Optimal. Leaf size=359 \[ -\frac {5 e^2 \sqrt {d+e x} (-a B e-7 A b e+8 b B d)}{64 b \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^4}+\frac {5 e \sqrt {d+e x} (-a B e-7 A b e+8 b B d)}{96 b (a+b x) \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^3}-\frac {\sqrt {d+e x} (-a B e-7 A b e+8 b B d)}{24 b (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^2}-\frac {\sqrt {d+e x} (A b-a B)}{4 b (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)}+\frac {5 e^3 (a+b x) (-a B e-7 A b e+8 b B d) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{64 b^{3/2} \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^{9/2}} \]
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Rubi [A] time = 0.35, antiderivative size = 359, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {770, 78, 51, 63, 208} \begin {gather*} -\frac {5 e^2 \sqrt {d+e x} (-a B e-7 A b e+8 b B d)}{64 b \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^4}+\frac {5 e^3 (a+b x) (-a B e-7 A b e+8 b B d) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{64 b^{3/2} \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^{9/2}}+\frac {5 e \sqrt {d+e x} (-a B e-7 A b e+8 b B d)}{96 b (a+b x) \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^3}-\frac {\sqrt {d+e x} (-a B e-7 A b e+8 b B d)}{24 b (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^2}-\frac {\sqrt {d+e x} (A b-a B)}{4 b (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)} \end {gather*}
Antiderivative was successfully verified.
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Rule 51
Rule 63
Rule 78
Rule 208
Rule 770
Rubi steps
\begin {align*} \int \frac {A+B x}{\sqrt {d+e x} \left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx &=\frac {\left (b^4 \left (a b+b^2 x\right )\right ) \int \frac {A+B x}{\left (a b+b^2 x\right )^5 \sqrt {d+e x}} \, dx}{\sqrt {a^2+2 a b x+b^2 x^2}}\\ &=-\frac {(A b-a B) \sqrt {d+e x}}{4 b (b d-a e) (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {\left (b^2 (8 b B d-7 A b e-a B e) \left (a b+b^2 x\right )\right ) \int \frac {1}{\left (a b+b^2 x\right )^4 \sqrt {d+e x}} \, dx}{8 (b d-a e) \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=-\frac {(A b-a B) \sqrt {d+e x}}{4 b (b d-a e) (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {(8 b B d-7 A b e-a B e) \sqrt {d+e x}}{24 b (b d-a e)^2 (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {\left (5 b e (8 b B d-7 A b e-a B e) \left (a b+b^2 x\right )\right ) \int \frac {1}{\left (a b+b^2 x\right )^3 \sqrt {d+e x}} \, dx}{48 (b d-a e)^2 \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=-\frac {(A b-a B) \sqrt {d+e x}}{4 b (b d-a e) (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {(8 b B d-7 A b e-a B e) \sqrt {d+e x}}{24 b (b d-a e)^2 (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {5 e (8 b B d-7 A b e-a B e) \sqrt {d+e x}}{96 b (b d-a e)^3 (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {\left (5 e^2 (8 b B d-7 A b e-a B e) \left (a b+b^2 x\right )\right ) \int \frac {1}{\left (a b+b^2 x\right )^2 \sqrt {d+e x}} \, dx}{64 (b d-a e)^3 \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=-\frac {5 e^2 (8 b B d-7 A b e-a B e) \sqrt {d+e x}}{64 b (b d-a e)^4 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {(A b-a B) \sqrt {d+e x}}{4 b (b d-a e) (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {(8 b B d-7 A b e-a B e) \sqrt {d+e x}}{24 b (b d-a e)^2 (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {5 e (8 b B d-7 A b e-a B e) \sqrt {d+e x}}{96 b (b d-a e)^3 (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {\left (5 e^3 (8 b B d-7 A b e-a B e) \left (a b+b^2 x\right )\right ) \int \frac {1}{\left (a b+b^2 x\right ) \sqrt {d+e x}} \, dx}{128 b (b d-a e)^4 \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=-\frac {5 e^2 (8 b B d-7 A b e-a B e) \sqrt {d+e x}}{64 b (b d-a e)^4 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {(A b-a B) \sqrt {d+e x}}{4 b (b d-a e) (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {(8 b B d-7 A b e-a B e) \sqrt {d+e x}}{24 b (b d-a e)^2 (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {5 e (8 b B d-7 A b e-a B e) \sqrt {d+e x}}{96 b (b d-a e)^3 (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {\left (5 e^2 (8 b B d-7 A b e-a B e) \left (a b+b^2 x\right )\right ) \operatorname {Subst}\left (\int \frac {1}{a b-\frac {b^2 d}{e}+\frac {b^2 x^2}{e}} \, dx,x,\sqrt {d+e x}\right )}{64 b (b d-a e)^4 \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=-\frac {5 e^2 (8 b B d-7 A b e-a B e) \sqrt {d+e x}}{64 b (b d-a e)^4 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {(A b-a B) \sqrt {d+e x}}{4 b (b d-a e) (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {(8 b B d-7 A b e-a B e) \sqrt {d+e x}}{24 b (b d-a e)^2 (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {5 e (8 b B d-7 A b e-a B e) \sqrt {d+e x}}{96 b (b d-a e)^3 (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {5 e^3 (8 b B d-7 A b e-a B e) (a+b x) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{64 b^{3/2} (b d-a e)^{9/2} \sqrt {a^2+2 a b x+b^2 x^2}}\\ \end {align*}
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Mathematica [C] time = 0.08, size = 114, normalized size = 0.32 \begin {gather*} \frac {\sqrt {d+e x} \left (-\frac {e^3 (a+b x)^4 (a B e+7 A b e-8 b B d) \, _2F_1\left (\frac {1}{2},4;\frac {3}{2};\frac {b (d+e x)}{b d-a e}\right )}{(b d-a e)^4}+a B-A b\right )}{4 b (a+b x)^3 \sqrt {(a+b x)^2} (b d-a e)} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 59.75, size = 511, normalized size = 1.42 \begin {gather*} \frac {(-a e-b e x) \left (\frac {e^3 \sqrt {d+e x} \left (15 a^4 B e^4-279 a^3 A b e^4-73 a^3 b B e^3 (d+e x)+219 a^3 b B d e^3-511 a^2 A b^2 e^3 (d+e x)+837 a^2 A b^2 d e^3-747 a^2 b^2 B d^2 e^2-55 a^2 b^2 B e^2 (d+e x)^2+730 a^2 b^2 B d e^2 (d+e x)-837 a A b^3 d^2 e^2-385 a A b^3 e^2 (d+e x)^2+1022 a A b^3 d e^2 (d+e x)+777 a b^3 B d^3 e-1241 a b^3 B d^2 e (d+e x)-15 a b^3 B e (d+e x)^3+495 a b^3 B d e (d+e x)^2+279 A b^4 d^3 e-511 A b^4 d^2 e (d+e x)-105 A b^4 e (d+e x)^3+385 A b^4 d e (d+e x)^2-264 b^4 B d^4+584 b^4 B d^3 (d+e x)-440 b^4 B d^2 (d+e x)^2+120 b^4 B d (d+e x)^3\right )}{192 b (b d-a e)^4 (-a e-b (d+e x)+b d)^4}-\frac {5 \left (-a B e^4-7 A b e^4+8 b B d e^3\right ) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x} \sqrt {a e-b d}}{b d-a e}\right )}{64 b^{3/2} (b d-a e)^4 \sqrt {a e-b d}}\right )}{e \sqrt {\frac {(a e+b e x)^2}{e^2}}} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 0.49, size = 1980, normalized size = 5.52
