (B a b - A b^{2}\right )} d e^{2} - {\left (B a^{2} - A a b\right )} e^{3}\right )} x\right )} \sqrt {e x + d}}{15 \, {\left (b^{3} d^{6} e - 3 \, a b^{2} d^{5} e^{2} + 3 \, a^{2} b d^{4} e^{3} - a^{3} d^{3} e^{4} + {\left (b^{3} d^{3} e^{4} - 3 \, a b^{2} d^{2} e^{5} + 3 \, a^{2} b d e^{6} - a^{3} e^{7}\right )} x^{3} + 3 \, {\left (b^{3} d^{4} e^{3} - 3 \, a b^{2} d^{3} e^{4} + 3 \, a^{2} b d^{2} e^{5} - a^{3} d e^{6}\right )} x^{2} + 3 \, {\left (b^{3} d^{5} e^{2} - 3 \, a b^{2} d^{4} e^{3} + 3 \, a^{2} b d^{3} e^{4} - a^{3} d^{2} e^{5}\right )} x\right )}}, \frac {2 \, {\left (15 \, {\left ({\left (B a b - A b^{2}\right )} e^{4} x^{3} + 3 \, {\left (B a b - A b^{2}\right )} d e^{3} x^{2} + 3 \, {\left (B a b - A b^{2}\right )} d^{2} e^{2} x + {\left (B a b - A b^{2}\right )} d^{3} e\right )} \sqrt {-\frac {b}{b d - a e}} \arctan \left (-\frac {{\left (b d - a e\right )} \sqrt {e x + d} \sqrt {-\frac {b}{b d - a e}}}{b e x + b d}\right ) - {\left (3 \, B b^{2} d^{3} - 3 \, A a^{2} e^{3} + 15 \, {\left (B a b - A b^{2}\right )} e^{3} x^{2} + {\left (14 \, B a b - 23 \, A b^{2}\right )} d^{2} e - {\left (2 \, B a^{2} - 11 \, A a b\right )} d e^{2} + 5 \, {\left (7 \, {\left (B a b - A b^{2}\right )} d e^{2} - {\left (B a^{2} - A a b\right )} e^{3}\right )} x\right )} \sqrt {e x + d}\right )}}{15 \, {\left (b^{3} d^{6} e - 3 \, a b^{2} d^{5} e^{2} + 3 \, a^{2} b d^{4} e^{3} - a^{3} d^{3} e^{4} + {\left (b^{3} d^{3} e^{4} - 3 \, a b^{2} d^{2} e^{5} + 3 \, a^{2} b d e^{6} - a^{3} e^{7}\right )} x^{3} + 3 \, {\left (b^{3} d^{4} e^{3} - 3 \, a b^{2} d^{3} e^{4} + 3 \, a^{2} b d^{2} e^{5} - a^{3} d e^{6}\right )} x^{2} + 3 \, {\left (b^{3} d^{5} e^{2} - 3 \, a b^{2} d^{4} e^{3} + 3 \, a^{2} b d^{3} e^{4} - a^{3} d^{2} e^{5}\right )} x\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 1.30, size = 284, normalized size = 1.88 \begin {gather*} -\frac {2 \, {\left (B a b^{2} - A b^{3}\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 {2 \, {\left (3 \, B b^{2} d^{3} + 15 \, {\left (x e + d\right )}^{2} B a b e - 15 \, {\left (x e + d\right )}^{2} A b^{2} e + 5 \, {\left (x e + d\right )} B a b d e - 5 \, {\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} + 5 \, {\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^{3} d^{3} e - 3 \, a b^{2} d^{2} e^{2} + 3 \, a^{2} b d e^{3} - a^{3} e^{4}\right )} {\left (x e + d\right )}^{\frac {5}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 234, normalized size = 1.55 \begin {gather*} -\frac {2 A \,b^{3} \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 a \,b^{2} \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 A \,b^{2}}{\left (a e -b d \right )^{3} \sqrt {e x +d}}+\frac {2 B a b}{\left (a e -b d \right )^{3} \sqrt {e x +d}}+\frac {2 A b}{3 \left (a e -b d \right )^{2} \left (e x +d \right )^{\frac {3}{2}}}-\frac {2 B a}{3 \left (a e -b d \right )^{2} \left (e x +d \right )^{\frac {3}{2}}}-\frac {2 A}{5 \left (a e -b d \right ) \left (e x +d \right )^{\frac {5}{2}}}+\frac {2 B d}{5 \left (a e -b d \right ) \left (e x +d \right )^{\frac {5}{2}} e} \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 = 0.20, size = 173, normalized size = 1.15 \begin {gather*} -\frac {\frac {2\,\left (A\,e-B\,d\right )}{5\,\left (a\,e-b\,d\right )}-\frac {2\,\left (A\,b\,e-B\,a\,e\right )\,\left (d+e\,x\right )}{3\,{\left (a\,e-b\,d\right )}^2}+\frac {2\,b\,\left (A\,b\,e-B\,a\,e\right )\,{\left (d+e\,x\right )}^2}{{\left (a\,e-b\,d\right )}^3}}{e\,{\left (d+e\,x\right )}^{5/2}}-\frac {2\,b^{3/2}\,\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 (A\,b-B\,a\right )}{{\left (a\,e-b\,d\right )}^{7/2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 51.98, size = 136, normalized size = 0.90 \begin {gather*} \frac {2 b \left (- A b + B a\right )}{\sqrt {d + e x} \left (a e - b d\right )^{3}} + \frac {2 b \left (- A b + B a\right ) \operatorname {atan}{\left (\frac {\sqrt {d + e x}}{\sqrt {\frac {a e - b d}{b}}} \right )}}{\sqrt {\frac {a e - b d}{b}} \left (a e - b d\right )^{3}} - \frac {2 \left (- A b + B a\right )}{3 \left (d + e x\right )^{\frac {3}{2}} \left (a e - b d\right )^{2}} + \frac {2 \left (- A e + B d\right )}{5 e \left (d + e x\right )^{\frac {5}{2}} \left (a e - b d\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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