Optimal. Leaf size=197 \[ -\frac {3 e (-5 a B e+A b e+4 b B d) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{4 b^{7/2} \sqrt {b d-a e}}+\frac {3 e \sqrt {d+e x} (-5 a B e+A b e+4 b B d)}{4 b^3 (b d-a e)}-\frac {(d+e x)^{3/2} (-5 a B e+A b e+4 b B d)}{4 b^2 (a+b x) (b d-a e)}-\frac {(d+e x)^{5/2} (A b-a B)}{2 b (a+b x)^2 (b d-a e)} \]
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Rubi [A] time = 0.15, antiderivative size = 197, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {78, 47, 50, 63, 208} \[ -\frac {(d+e x)^{3/2} (-5 a B e+A b e+4 b B d)}{4 b^2 (a+b x) (b d-a e)}+\frac {3 e \sqrt {d+e x} (-5 a B e+A b e+4 b B d)}{4 b^3 (b d-a e)}-\frac {3 e (-5 a B e+A b e+4 b B d) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{4 b^{7/2} \sqrt {b d-a e}}-\frac {(d+e x)^{5/2} (A b-a B)}{2 b (a+b x)^2 (b d-a e)} \]
Antiderivative was successfully verified.
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Rule 47
Rule 50
Rule 63
Rule 78
Rule 208
Rubi steps
\begin {align*} \int \frac {(A+B x) (d+e x)^{3/2}}{(a+b x)^3} \, dx &=-\frac {(A b-a B) (d+e x)^{5/2}}{2 b (b d-a e) (a+b x)^2}+\frac {(4 b B d+A b e-5 a B e) \int \frac {(d+e x)^{3/2}}{(a+b x)^2} \, dx}{4 b (b d-a e)}\\ &=-\frac {(4 b B d+A b e-5 a B e) (d+e x)^{3/2}}{4 b^2 (b d-a e) (a+b x)}-\frac {(A b-a B) (d+e x)^{5/2}}{2 b (b d-a e) (a+b x)^2}+\frac {(3 e (4 b B d+A b e-5 a B e)) \int \frac {\sqrt {d+e x}}{a+b x} \, dx}{8 b^2 (b d-a e)}\\ &=\frac {3 e (4 b B d+A b e-5 a B e) \sqrt {d+e x}}{4 b^3 (b d-a e)}-\frac {(4 b B d+A b e-5 a B e) (d+e x)^{3/2}}{4 b^2 (b d-a e) (a+b x)}-\frac {(A b-a B) (d+e x)^{5/2}}{2 b (b d-a e) (a+b x)^2}+\frac {(3 e (4 b B d+A b e-5 a B e)) \int \frac {1}{(a+b x) \sqrt {d+e x}} \, dx}{8 b^3}\\ &=\frac {3 e (4 b B d+A b e-5 a B e) \sqrt {d+e x}}{4 b^3 (b d-a e)}-\frac {(4 b B d+A b e-5 a B e) (d+e x)^{3/2}}{4 b^2 (b d-a e) (a+b x)}-\frac {(A b-a B) (d+e x)^{5/2}}{2 b (b d-a e) (a+b x)^2}+\frac {(3 (4 b B d+A b e-5 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 )}{4 b^3}\\ &=\frac {3 e (4 b B d+A b e-5 a B e) \sqrt {d+e x}}{4 b^3 (b d-a e)}-\frac {(4 b B d+A b e-5 a B e) (d+e x)^{3/2}}{4 b^2 (b d-a e) (a+b x)}-\frac {(A b-a B) (d+e x)^{5/2}}{2 b (b d-a e) (a+b x)^2}-\frac {3 e (4 b B d+A b e-5 a B e) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{4 b^{7/2} \sqrt {b d-a e}}\\ \end {align*}
