Optimal. Leaf size=313 \[ -\frac {3 e^4 (-a B e-A b e+2 b B d) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{128 b^{7/2} (b d-a e)^{7/2}}+\frac {3 e^3 \sqrt {d+e x} (-a B e-A b e+2 b B d)}{128 b^3 (a+b x) (b d-a e)^3}-\frac {e^2 \sqrt {d+e x} (-a B e-A b e+2 b B d)}{64 b^3 (a+b x)^2 (b d-a e)^2}-\frac {e \sqrt {d+e x} (-a B e-A b e+2 b B d)}{16 b^3 (a+b x)^3 (b d-a e)}-\frac {(d+e x)^{3/2} (-a B e-A b e+2 b B d)}{8 b^2 (a+b x)^4 (b d-a e)}-\frac {(d+e x)^{5/2} (A b-a B)}{5 b (a+b x)^5 (b d-a e)} \]
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Rubi [A] time = 0.27, antiderivative size = 313, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {27, 78, 47, 51, 63, 208} \begin {gather*} \frac {3 e^3 \sqrt {d+e x} (-a B e-A b e+2 b B d)}{128 b^3 (a+b x) (b d-a e)^3}-\frac {e^2 \sqrt {d+e x} (-a B e-A b e+2 b B d)}{64 b^3 (a+b x)^2 (b d-a e)^2}-\frac {3 e^4 (-a B e-A b e+2 b B d) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{128 b^{7/2} (b d-a e)^{7/2}}-\frac {e \sqrt {d+e x} (-a B e-A b e+2 b B d)}{16 b^3 (a+b x)^3 (b d-a e)}-\frac {(d+e x)^{3/2} (-a B e-A b e+2 b B d)}{8 b^2 (a+b x)^4 (b d-a e)}-\frac {(d+e x)^{5/2} (A b-a B)}{5 b (a+b x)^5 (b d-a e)} \end {gather*}
Antiderivative was successfully verified.
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Rule 27
Rule 47
Rule 51
Rule 63
Rule 78
Rule 208
Rubi steps
\begin {align*} \int \frac {(A+B x) (d+e x)^{3/2}}{\left (a^2+2 a b x+b^2 x^2\right )^3} \, dx &=\int \frac {(A+B x) (d+e x)^{3/2}}{(a+b x)^6} \, dx\\ &=-\frac {(A b-a B) (d+e x)^{5/2}}{5 b (b d-a e) (a+b x)^5}+\frac {(2 b B d-A b e-a B e) \int \frac {(d+e x)^{3/2}}{(a+b x)^5} \, dx}{2 b (b d-a e)}\\ &=-\frac {(2 b B d-A b e-a B e) (d+e x)^{3/2}}{8 b^2 (b d-a e) (a+b x)^4}-\frac {(A b-a B) (d+e x)^{5/2}}{5 b (b d-a e) (a+b x)^5}+\frac {(3 e (2 b B d-A b e-a B e)) \int \frac {\sqrt {d+e x}}{(a+b x)^4} \, dx}{16 b^2 (b d-a e)}\\ &=-\frac {e (2 b B d-A b e-a B e) \sqrt {d+e x}}{16 b^3 (b d-a e) (a+b x)^3}-\frac {(2 b B d-A b e-a B e) (d+e x)^{3/2}}{8 b^2 (b d-a e) (a+b x)^4}-\frac {(A b-a B) (d+e x)^{5/2}}{5 b (b d-a e) (a+b x)^5}+\frac {\left (e^2 (2 b B d-A b e-a B e)\right ) \int \frac {1}{(a+b x)^3 \sqrt {d+e x}} \, dx}{32 b^3 (b d-a e)}\\ &=-\frac {e (2 b B d-A b e-a B e) \sqrt {d+e x}}{16 b^3 (b d-a e) (a+b x)^3}-\frac {e^2 (2 b B d-A b e-a B e) \sqrt {d+e x}}{64 b^3 (b d-a e)^2 (a+b x)^2}-\frac {(2 b B d-A b e-a B e) (d+e x)^{3/2}}{8 b^2 (b d-a e) (a+b x)^4}-\frac {(A b-a B) (d+e x)^{5/2}}{5 b (b d-a e) (a+b x)^5}-\frac {\left (3 e^3 (2 b B d-A b e-a B