Optimal. Leaf size=274 \[ -\frac {7 e (b d-a e)^{3/2} (-9 a B e+5 A b e+4 b B d) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{4 b^{11/2}}+\frac {7 e \sqrt {d+e x} (b d-a e) (-9 a B e+5 A b e+4 b B d)}{4 b^5}+\frac {7 e (d+e x)^{3/2} (-9 a B e+5 A b e+4 b B d)}{12 b^4}+\frac {7 e (d+e x)^{5/2} (-9 a B e+5 A b e+4 b B d)}{20 b^3 (b d-a e)}-\frac {(d+e x)^{7/2} (-9 a B e+5 A b e+4 b B d)}{4 b^2 (a+b x) (b d-a e)}-\frac {(d+e x)^{9/2} (A b-a B)}{2 b (a+b x)^2 (b d-a e)} \]
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Rubi [A] time = 0.24, antiderivative size = 274, normalized size of antiderivative = 1.00, number of steps used = 7, 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} \begin {gather*} -\frac {(d+e x)^{7/2} (-9 a B e+5 A b e+4 b B d)}{4 b^2 (a+b x) (b d-a e)}+\frac {7 e (d+e x)^{5/2} (-9 a B e+5 A b e+4 b B d)}{20 b^3 (b d-a e)}+\frac {7 e (d+e x)^{3/2} (-9 a B e+5 A b e+4 b B d)}{12 b^4}+\frac {7 e \sqrt {d+e x} (b d-a e) (-9 a B e+5 A b e+4 b B d)}{4 b^5}-\frac {7 e (b d-a e)^{3/2} (-9 a B e+5 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^{11/2}}-\frac {(d+e x)^{9/2} (A b-a B)}{2 b (a+b x)^2 (b d-a e)} \end {gather*}
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)^{7/2}}{(a+b x)^3} \, dx &=-\frac {(A b-a B) (d+e x)^{9/2}}{2 b (b d-a e) (a+b x)^2}+\frac {(4 b B d+5 A b e-9 a B e) \int \frac {(d+e x)^{7/2}}{(a+b x)^2} \, dx}{4 b (b d-a e)}\\ &=-\frac {(4 b B d+5 A b e-9 a B e) (d+e x)^{7/2}}{4 b^2 (b d-a e) (a+b x)}-\frac {(A b-a B) (d+e x)^{9/2}}{2 b (b d-a e) (a+b x)^2}+\frac {(7 e (4 b B d+5 A b e-9 a B e)) \int \frac {(d+e x)^{5/2}}{a+b x} \, dx}{8 b^2 (b d-a e)}\\ &=\frac {7 e (4 b B d+5 A b e-9 a B e) (d+e x)^{5/2}}{20 b^3 (b d-a e)}-\frac {(4 b B d+5 A b e-9 a B e) (d+e x)^{7/2}}{4 b^2 (b d-a e) (a+b x)}-\frac {(A b-a B) (d+e x)^{9/2}}{2 b (b d-a e) (a+b x)^2}+\frac {(7 e (4 b B d+5 A b e-9 a B e)) \int \frac {(d+e x)^{3/2}}{a+b x} \, dx}{8 b^3}\\ &=\frac {7 e (4 b B d+5 A b e-9 a B e) (d+e x)^{3/2}}{12 b^4}+\frac {7 e (4 b B d+5 A b e-9 a B e) (d+e x)^{5/2}}{20 b^3 (b d-a e)}-\frac {(4 b B d+5 A b e-9 a B e) (d+e x)^{7/2}}{4 b^2 (b d-a e) (a+b x)}-\frac {(A b-a B) (d+e x)^{9/2}}{2 b (b d-a e) (a+b x)^2}+\frac {(7 e (b d-a e) (4 b B d+5 A b e-9 a B e)) \int \frac {\sqrt {d+e x}}{a+b x} \, dx}{8 b^4}\\ &=\frac {7 e (b d-a e) (4 b B d+5 A b e-9 a B e) \sqrt {d+e x}}{4 b^5}+\frac {7 e (4 b B d+5 A b e-9 a B e) (d+e x)^{3/2}}{12 b^4}+\frac {7 e (4 b B d+5 A b e-9 a B e) (d+e x)^{5/2}}{20 b^3 (b d-a e)}-\frac {(4 b B d+5 A b e-9 a B e) (d+e x)^{7/2}}{4 b^2 (b d-a e) (a+b x)}-\frac {(A b-a B) (d+e x)^{9/2}}{2 b (b d-a e) (a+b x)^2}+\frac {\left (7 e (b d-a e)^2 (4 b B d+5 A b e-9 a B e)\right ) \int \frac {1}{(a+b x) \sqrt {d+e x}} \, dx}{8 b^5}\\ &=\frac {7 e (b d-a e) (4 b B d+5 A b e-9 a B e) \sqrt {d+e x}}{4 b^5}+\frac {7 e (4 b B d+5 A