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Mathematica [C] time = 0.09, size = 96, normalized size = 0.49 \[ \frac {(d+e x)^{5/2} \left (\frac {e (-5 a B e+A b e+4 b B d) \, _2F_1\left (2,\frac {5}{2};\frac {7}{2};\frac {b (d+e x)}{b d-a e}\right )}{(b d-a e)^2}+\frac {5 a B-5 A b}{(a+b x)^2}\right )}{10 b (b d-a e)} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.70, size = 703, normalized size = 3.57 \[ \left [\frac {3 \, {\left (4 \, B a^{2} b d e - {\left (5 \, B a^{3} - A a^{2} b\right )} e^{2} + {\left (4 \, B b^{3} d e - {\left (5 \, B a b^{2} - A b^{3}\right )} e^{2}\right )} x^{2} + 2 \, {\left (4 \, B a b^{2} d e - {\left (5 \, B a^{2} b - A a b^{2}\right )} e^{2}\right )} x\right )} \sqrt {b^{2} d - a b e} \log \left (\frac {b e x + 2 \, b d - a e - 2 \, \sqrt {b^{2} d - a b e} \sqrt {e x + d}}{b x + a}\right ) - 2 \, {\left (2 \, {\left (B a b^{3} + A b^{4}\right )} d^{2} - {\left (17 \, B a^{2} b^{2} - A a b^{3}\right )} d e + 3 \, {\left (5 \, B a^{3} b - A a^{2} b^{2}\right )} e^{2} - 8 \, {\left (B b^{4} d e - B a b^{3} e^{2}\right )} x^{2} + {\left (4 \, B b^{4} d^{2} - {\left (29 \, B a b^{3} - 5 \, A b^{4}\right )} d e + 5 \, {\left (5 \, B a^{2} b^{2} - A a b^{3}\right )} e^{2}\right )} x\right )} \sqrt {e x + d}}{8 \, {\left (a^{2} b^{5} d - a^{3} b^{4} e + {\left (b^{7} d - a b^{6} e\right )} x^{2} + 2 \, {\left (a b^{6} d - a^{2} b^{5} e\right )} x\right )}}, \frac {3 \, {\left (4 \, B a^{2} b d e - {\left (5 \, B a^{3} - A a^{2} b\right )} e^{2} + {\left (4 \, B b^{3} d e - {\left (5 \, B a b^{2} - A b^{3}\right )} e^{2}\right )} x^{2} + 2 \, {\left (4 \, B a b^{2} d e - {\left (5 \, B a^{2} b - A a b^{2}\right )} e^{2}\right )} x\right )} \sqrt {-b^{2} d + a b e} \arctan \left (\frac {\sqrt {-b^{2} d + a b e} \sqrt {e x + d}}{b e x + b d}\right ) - {\left (2 \, {\left (B a b^{3} + A b^{4}\right )} d^{2} - {\left (17 \, B a^{2} b^{2} - A a b^{3}\right )} d e + 3 \, {\left (5 \, B a^{3} b - A a^{2} b^{2}\right )} e^{2} - 8 \, {\left (B b^{4} d e - B a b^{3} e^{2}\right )} x^{2} + {\left (4 \, B b^{4} d^{2} - {\left (29 \, B a b^{3} - 5 \, A b^{4}\right )} d e + 5 \, {\left (5 \, B a^{2} b^{2} - A a b^{3}\right )} e^{2}\right )} x\right )} \sqrt {e x + d}}{4 \, {\left (a^{2} b^{5} d - a^{3} b^{4} e + {\left (b^{7} d - a b^{6} e\right )} x^{2} + 2 \, {\left (a b^{6} d - a^{2} b^{5} e\right )} x\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.31, size = 236, normalized size = 1.20 \[ \frac {2 \, \sqrt {x e + d} B e}{b^{3}} + \frac {3 \, {\left (4 \, B b d e - 5 \, B a e^{2} + A b e^{2}\right )} \arctan \left (\frac {\sqrt {x e + d} b}{\sqrt {-b^{2} d + a b e}}\right )}{4 \, \sqrt {-b^{2} d + a b e} b^{3}} - \frac {4 \, {\left (x e + d\right )}^{\frac {3}{2}} B b^{2} d e - 4 \, \sqrt {x e + d} B b^{2} d^{2} e - 9 \, {\left (x e + d\right )}^{\frac {3}{2}} B a b e^{2} + 5 \, {\left (x e + d\right )}^{\frac {3}{2}} A b^{2} e^{2} + 11 \, \sqrt {x e + d} B a b d e^{2} - 3 \, \sqrt {x e + d} A b^{2} d e^{2} - 7 \, \sqrt {x e + d} B a^{2} e^{3} + 3 \, \sqrt {x e + d} A a b e^{3}}{4 \, {\left ({\left (x e + d\right )} b - b d + a e\right )}^{2} b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.02, size = 360, normalized size = 1.83 \[ -\frac {3 \sqrt {e x +d}\, A a \,e^{3}}{4 \left (b x e +a e \right )^{2} b^{2}}+\frac {3 \sqrt {e x +d}\, A d \,e^{2}}{4 \left (b x e +a e \right )^{2} b}+\frac {7 \sqrt {e x +d}\, B \,a^{2} e^{3}}{4 \left (b x e +a e \right )^{2} b^{3}}-\frac {11 \sqrt {e x +d}\, B a d \,e^{2}}{4 \left (b x e +a e \right )^{2} b^{2}}+\frac {\sqrt {e x +d}\, B \,d^{2} e}{\left (b x e +a e \right )^{2} b}-\frac {5 \left (e x +d \right )^{\frac {3}{2}} A \,e^{2}}{4 \left (b x e +a e \right )^{2} b}+\frac {3 A \,e^{2} \arctan \left (\frac {\sqrt {e x +d}\, b}{\sqrt {\left (a e -b d \right ) b}}\right )}{4 \sqrt {\left (a e -b d \right ) b}\, b^{2}}+\frac {9 \left (e x +d \right )^{\frac {3}{2}} B a \,e^{2}}{4 \left (b x e +a e \right )^{2} b^{2}}-\frac {15 B a \,e^{2} \arctan \left (\frac {\sqrt {e x +d}\, b}{\sqrt {\left (a e -b d \right ) b}}\right )}{4 \sqrt {\left (a e -b d \right ) b}\, b^{3}}-\frac {\left (e x +d \right )^{\frac {3}{2}} B d e}{\left (b x e +a e \right )^{2} b}+\frac {3 B d e \arctan \left (\frac {\sqrt {e x +d}\, b}{\sqrt {\left (a e -b d \right ) b}}\right )}{\sqrt {\left (a e -b d \right ) b}\, b^{2}}+\frac {2 \sqrt {e x +d}\, B e}{b^{3}} \]
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 = 1.30, size = 256, normalized size = 1.30 \[ \frac {\sqrt {d+e\,x}\,\left (\frac {7\,B\,a^2\,e^3}{4}-\frac {11\,B\,a\,b\,d\,e^2}{4}-\frac {3\,A\,a\,b\,e^3}{4}+B\,b^2\,d^2\,e+\frac {3\,A\,b^2\,d\,e^2}{4}\right )-{\left (d+e\,x\right )}^{3/2}\,\left (\frac {5\,A\,b^2\,e^2}{4}+B\,d\,b^2\,e-\frac {9\,B\,a\,b\,e^2}{4}\right )}{b^5\,{\left (d+e\,x\right )}^2-\left (2\,b^5\,d-2\,a\,b^4\,e\right )\,\left (d+e\,x\right )+b^5\,d^2+a^2\,b^3\,e^2-2\,a\,b^4\,d\,e}+\frac {2\,B\,e\,\sqrt {d+e\,x}}{b^3}+\frac {3\,e\,\mathrm {atan}\left (\frac {\sqrt {b}\,e\,\sqrt {d+e\,x}\,\left (A\,b\,e-5\,B\,a\,e+4\,B\,b\,d\right )}{\sqrt {a\,e-b\,d}\,\left (A\,b\,e^2-5\,B\,a\,e^2+4\,B\,b\,d\,e\right )}\right )\,\left (A\,b\,e-5\,B\,a\,e+4\,B\,b\,d\right )}{4\,b^{7/2}\,\sqrt {a\,e-b\,d}} \]
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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