e)\right ) \int \frac {1}{(a+b x)^2 \sqrt {d+e x}} \, dx}{128 b^3 (b d-a e)^2}\\ &=-\frac {e (2 b B d-A b e-a B e) \sqrt {d+e x}}{16 b^3 (b d-a e) (a+b x)^3}-\frac {e^2 (2 b B d-A b e-a B e) \sqrt {d+e x}}{64 b^3 (b d-a e)^2 (a+b x)^2}+\frac {3 e^3 (2 b B d-A b e-a B e) \sqrt {d+e x}}{128 b^3 (b d-a e)^3 (a+b x)}-\frac {(2 b B d-A b e-a B e) (d+e x)^{3/2}}{8 b^2 (b d-a e) (a+b x)^4}-\frac {(A b-a B) (d+e x)^{5/2}}{5 b (b d-a e) (a+b x)^5}+\frac {\left (3 e^4 (2 b B d-A b e-a B e)\right ) \int \frac {1}{(a+b x) \sqrt {d+e x}} \, dx}{256 b^3 (b d-a e)^3}\\ &=-\frac {e (2 b B d-A b e-a B e) \sqrt {d+e x}}{16 b^3 (b d-a e) (a+b x)^3}-\frac {e^2 (2 b B d-A b e-a B e) \sqrt {d+e x}}{64 b^3 (b d-a e)^2 (a+b x)^2}+\frac {3 e^3 (2 b B d-A b e-a B e) \sqrt {d+e x}}{128 b^3 (b d-a e)^3 (a+b x)}-\frac {(2 b B d-A b e-a B e) (d+e x)^{3/2}}{8 b^2 (b d-a e) (a+b x)^4}-\frac {(A b-a B) (d+e x)^{5/2}}{5 b (b d-a e) (a+b x)^5}+\frac {\left (3 e^3 (2 b B d-A b e-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 )}{128 b^3 (b d-a e)^3}\\ &=-\frac {e (2 b B d-A b e-a B e) \sqrt {d+e x}}{16 b^3 (b d-a e) (a+b x)^3}-\frac {e^2 (2 b B d-A b e-a B e) \sqrt {d+e x}}{64 b^3 (b d-a e)^2 (a+b x)^2}+\frac {3 e^3 (2 b B d-A b e-a B e) \sqrt {d+e x}}{128 b^3 (b d-a e)^3 (a+b x)}-\frac {(2 b B d-A b e-a B e) (d+e x)^{3/2}}{8 b^2 (b d-a e) (a+b x)^4}-\frac {(A b-a B) (d+e x)^{5/2}}{5 b (b d-a e) (a+b x)^5}-\frac {3 e^4 (2 b B d-A b e-a B e) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{128 b^{7/2} (b d-a e)^{7/2}}\\ \end {align*}
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Mathematica [C] time = 0.06, size = 98, normalized size = 0.31 \begin {gather*} \frac {(d+e x)^{5/2} \left (\frac {5 e^4 (a B e+A b e-2 b B d) \, _2F_1\left (\frac {5}{2},5;\frac {7}{2};\frac {b (d+e x)}{b d-a e}\right )}{(b d-a e)^5}+\frac {5 a B-5 A b}{(a+b x)^5}\right )}{25 b (b d-a e)} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [B] time = 3.19, size = 660, normalized size = 2.11 \begin {gather*} -\frac {e^4 \sqrt {d+e x} \left (15 a^5 B e^5+15 a^4 A b e^5+70 a^4 b B e^4 (d+e x)-90 a^4 b B d e^4+70 a^3 A b^2 e^4 (d+e x)-60 a^3 A b^2 d e^4+210 a^3 b^2 B d^2 e^3+128 a^3 b^2 B e^3 (d+e x)^2-350 a^3 b^2 B d e^3 (d+e x)+90 a^2 A b^3 d^2 e^3-128 a^2 A b^3 e^3 (d+e x)^2-210 a^2 A b^3 d e^3 (d+e x)-240 a^2 b^3 B d^3 e^2+630 a^2 b^3 B d^2 e^2 (d+e x)-70 a^2 b^3 B e^2 (d+e x)^3-256 a^2 b^3 B d e^2 (d+e x)^2-60 a A b^4 d^3 e^2+210 a A b^4 d^2 e^2 (d+e x)-70 a A b^4 e^2 (d+e x)^3+256 a A b^4 d e^2 (d+e x)^2+135 a b^4 B d^4 e-490 a b^4 B