Optimal. Leaf size=233 \[ -\frac {5 e \sqrt {b d-a e} (-7 a B e+3 A b e+4 b B d) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{4 b^{9/2}}+\frac {5 e \sqrt {d+e x} (-7 a B e+3 A b e+4 b B d)}{4 b^4}+\frac {5 e (d+e x)^{3/2} (-7 a B e+3 A b e+4 b B d)}{12 b^3 (b d-a e)}-\frac {(d+e x)^{5/2} (-7 a B e+3 A b e+4 b B d)}{4 b^2 (a+b x) (b d-a e)}-\frac {(d+e x)^{7/2} (A b-a B)}{2 b (a+b x)^2 (b d-a e)} \]
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Rubi [A] time = 0.19, antiderivative size = 233, normalized size of antiderivative = 1.00, number of steps used = 6, 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} \[ -\frac {(d+e x)^{5/2} (-7 a B e+3 A b e+4 b B d)}{4 b^2 (a+b x) (b d-a e)}+\frac {5 e (d+e x)^{3/2} (-7 a B e+3 A b e+4 b B d)}{12 b^3 (b d-a e)}+\frac {5 e \sqrt {d+e x} (-7 a B e+3 A b e+4 b B d)}{4 b^4}-\frac {5 e \sqrt {b d-a e} (-7 a B e+3 A b e+4 b B d) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{4 b^{9/2}}-\frac {(d+e x)^{7/2} (A b-a B)}{2 b (a+b x)^2 (b d-a e)} \]
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)^{5/2}}{(a+b x)^3} \, dx &=-\frac {(A b-a B) (d+e x)^{7/2}}{2 b (b d-a e) (a+b x)^2}+\frac {(4 b B d+3 A b e-7 a B e) \int \frac {(d+e x)^{5/2}}{(a+b x)^2} \, dx}{4 b (b d-a e)}\\ &=-\frac {(4 b B d+3 A b e-7 a B e) (d+e x)^{5/2}}{4 b^2 (b d-a e) (a+b x)}-\frac {(A b-a B) (d+e x)^{7/2}}{2 b (b d-a e) (a+b x)^2}+\frac {(5 e (4 b B d+3 A b e-7 a B e)) \int \frac {(d+e x)^{3/2}}{a+b x} \, dx}{8 b^2 (b d-a e)}\\ &=\frac {5 e (4 b B d+3 A b e-7 a B e) (d+e x)^{3/2}}{12 b^3 (b d-a e)}-\frac {(4 b B d+3 A b e-7 a B e) (d+e x)^{5/2}}{4 b^2 (b d-a e) (a+b x)}-\frac {(A b-a B) (d+e x)^{7/2}}{2 b (b d-a e) (a+b x)^2}+\frac {(5 e (4 b B d+3 A b e-7 a B e)) \int \frac {\sqrt {d+e x}}{a+b x} \, dx}{8 b^3}\\ &=\frac {5 e (4 b B d+3 A b e-7 a B e) \sqrt {d+e x}}{4 b^4}+\frac {5 e (4 b B d+3 A b e-7 a B e) (d+e x)^{3/2}}{12 b^3 (b d-a e)}-\frac {(4 b B d+3 A b e-7 a B e) (d+e x)^{5/2}}{4 b^2 (b d-a e) (a+b x)}-\frac {(A b-a B) (d+e x)^{7/2}}{2 b (b d-a e) (a+b x)^2}+\frac {(5 e (b d-a e) (4 b B d+3 A b e-7 a B e)) \int \frac {1}{(a+b x) \sqrt {d+e x}} \, dx}{8 b^4}\\ &=\frac {5 e (4 b B d+3 A b e-7 a B e) \sqrt {d+e x}}{4 b^4}+\frac {5 e (4 b B d+3 A b e-7 a B e) (d+e x)^{3/2}}{12 b^3 (b d-a e)}-\frac {(4 b B d+3 A b e-7 a B e) (d+e x)^{5/2}}{4 b^2 (b d-a e) (a+b x)}-\frac {(A b-a B) (d+e x)^{7/2}}{2 b (b d-a e) (a+b x)^2}+\frac {(5 (b d-a e) (4 b B d+3 A b e-7 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^4}\\ &=\frac {5 e (4 b B d+3 A b e-7 a B e) \sqrt {d+e x}}{4 b^4}+\frac {5 e (4 b B d+3 A b e-7 a B e) (d+e x)^{3/2}}{12 b^3 (b d-a e)}-\frac {(4 b B d+3 A b e-7 a B e) (d+e x)^{5/2}}{4 b^2 (b d-a e) (a+b x)}-\frac {(A b-a B) (d+e x)^{7/2}}{2 b (b d-a e) (a+b x)^2}-\frac {5 e \sqrt {b d-a e} (4 b B d+3 A b e-7 a B e) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{4 b^{9/2}}\\ \end {align*}