b e-9 a B e) (d+e x)^{3/2}}{12 b^4}+\frac {7 e (4 b B d+5 A b e-9 a B e) (d+e x)^{5/2}}{20 b^3 (b d-a e)}-\frac {(4 b B d+5 A b e-9 a B e) (d+e x)^{7/2}}{4 b^2 (b d-a e) (a+b x)}-\frac {(A b-a B) (d+e x)^{9/2}}{2 b (b d-a e) (a+b x)^2}+\frac {\left (7 (b d-a e)^2 (4 b B d+5 A b e-9 a B e)\right ) \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^5}\\ &=\frac {7 e (b d-a e) (4 b B d+5 A b e-9 a B e) \sqrt {d+e x}}{4 b^5}+\frac {7 e (4 b B d+5 A b e-9 a B e) (d+e x)^{3/2}}{12 b^4}+\frac {7 e (4 b B d+5 A b e-9 a B e) (d+e x)^{5/2}}{20 b^3 (b d-a e)}-\frac {(4 b B d+5 A b e-9 a B e) (d+e x)^{7/2}}{4 b^2 (b d-a e) (a+b x)}-\frac {(A b-a B) (d+e x)^{9/2}}{2 b (b d-a e) (a+b x)^2}-\frac {7 e (b d-a e)^{3/2} (4 b B d+5 A b e-9 a B e) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{4 b^{11/2}}\\ \end {align*}
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Mathematica [C] time = 0.12, size = 97, normalized size = 0.35 \begin {gather*} \frac {(d+e x)^{9/2} \left (\frac {e (-9 a B e+5 A b e+4 b B d) \, _2F_1\left (2,\frac {9}{2};\frac {11}{2};\frac {b (d+e x)}{b d-a e}\right )}{(b d-a e)^2}+\frac {9 a B-9 A b}{(a+b x)^2}\right )}{18 b (b d-a e)} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 1.14, size = 477, normalized size = 1.74 \begin {gather*} \frac {e \sqrt {d+e x} \left (945 a^4 B e^4-525 a^3 A b e^4+1575 a^3 b B e^3 (d+e x)-3255 a^3 b B d e^3-875 a^2 A b^2 e^3 (d+e x)+1575 a^2 A b^2 d e^3+4095 a^2 b^2 B d^2 e^2+504 a^2 b^2 B e^2 (d+e x)^2-3850 a^2 b^2 B d e^2 (d+e x)-1575 a A b^3 d^2 e^2-280 a A b^3 e^2 (d+e x)^2+1750 a A b^3 d e^2 (d+e x)-2205 a b^3 B d^3 e+2975 a b^3 B d^2 e (d+e x)-72 a b^3 B e (d+e x)^3-728 a b^3 B d e (d+e x)^2+525 A b^4 d^3 e-875 A b^4 d^2 e (d+e x)+40 A b^4 e (d+e x)^3+280 A b^4 d e (d+e x)^2+420 b^4 B d^4-700 b^4 B d^3 (d+e x)+224 b^4 B d^2 (d+e x)^2+24 b^4 B (d+e x)^4+32 b^4 B d (d+e x)^3\right )}{60 b^5 (a e+b (d+e x)-b d)^2}-\frac {7 (b d-a e)^2 \left (-9 a B e^2+5 A b e^2+4 b B d e\right ) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x} \sqrt {a e-b d}}{b d-a e}\right )}{4 b^{11/2} \sqrt {a e-b d}} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 1.68, size = 1060, normalized size = 3.87
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 1.39, size = 607, normalized size = 2.22 \begin {gather*} \frac {7 \, {\left (4 \, B b^{3} d^{3} e - 17 \, B a b^{2} d^{2} e^{2} + 5 \, A b^{3} d^{2} e^{2} + 22 \, B a^{2} b d e^{3} - 10 \, A a b^{2} d e^{3} - 9 \, B a^{3} e^{4} + 5 \, A a^{2} b e^{4}\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^{5}} - \frac {4 \, {\left (x e + d\right )}^{\frac {3}{2}} B b^{4} d^{3} e - 4 \, \sqrt {x e + d} B b^{4} d^{4} e - 25 \, {\left (x e + d\right )}^{\frac {3}{2}} B a b^{3} d^{2} e^{2} + 13 \, {\left (x e + d\right )}^{\frac {3}{2}} A b^{4} d^{2} e^{2} + 27 \, \sqrt {x e + d} B a b^{3} d^{3} e^{2} - 