Optimal. Leaf size=119 \[ -\frac {2 (B d-A e)}{3 e (b d-a e) (d+e x)^{3/2}}+\frac {2 (A b-a B)}{(b d-a e)^2 \sqrt {d+e x}}-\frac {2 \sqrt {b} (A b-a B) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{(b d-a e)^{5/2}} \]
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Rubi [A]
time = 0.05, antiderivative size = 119, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {79, 53, 65, 214}
\begin {gather*} \frac {2 (A b-a B)}{\sqrt {d+e x} (b d-a e)^2}-\frac {2 (B d-A e)}{3 e (d+e x)^{3/2} (b d-a e)}-\frac {2 \sqrt {b} (A b-a B) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{(b d-a e)^{5/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 53
Rule 65
Rule 79
Rule 214
Rubi steps
\begin {align*} \int \frac {A+B x}{(a+b x) (d+e x)^{5/2}} \, dx &=-\frac {2 (B d-A e)}{3 e (b d-a e) (d+e x)^{3/2}}+\frac {(A b-a B) \int \frac {1}{(a+b x) (d+e x)^{3/2}} \, dx}{b d-a e}\\ &=-\frac {2 (B d-A e)}{3 e (b d-a e) (d+e x)^{3/2}}+\frac {2 (A b-a B)}{(b d-a e)^2 \sqrt {d+e x}}+\frac {(b (A b-a B)) \int \frac {1}{(a+b x) \sqrt {d+e x}} \, dx}{(b d-a e)^2}\\ &=-\frac {2 (B d-A e)}{3 e (b d-a e) (d+e x)^{3/2}}+\frac {2 (A b-a B)}{(b d-a e)^2 \sqrt {d+e x}}+\frac {(2 b (A b-a B)) \text {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)^2}\\ &=-\frac {2 (B d-A e)}{3 e (b d-a e) (d+e x)^{3/2}}+\frac {2 (A b-a B)}{(b d-a e)^2 \sqrt {d+e x}}-\frac {2 \sqrt {b} (A b-a B) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{(b d-a e)^{5/2}}\\ \end {align*}
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Mathematica [A]
time = 0.27, size = 118, normalized size = 0.99 \begin {gather*} -\frac {2 \left (b B d^2-A b e (4 d+3 e x)+a e (2 B d+A e+3 B e x)\right )}{3 e (b d-a e)^2 (d+e x)^{3/2}}+\frac {2 \sqrt {b} (A b-a B) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {-b d+a e}}\right )}{(-b d+a e)^{5/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.09, size = 116, normalized size = 0.97
method | result | size |
derivativedivides | \(\frac {-\frac {2 \left (A e -B d \right )}{3 \left (a e -b d \right ) \left (e x +d \right )^{\frac {3}{2}}}+\frac {2 e \left (A b -B a \right )}{\left (a e -b d \right )^{2} \sqrt {e x +d}}+\frac {2 b e \left (A b -B a \right ) \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (a e -b d \right ) b}}\right )}{\left (a e -b d \right )^{2} \sqrt {\left (a e -b d \right ) b}}}{e}\) | \(116\) |
default | \(\frac {-\frac {2 \left (A e -B d \right )}{3 \left (a e -b d \right ) \left (e x +d \right )^{\frac {3}{2}}}+\frac {2 e \left (A b -B a \right )}{\left (a e -b d \right )^{2} \sqrt {e x +d}}+\frac {2 b e \left (A b -B a \right ) \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (a e -b d \right ) b}}\right )}{\left (a e -b d \right )^{2} \sqrt {\left (a e -b d \right ) b}}}{e}\) | \(116\) |
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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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 248 vs.