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Mathematica [C] time = 0.10, size = 97, normalized size = 0.42 \[ \frac {(d+e x)^{7/2} \left (\frac {e (-7 a B e+3 A b e+4 b B d) \, _2F_1\left (2,\frac {7}{2};\frac {9}{2};\frac {b (d+e x)}{b d-a e}\right )}{(b d-a e)^2}+\frac {7 a B-7 A b}{(a+b x)^2}\right )}{14 b (b d-a e)} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.87, size = 680, normalized size = 2.92 \[ \left [-\frac {15 \, {\left (4 \, B a^{2} b d e - {\left (7 \, B a^{3} - 3 \, A a^{2} b\right )} e^{2} + {\left (4 \, B b^{3} d e - {\left (7 \, B a b^{2} - 3 \, A b^{3}\right )} e^{2}\right )} x^{2} + 2 \, {\left (4 \, B a b^{2} d e - {\left (7 \, B a^{2} b - 3 \, A a b^{2}\right )} e^{2}\right )} x\right )} \sqrt {\frac {b d - a e}{b}} \log \left (\frac {b e x + 2 \, b d - a e + 2 \, \sqrt {e x + d} b \sqrt {\frac {b d - a e}{b}}}{b x + a}\right ) - 2 \, {\left (8 \, B b^{3} e^{2} x^{3} - 6 \, {\left (B a b^{2} + A b^{3}\right )} d^{2} + 5 \, {\left (19 \, B a^{2} b - 3 \, A a b^{2}\right )} d e - 15 \, {\left (7 \, B a^{3} - 3 \, A a^{2} b\right )} e^{2} + 8 \, {\left (7 \, B b^{3} d e - {\left (7 \, B a b^{2} - 3 \, A b^{3}\right )} e^{2}\right )} x^{2} - {\left (12 \, B b^{3} d^{2} - {\left (163 \, B a b^{2} - 27 \, A b^{3}\right )} d e + 25 \, {\left (7 \, B a^{2} b - 3 \, A a b^{2}\right )} e^{2}\right )} x\right )} \sqrt {e x + d}}{24 \, {\left (b^{6} x^{2} + 2 \, a b^{5} x + a^{2} b^{4}\right )}}, -\frac {15 \, {\left (4 \, B a^{2} b d e - {\left (7 \, B a^{3} - 3 \, A a^{2} b\right )} e^{2} + {\left (4 \, B b^{3} d e - {\left (7 \, B a b^{2} - 3 \, A b^{3}\right )} e^{2}\right )} x^{2} + 2 \, {\left (4 \, B a b^{2} d e - {\left (7 \, B a^{2} b - 3 \, A a b^{2}\right )} e^{2}\right )} x\right )} \sqrt {-\frac {b d - a e}{b}} \arctan \left (-\frac {\sqrt {e x + d} b \sqrt {-\frac {b d - a e}{b}}}{b d - a e}\right ) - {\left (8 \, B b^{3} e^{2} x^{3} - 6 \, {\left (B a b^{2} + A b^{3}\right )} d^{2} + 5 \, {\left (19 \, B a^{2} b - 3 \, A a b^{2}\right )} d e - 15 \, {\left (7 \, B a^{3} - 3 \, A a^{2} b\right )} e^{2} + 8 \, {\left (7 \, B b^{3} d e - {\left (7 \, B a b^{2} - 3 \, A b^{3}\right )} e^{2}\right )} x^{2} - {\left (12 \, B b^{3} d^{2} - {\left (163 \, B a b^{2} - 27 \, A b^{3}\right )} d e + 25 \, {\left (7 \, B a^{2} b - 3 \, A a b^{2}\right )} e^{2}\right )} x\right )} \sqrt {e x + d}}{12 \, {\left (b^{6} x^{2} + 2 \, a b^{5} x + a^{2} b^{4}\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.28, size = 400, normalized size = 1.72 \[ \frac {5 \, {\left (4 \, B b^{2} d^{2} e - 11 \, B a b d e^{2} + 3 \, A b^{2} d e^{2} + 7 \, B a^{2} e^{3} - 3 \, A a b e^{3}\right )} \arctan \left (\frac {\sqrt {x e + d} b}{\sqrt {-b^{2} d + a b e}}\right )}{4 \, \sqrt {-b^{2} d + a b e} b^{4}} - \frac {4 \, {\left (x e + d\right )}^{\frac {3}{2}} B b^{3} d^{2} e - 4 \, \sqrt {x e + d} B b^{3} d^{3} e - 17 \, {\left (x e + d\right )}^{\frac {3}{2}} B a b^{2} d e^{2} + 9 \, {\left (x e + d\right )}^{\frac {3}{2}} A b^{3} d e^{2} + 19 \, \sqrt {x e + d} B a b^{2} d^{2} e^{2} - 7 \, \sqrt {x e + d} A b^{3} d^{2} e^{2} + 13 \, {\left (x e + d\right )}^{\frac {3}{2}} B a^{2} b e^{3} - 9 \, {\left (x e + d\right )}^{\frac {3}{2}} A a b^{2} e^{3} - 26 \, \sqrt {x e + d} B a^{2} b d e^{3} + 14 \, \sqrt {x e + d} A a b^{2} d e^{3} + 11 \, \sqrt {x e + d} B