11 \, \sqrt {x e + d} A b^{4} d^{3} e^{2} + 38 \, {\left (x e + d\right )}^{\frac {3}{2}} B a^{2} b^{2} d e^{3} - 26 \, {\left (x e + d\right )}^{\frac {3}{2}} A a b^{3} d e^{3} - 57 \, \sqrt {x e + d} B a^{2} b^{2} d^{2} e^{3} + 33 \, \sqrt {x e + d} A a b^{3} d^{2} e^{3} - 17 \, {\left (x e + d\right )}^{\frac {3}{2}} B a^{3} b e^{4} + 13 \, {\left (x e + d\right )}^{\frac {3}{2}} A a^{2} b^{2} e^{4} + 49 \, \sqrt {x e + d} B a^{3} b d e^{4} - 33 \, \sqrt {x e + d} A a^{2} b^{2} d e^{4} - 15 \, \sqrt {x e + d} B a^{4} e^{5} + 11 \, \sqrt {x e + d} A a^{3} b e^{5}}{4 \, {\left ({\left (x e + d\right )} b - b d + a e\right )}^{2} b^{5}} + \frac {2 \, {\left (3 \, {\left (x e + d\right )}^{\frac {5}{2}} B b^{12} e + 10 \, {\left (x e + d\right )}^{\frac {3}{2}} B b^{12} d e + 45 \, \sqrt {x e + d} B b^{12} d^{2} e - 15 \, {\left (x e + d\right )}^{\frac {3}{2}} B a b^{11} e^{2} + 5 \, {\left (x e + d\right )}^{\frac {3}{2}} A b^{12} e^{2} - 135 \, \sqrt {x e + d} B a b^{11} d e^{2} + 45 \, \sqrt {x e + d} A b^{12} d e^{2} + 90 \, \sqrt {x e + d} B a^{2} b^{10} e^{3} - 45 \, \sqrt {x e + d} A a b^{11} e^{3}\right )}}{15 \, b^{15}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.03, size = 940, normalized size = 3.43 \begin {gather*} -\frac {11 \sqrt {e x +d}\, A \,a^{3} e^{5}}{4 \left (b x e +a e \right )^{2} b^{4}}+\frac {33 \sqrt {e x +d}\, A \,a^{2} d \,e^{4}}{4 \left (b x e +a e \right )^{2} b^{3}}-\frac {33 \sqrt {e x +d}\, A a \,d^{2} e^{3}}{4 \left (b x e +a e \right )^{2} b^{2}}+\frac {11 \sqrt {e x +d}\, A \,d^{3} e^{2}}{4 \left (b x e +a e \right )^{2} b}+\frac {15 \sqrt {e x +d}\, B \,a^{4} e^{5}}{4 \left (b x e +a e \right )^{2} b^{5}}-\frac {49 \sqrt {e x +d}\, B \,a^{3} d \,e^{4}}{4 \left (b x e +a e \right )^{2} b^{4}}+\frac {57 \sqrt {e x +d}\, B \,a^{2} d^{2} e^{3}}{4 \left (b x e +a e \right )^{2} b^{3}}-\frac {27 \sqrt {e x +d}\, B a \,d^{3} e^{2}}{4 \left (b x e +a e \right )^{2} b^{2}}+\frac {\sqrt {e x +d}\, B \,d^{4} e}{\left (b x e +a e \right )^{2} b}-\frac {13 \left (e x +d \right )^{\frac {3}{2}} A \,a^{2} e^{4}}{4 \left (b x e +a e \right )^{2} b^{3}}+\frac {35 A \,a^{2} e^{4} \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 {13 \left (e x +d \right )^{\frac {3}{2}} A a d \,e^{3}}{2 \left (b x e +a e \right )^{2} b^{2}}-\frac {35 A a d \,e^{3} \arctan \left (\frac {\sqrt {e x +d}\, b}{\sqrt {\left (a e -b d \right ) b}}\right )}{2 \sqrt {\left (a e -b d \right ) b}\, b^{3}}-\frac {13 \left (e x +d \right )^{\frac {3}{2}} A \,d^{2} e^{2}}{4 \left (b x e +a e \right )^{2} b}+\frac {35 A \,d^{2} 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 {17 \left (e x +d \right )^{\frac {3}{2}} B \,a^{3} e^{4}}{4 \left (b x e +a e \right )^{2} b^{4}}-\frac {63 B \,a^{3} e^{4} \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^{5}}-\frac {19 \left (e x +d \right )^{\frac {3}{2}} B \,a^{2} d \,e^{3}}{2 \left (b x e +a e \right )^{2} b^{3}}+\frac {77 B \,a^{2} d \,e^{3} \arctan \left (\frac {\sqrt {e x +d}\, b}{\sqrt {\left (a e -b d \right ) b}}\right )}{2 \sqrt {\left (a e -b d \right ) b}\, b^{4}}+\frac {25 \left (e x +d \right )^{\frac {3}{2}} B a \,d^{2} e^{2}}{4 \left (b x e +a e \right )^{2} b^{2}}-\frac {119 B a \,d^{2} 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^{3} e}{\left (b x e +a e \right )^{2} b}+\frac {7 B \,d^{3} 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 {6 \sqrt {e x +d}\, A a \,e^{3}}{b^{4}}+\frac {6 \sqrt {e x +d}\, A d \,e^{2}}{b^{3}}+\frac {12 \sqrt {e x +d}\, B \,a^{2} e^{3}}{b^{5}}-\frac {18 \sqrt {e x +d}\, B a d \,e^{2}}{b^{4}}+\frac {6 \sqrt {e x +d}\, B \,d^{2} e}{b^{3}}+\frac {2 \left (e x +d \right )^{\frac {3}{2}} A \,e^{2}}{3 b^{3}}-\frac {2 \left (e x +d \right )^{\frac {3}{2}} B a \,e^{2}}{b^{4}}+\frac {4 \left (e x +d \right )^{\frac {3}{2}} B d e}{3 b^{3}}+\frac {2 \left (e x +d \right )^{\frac {5}{2}} B e}{5 b^{3}} \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 = 1.33, size = 561, normalized size = 2.05 \begin {gather*} \left (\frac {2\,A\,e^2-2\,B\,d\,e}{3\,b^3}+\frac {2\,B\,e\,\left (3\,b^3\,d-3\,a\,b^2\,e\right )}{3\,b^6}\right )\,{\left (d+e\,x\right )}^{3/2}+\left (\frac {\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 )\,\left (3\,b^3\,d-3\,a\,b^2\,e\right )}{b^3}-\frac {6\,B\,e\,{\left (a\,e-b\,d\right )}^2}{b^5}\right )\,\sqrt {d+e\,x}-\frac {{\left (d+e\,x\right )}^{3/2}\,\left (-\frac {17\,B\,a^3\,b\,e^4}{4}+\frac {19\,B\,a^2\,b^2\,d\,e^3}{2}+\frac {13\,A\,a^2\,b^2\,e^4}{4}-\frac {25\,B\,a\,b^3\,d^2\,e^2}{4}-\frac {13\,A\,a\,b^3\,d\,e^3}{2}+B\,b^4\,d^3\,e+\frac {13\,A\,b^4\,d^2\,e^2}{4}\right )-\sqrt {d+e\,x}\,\left (\frac {15\,B\,a^4\,e^5}{4}-\frac {49\,B\,a^3\,b\,d\,e^4}{4}-\frac {11\,A\,a^3\,b\,e^5}{4}+\frac {57\,B\,a^2\,b^2\,d^2\,e^3}{4}+\frac {33\,A\,a^2\,b^2\,d\,e^4}{4}-\frac {27\,B\,a\,b^3\,d^3\,e^2}{4}-\frac {33\,A\,a\,b^3\,d^2\,e^3}{4}+B\,b^4\,d^4\,e+\frac {11\,A\,b^4\,d^3\,e^2}{4}\right )}{b^7\,{\left (d+e\,x\right )}^2-\left (2\,b^7\,d-2\,a\,b^6\,e\right )\,\left (d+e\,x\right )+b^7\,d^2+a^2\,b^5\,e^2-2\,a\,b^6\,d\,e}+\frac {2\,B\,e\,{\left (d+e\,x\right )}^{5/2}}{5\,b^3}+\frac {7\,e\,\mathrm {atan}\left (\frac {\sqrt {b}\,e\,{\left (a\,e-b\,d\right )}^{3/2}\,\sqrt {d+e\,x}\,\left (5\,A\,b\,e-9\,B\,a\,e+4\,B\,b\,d\right )}{-9\,B\,a^3\,e^4+22\,B\,a^2\,b\,d\,e^3+5\,A\,a^2\,b\,e^4-17\,B\,a\,b^2\,d^2\,e^2-10\,A\,a\,b^2\,d\,e^3+4\,B\,b^3\,d^3\,e+5\,A\,b^3\,d^2\,e^2}\right )\,{\left (a\,e-b\,d\right )}^{3/2}\,\left (5\,A\,b\,e-9\,B\,a\,e+4\,B\,b\,d\right )}{4\,b^{11/2}} \end {gather*}
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 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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