d^3 e (d+e x)+128 a b^4 B d^2 e (d+e x)^2-15 a b^4 B e (d+e x)^4+210 a b^4 B d e (d+e x)^3+15 A b^5 d^4 e-70 A b^5 d^3 e (d+e x)-128 A b^5 d^2 e (d+e x)^2-15 A b^5 e (d+e x)^4+70 A b^5 d e (d+e x)^3-30 b^5 B d^5+140 b^5 B d^4 (d+e x)-140 b^5 B d^2 (d+e x)^3+30 b^5 B d (d+e x)^4\right )}{640 b^3 (b d-a e)^3 (-a e-b (d+e x)+b d)^5}-\frac {3 \left (-a B e^5-A b e^5+2 b B d e^4\right ) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x} \sqrt {a e-b d}}{b d-a e}\right )}{128 b^{7/2} (b d-a e)^3 \sqrt {a e-b d}} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 0.51, size = 2349, normalized size = 7.50
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.27, size = 814, normalized size = 2.60 \begin {gather*} \frac {3 \, {\left (2 \, B b d e^{4} - B a e^{5} - A b e^{5}\right )} \arctan \left (\frac {\sqrt {x e + d} b}{\sqrt {-b^{2} d + a b e}}\right )}{128 \, {\left (b^{6} d^{3} - 3 \, a b^{5} d^{2} e + 3 \, a^{2} b^{4} d e^{2} - a^{3} b^{3} e^{3}\right )} \sqrt {-b^{2} d + a b e}} + \frac {30 \, {\left (x e + d\right )}^{\frac {9}{2}} B b^{5} d e^{4} - 140 \, {\left (x e + d\right )}^{\frac {7}{2}} B b^{5} d^{2} e^{4} + 140 \, {\left (x e + d\right )}^{\frac {3}{2}} B b^{5} d^{4} e^{4} - 30 \, \sqrt {x e + d} B b^{5} d^{5} e^{4} - 15 \, {\left (x e + d\right )}^{\frac {9}{2}} B a b^{4} e^{5} - 15 \, {\left (x e + d\right )}^{\frac {9}{2}} A b^{5} e^{5} + 210 \, {\left (x e + d\right )}^{\frac {7}{2}} B a b^{4} d e^{5} + 70 \, {\left (x e + d\right )}^{\frac {7}{2}} A b^{5} d e^{5} + 128 \, {\left (x e + d\right )}^{\frac {5}{2}} B a b^{4} d^{2} e^{5} - 128 \, {\left (x e + d\right )}^{\frac {5}{2}} A b^{5} d^{2} e^{5} - 490 \, {\left (x e + d\right )}^{\frac {3}{2}} B a b^{4} d^{3} e^{5} - 70 \, {\left (x e + d\right )}^{\frac {3}{2}} A b^{5} d^{3} e^{5} + 135 \, \sqrt {x e + d} B a b^{4} d^{4} e^{5} + 15 \, \sqrt {x e + d} A b^{5} d^{4} e^{5} - 70 \, {\left (x e + d\right )}^{\frac {7}{2}} B a^{2} b^{3} e^{6} - 70 \, {\left (x e + d\right )}^{\frac {7}{2}} A a b^{4} e^{6} - 256 \, {\left (x e + d\right )}^{\frac {5}{2}} B a^{2} b^{3} d e^{6} + 256 \, {\left (x e + d\right )}^{\frac {5}{2}} A a b^{4} d e^{6} + 630 \, {\left (x e + d\right )}^{\frac {3}{2}} B a^{2} b^{3} d^{2} e^{6} + 210 \, {\left (x e + d\right )}^{\frac {3}{2}} A a b^{4} d^{2} e^{6} - 240 \, \sqrt {x e + d} B a^{2} b^{3} d^{3} e^{6} - 60 \, \sqrt {x e + d} A a b^{4} d^{3} e^{6} + 128 \, {\left (x e + d\right )}^{\frac {5}{2}} B a^{3} b^{2} e^{7} - 128 \, {\left (x e + d\right )}^{\frac {5}{2}} A a^{2} b^{3} e^{7} - 350 \, {\left (x e + d\right )}^{\frac {3}{2}} B a^{3} b^{2} d e^{7} - 210 \, {\left (x e + d\right )}^{\frac {3}{2}} A a^{2} b^{3} d e^{7} + 210 \, \sqrt {x e + d} B a^{3} b^{2} d^{2} e^{7} + 90 \, \sqrt {x e + d} A a^{2} b^{3} d^{2} e^{7} + 70 \, {\left (x e + d\right )}^{\frac {3}{2}} B a^{4} b e^{8} + 70 \, {\left (x e + d\right )}^{\frac {3}{2}} A a^{3} b^{2} e^{8} - 90 \, \sqrt {x e + d} B a^{4} b d e^{8} - 60 \, \sqrt {x e + d} A a^{3} b^{2} d e^{8} + 15 \, \sqrt {x e + d} B a^{5} e^{9} + 15 \, \sqrt {x e + d} A a^{4} b e^{9}}{640 \, {\left (b^{6} d^{3} - 3 \, a b^{5} d^{2} e + 3 \, a^{2} b^{4} d e^{2} - a^{3} b^{3} e^{3}\right )} {\left ({\left (x e + d\right )} b - b d + a e\right )}^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.09, size = 871, normalized size = 2.78 \begin {gather*} \frac {3 \left (e x +d \right )^{\frac {9}{2}} A \,b^{2} e^{5}}{128 \left (b e x +a e \right )^{5} \left (a^{3} e^{3}-3 a^{2} b d \,e^{2}+3 a \,b^{2} d^{2} e -b^{3} d^{3}\right )}+\frac {3 \left (e x +d \right )^{\frac {9}{2}} B a b \,e^{5}}{128 \left (b e x +a e \right )^{5} \left (a^{3} e^{3}-3 a^{2} b d \,e^{2}+3 a \,b^{2} d^{2} e -b^{3} d^{3}\right )}-\frac {3 \left (e x +d \right )^{\frac {9}{2}} B \,b^{2} d \,e^{4}}{64 \left (b e x +a e \right )^{5} \left (a^{3} e^{3}-3 a^{2} b d \,e^{2}+3 a \,b^{2} d^{2} e -b^{3} d^{3}\right )}+\frac {7 \left (e x +d \right )^{\frac {7}{2}} A b \,e^{5}}{64 \left (b e x +a e \right )^{5} \left (a^{2} e^{2}-2 a b d e +b^{2} d^{2}\right )}+\frac {7 \left (e x +d \right )^{\frac {7}{2}} B a \,e^{5}}{64 \left (b e x +a e \right )^{5} \left (a^{2} e^{2}-2 a b d e +b^{2} d^{2}\right )}-\frac {7 \left (e x +d \right )^{\frac {7}{2}} B b d \,e^{4}}{32 \left (b e x +a e \right )^{5} \left (a^{2} e^{2}-2 a b d e +b^{2} d^{2}\right )}-\frac {3 \sqrt {e x +d}\, A a \,e^{6}}{128 \left (b e x +a e \right )^{5} b^{2}}+\frac {3 \sqrt {e x +d}\, A d \,e^{5}}{128 \left (b e x +a e \right )^{5} b}+\frac {\left (e x +d \right )^{\frac {5}{2}} A \,e^{5}}{5 \left (b e x +a e \right )^{5} \left (a e -b d \right )}-\frac {3 \sqrt {e x +d}\, B \,a^{2} e^{6}}{128 \left (b e x +a e \right )^{5} b^{3}}-\frac {\left (e x +d \right )^{\frac {5}{2}} B a \,e^{5}}{5 \left (b e x +a e \right )^{5} \left (a e -b d \right ) b}+\frac {9 \sqrt {e x +d}\, B a d \,e^{5}}{128 \left (b e x +a e \right )^{5} b^{2}}-\frac {3 \sqrt {e x +d}\, B \,d^{2} e^{4}}{64 \left (b e x +a e \right )^{5} b}-\frac {7 \left (e x +d \right )^{\frac {3}{2}} A \,e^{5}}{64 \left (b e x +a e \right )^{5} b}+\frac {3 A \,e^{5} \arctan \left (\frac {\sqrt {e x +d}\, b}{\sqrt {\left (a e -b d \right ) b}}\right )}{128 \left (a^{3} e^{3}-3 a^{2} b d \,e^{2}+3 a \,b^{2} d^{2} e -b^{3} d^{3}\right ) \sqrt {\left (a e -b d \right ) b}\, b^{2}}-\frac {7 \left (e x +d \right )^{\frac {3}{2}} B a \,e^{5}}{64 \left (b e x +a e \right )^{5} b^{2}}+\frac {3 B a \,e^{5} \arctan \left (\frac {\sqrt {e x +d}\, b}{\sqrt {\left (a e -b d \right ) b}}\right )}{128 \left (a^{3} e^{3}-3 a^{2} b d \,e^{2}+3 a \,b^{2} d^{2} e -b^{3} d^{3}\right ) \sqrt {\left (a e -b d \right ) b}\, b^{3}}+\frac {7 \left (e x +d \right )^{\frac {3}{2}} B d \,e^{4}}{32 \left (b e x +a e \right )^{5} b}-\frac {3 B d \,e^{4} \arctan \left (\frac {\sqrt {e x +d}\, b}{\sqrt {\left (a e -b d \right ) b}}\right )}{64 \left (a^{3} e^{3}-3 a^{2} b d \,e^{2}+3 a \,b^{2} d^{2} e -b^{3} d^{3}\right ) \sqrt {\left (a e -b d \right ) b}\, b^{2}} \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 = 2.23, size = 535, normalized size = 1.71 \begin {gather*} \frac {\frac {7\,{\left (d+e\,x\right )}^{7/2}\,\left (A\,b\,e^5+B\,a\,e^5-2\,B\,b\,d\,e^4\right )}{64\,{\left (a\,e-b\,d\right )}^2}-\frac {7\,{\left (d+e\,x\right )}^{3/2}\,\left (A\,b\,e^5+B\,a\,e^5-2\,B\,b\,d\,e^4\right )}{64\,b^2}-\frac {3\,\left (a\,e-b\,d\right )\,\sqrt {d+e\,x}\,\left (A\,b\,e^5+B\,a\,e^5-2\,B\,b\,d\,e^4\right )}{128\,b^3}+\frac {3\,b\,{\left (d+e\,x\right )}^{9/2}\,\left (A\,b\,e^5+B\,a\,e^5-2\,B\,b\,d\,e^4\right )}{128\,{\left (a\,e-b\,d\right )}^3}+\frac {\left (A\,b\,e^5-B\,a\,e^5\right )\,{\left (d+e\,x\right )}^{5/2}}{5\,b\,\left (a\,e-b\,d\right )}}{\left (d+e\,x\right )\,\left (5\,a^4\,b\,e^4-20\,a^3\,b^2\,d\,e^3+30\,a^2\,b^3\,d^2\,e^2-20\,a\,b^4\,d^3\,e+5\,b^5\,d^4\right )-{\left (d+e\,x\right )}^2\,\left (-10\,a^3\,b^2\,e^3+30\,a^2\,b^3\,d\,e^2-30\,a\,b^4\,d^2\,e+10\,b^5\,d^3\right )+b^5\,{\left (d+e\,x\right )}^5-\left (5\,b^5\,d-5\,a\,b^4\,e\right )\,{\left (d+e\,x\right )}^4+a^5\,e^5-b^5\,d^5+{\left (d+e\,x\right )}^3\,\left (10\,a^2\,b^3\,e^2-20\,a\,b^4\,d\,e+10\,b^5\,d^2\right )-10\,a^2\,b^3\,d^3\,e^2+10\,a^3\,b^2\,d^2\,e^3+5\,a\,b^4\,d^4\,e-5\,a^4\,b\,d\,e^4}+\frac {3\,e^4\,\mathrm {atan}\left (\frac {\sqrt {b}\,e^4\,\sqrt {d+e\,x}\,\left (A\,b\,e+B\,a\,e-2\,B\,b\,d\right )}{\sqrt {a\,e-b\,d}\,\left (A\,b\,e^5+B\,a\,e^5-2\,B\,b\,d\,e^4\right )}\right )\,\left (A\,b\,e+B\,a\,e-2\,B\,b\,d\right )}{128\,b^{7/2}\,{\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 [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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