a^{3} e^{4} - 7 \, \sqrt {x e + d} A a^{2} b e^{4}}{4 \, {\left ({\left (x e + d\right )} b - b d + a e\right )}^{2} b^{4}} + \frac {2 \, {\left ({\left (x e + d\right )}^{\frac {3}{2}} B b^{6} e + 6 \, \sqrt {x e + d} B b^{6} d e - 9 \, \sqrt {x e + d} B a b^{5} e^{2} + 3 \, \sqrt {x e + d} A b^{6} e^{2}\right )}}{3 \, b^{9}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.02, size = 626, normalized size = 2.69 \[ \frac {7 \sqrt {e x +d}\, A \,a^{2} e^{4}}{4 \left (b x e +a e \right )^{2} b^{3}}-\frac {7 \sqrt {e x +d}\, A a d \,e^{3}}{2 \left (b x e +a e \right )^{2} b^{2}}+\frac {7 \sqrt {e x +d}\, A \,d^{2} e^{2}}{4 \left (b x e +a e \right )^{2} b}-\frac {11 \sqrt {e x +d}\, B \,a^{3} e^{4}}{4 \left (b x e +a e \right )^{2} b^{4}}+\frac {13 \sqrt {e x +d}\, B \,a^{2} d \,e^{3}}{2 \left (b x e +a e \right )^{2} b^{3}}-\frac {19 \sqrt {e x +d}\, B a \,d^{2} e^{2}}{4 \left (b x e +a e \right )^{2} b^{2}}+\frac {\sqrt {e x +d}\, B \,d^{3} e}{\left (b x e +a e \right )^{2} b}+\frac {9 \left (e x +d \right )^{\frac {3}{2}} A a \,e^{3}}{4 \left (b x e +a e \right )^{2} b^{2}}-\frac {15 A a \,e^{3} \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 {9 \left (e x +d \right )^{\frac {3}{2}} A d \,e^{2}}{4 \left (b x e +a e \right )^{2} b}+\frac {15 A d \,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 {13 \left (e x +d \right )^{\frac {3}{2}} B \,a^{2} e^{3}}{4 \left (b x e +a e \right )^{2} b^{3}}+\frac {35 B \,a^{2} e^{3} \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^{4}}+\frac {17 \left (e x +d \right )^{\frac {3}{2}} B a d \,e^{2}}{4 \left (b x e +a e \right )^{2} b^{2}}-\frac {55 B a d \,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^{2} e}{\left (b x e +a e \right )^{2} b}+\frac {5 B \,d^{2} 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}\, A \,e^{2}}{b^{3}}-\frac {6 \sqrt {e x +d}\, B a \,e^{2}}{b^{4}}+\frac {4 \sqrt {e x +d}\, B d e}{b^{3}}+\frac {2 \left (e x +d \right )^{\frac {3}{2}} B e}{3 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 = 1.35, size = 325, normalized size = 1.39 \[ \left (\frac {2\,A\,e^2-2\,B\,d\,e}{b^3}+\frac {2\,B\,e\,\left (3\,b^3\,d-3\,a\,b^2\,e\right )}{b^6}\right )\,\sqrt {d+e\,x}-\frac {{\left (d+e\,x\right )}^{3/2}\,\left (\frac {13\,B\,a^2\,b\,e^3}{4}-\frac {17\,B\,a\,b^2\,d\,e^2}{4}-\frac {9\,A\,a\,b^2\,e^3}{4}+B\,b^3\,d^2\,e+\frac {9\,A\,b^3\,d\,e^2}{4}\right )-\sqrt {d+e\,x}\,\left (-\frac {11\,B\,a^3\,e^4}{4}+\frac {13\,B\,a^2\,b\,d\,e^3}{2}+\frac {7\,A\,a^2\,b\,e^4}{4}-\frac {19\,B\,a\,b^2\,d^2\,e^2}{4}-\frac {7\,A\,a\,b^2\,d\,e^3}{2}+B\,b^3\,d^3\,e+\frac {7\,A\,b^3\,d^2\,e^2}{4}\right )}{b^6\,{\left (d+e\,x\right )}^2-\left (2\,b^6\,d-2\,a\,b^5\,e\right )\,\left (d+e\,x\right )+b^6\,d^2+a^2\,b^4\,e^2-2\,a\,b^5\,d\,e}+\frac {2\,B\,e\,{\left (d+e\,x\right )}^{3/2}}{3\,b^3}+\frac {e\,\mathrm {atan}\left (\frac {\sqrt {b}\,\sqrt {d+e\,x}\,1{}\mathrm {i}}{\sqrt {b\,d-a\,e}}\right )\,\sqrt {b\,d-a\,e}\,\left (3\,A\,b\,e-7\,B\,a\,e+4\,B\,b\,d\right )\,5{}\mathrm {i}}{4\,b^{9/2}} \]
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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