\(2 (110) = 220\).
time = 1.15, size = 507, normalized size = 4.26 \begin {gather*} \left [-\frac {3 \, {\left ({\left (B a - A b\right )} x^{2} e^{3} + 2 \, {\left (B a - A b\right )} d x e^{2} + {\left (B a - A b\right )} d^{2} e\right )} \sqrt {\frac {b}{b d - a e}} \log \left (\frac {2 \, b d - 2 \, {\left (b d - a e\right )} \sqrt {x e + d} \sqrt {\frac {b}{b d - a e}} + {\left (b x - a\right )} e}{b x + a}\right ) + 2 \, {\left (B b d^{2} + 2 \, {\left (B a - 2 \, A b\right )} d e + {\left (A a + 3 \, {\left (B a - A b\right )} x\right )} e^{2}\right )} \sqrt {x e + d}}{3 \, {\left (b^{2} d^{4} e + a^{2} x^{2} e^{5} - 2 \, {\left (a b d x^{2} - a^{2} d x\right )} e^{4} + {\left (b^{2} d^{2} x^{2} - 4 \, a b d^{2} x + a^{2} d^{2}\right )} e^{3} + 2 \, {\left (b^{2} d^{3} x - a b d^{3}\right )} e^{2}\right )}}, \frac {2 \, {\left (3 \, {\left ({\left (B a - A b\right )} x^{2} e^{3} + 2 \, {\left (B a - A b\right )} d x e^{2} + {\left (B a - A b\right )} d^{2} e\right )} \sqrt {-\frac {b}{b d - a e}} \arctan \left (-\frac {{\left (b d - a e\right )} \sqrt {x e + d} \sqrt {-\frac {b}{b d - a e}}}{b x e + b d}\right ) - {\left (B b d^{2} + 2 \, {\left (B a - 2 \, A b\right )} d e + {\left (A a + 3 \, {\left (B a - A b\right )} x\right )} e^{2}\right )} \sqrt {x e + d}\right )}}{3 \, {\left (b^{2} d^{4} e + a^{2} x^{2} e^{5} - 2 \, {\left (a b d x^{2} - a^{2} d x\right )} e^{4} + {\left (b^{2} d^{2} x^{2} - 4 \, a b d^{2} x + a^{2} d^{2}\right )} e^{3} + 2 \, {\left (b^{2} d^{3} x - a b d^{3}\right )} e^{2}\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 18.08, size = 105, normalized size = 0.88 \begin {gather*} - \frac {2 \left (- A b + B a\right )}{\sqrt {d + e x} \left (a e - b d\right )^{2}} - \frac {2 \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 )^{2}} + \frac {2 \left (- A e + B d\right )}{3 e \left (d + e x\right )^{\frac {3}{2}} \left (a e - b d\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.01, size = 161, normalized size = 1.35 \begin {gather*} -\frac {2 \, {\left (B a b - A b^{2}\right )} \arctan \left (\frac {\sqrt {x e + d} b}{\sqrt {-b^{2} d + a b e}}\right )}{{\left (b^{2} d^{2} - 2 \, a b d e + a^{2} e^{2}\right )} \sqrt {-b^{2} d + a b e}} - \frac {2 \, {\left (B b d^{2} + 3 \, {\left (x e + d\right )} B a e - 3 \, {\left (x e + d\right )} A b e - B a d e - A b d e + A a e^{2}\right )}}{3 \, {\left (b^{2} d^{2} e - 2 \, a b d e^{2} + a^{2} e^{3}\right )} {\left (x e + d\right )}^{\frac {3}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.31, size = 128, normalized size = 1.08 \begin {gather*} \frac {2\,\sqrt {b}\,\mathrm {atan}\left (\frac {\sqrt {b}\,\sqrt {d+e\,x}\,\left (a^2\,e^2-2\,a\,b\,d\,e+b^2\,d^2\right )}{{\left (a\,e-b\,d\right )}^{5/2}}\right )\,\left (A\,b-B\,a\right )}{{\left (a\,e-b\,d\right )}^{5/2}}-\frac {\frac {2\,\left (A\,e-B\,d\right )}{3\,\left (a\,e-b\,d\right )}-\frac {2\,\left (A\,b\,e-B\,a\,e\right )\,\left (d+e\,x\right )}{{\left (a\,e-b\,d\right )}^2}}{e\,{\left (d+e\,x\right )}^{3/2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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