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.41, size = 851, normalized size = 2.37 \begin {gather*} -\frac {5 \, {\left (8 \, B b d e^{3} - B a e^{4} - 7 \, A b e^{4}\right )} \arctan \left (\frac {\sqrt {x e + d} b}{\sqrt {-b^{2} d + a b e}}\right )}{64 \, {\left (b^{5} d^{4} \mathrm {sgn}\left ({\left (x e + d\right )} b e - b d e + a e^{2}\right ) - 4 \, a b^{4} d^{3} e \mathrm {sgn}\left ({\left (x e + d\right )} b e - b d e + a e^{2}\right ) + 6 \, a^{2} b^{3} d^{2} e^{2} \mathrm {sgn}\left ({\left (x e + d\right )} b e - b d e + a e^{2}\right ) - 4 \, a^{3} b^{2} d e^{3} \mathrm {sgn}\left ({\left (x e + d\right )} b e - b d e + a e^{2}\right ) + a^{4} b e^{4} \mathrm {sgn}\left ({\left (x e + d\right )} b e - b d e + a e^{2}\right )\right )} \sqrt {-b^{2} d + a b e}} - \frac {120 \, {\left (x e + d\right )}^{\frac {7}{2}} B b^{4} d e^{3} - 440 \, {\left (x e + d\right )}^{\frac {5}{2}} B b^{4} d^{2} e^{3} + 584 \, {\left (x e + d\right )}^{\frac {3}{2}} B b^{4} d^{3} e^{3} - 264 \, \sqrt {x e + d} B b^{4} d^{4} e^{3} - 15 \, {\left (x e + d\right )}^{\frac {7}{2}} B a b^{3} e^{4} - 105 \, {\left (x e + d\right )}^{\frac {7}{2}} A b^{4} e^{4} + 495 \, {\left (x e + d\right )}^{\frac {5}{2}} B a b^{3} d e^{4} + 385 \, {\left (x e + d\right )}^{\frac {5}{2}} A b^{4} d e^{4} - 1241 \, {\left (x e + d\right )}^{\frac {3}{2}} B a b^{3} d^{2} e^{4} - 511 \, {\left (x e + d\right )}^{\frac {3}{2}} A b^{4} d^{2} e^{4} + 777 \, \sqrt {x e + d} B a b^{3} d^{3} e^{4} + 279 \, \sqrt {x e + d} A b^{4} d^{3} e^{4} - 55 \, {\left (x e + d\right )}^{\frac {5}{2}} B a^{2} b^{2} e^{5} - 385 \, {\left (x e + d\right )}^{\frac {5}{2}} A a b^{3} e^{5} + 730 \, {\left (x e + d\right )}^{\frac {3}{2}} B a^{2} b^{2} d e^{5} + 1022 \, {\left (x e + d\right )}^{\frac {3}{2}} A a b^{3} d e^{5} - 747 \, \sqrt {x e + d} B a^{2} b^{2} d^{2} e^{5} - 837 \, \sqrt {x e + d} A a b^{3} d^{2} e^{5} - 73 \, {\left (x e + d\right )}^{\frac {3}{2}} B a^{3} b e^{6} - 511 \, {\left (x e + d\right )}^{\frac {3}{2}} A a^{2} b^{2} e^{6} + 219 \, \sqrt {x e + d} B a^{3} b d e^{6} + 837 \, \sqrt {x e + d} A a^{2} b^{2} d e^{6} + 15 \, \sqrt {x e + d} B a^{4} e^{7} - 279 \, \sqrt {x e + d} A a^{3} b e^{7}}{192 \, {\left (b^{5} d^{4} \mathrm {sgn}\left ({\left (x e + d\right )} b e - b d e + a e^{2}\right ) - 4 \, a b^{4} d^{3} e \mathrm {sgn}\left ({\left (x e + d\right )} b e - b d e + a e^{2}\right ) + 6 \, a^{2} b^{3} d^{2} e^{2} \mathrm {sgn}\left ({\left (x e + d\right )} b e - b d e + a e^{2}\right ) - 4 \, a^{3} b^{2} d e^{3} \mathrm {sgn}\left ({\left (x e + d\right )} b e - b d e + a e^{2}\right ) + a^{4} b e^{4} \mathrm {sgn}\left ({\left (x e + d\right )} b e - b d e + a e^{2}\right )\right )} {\left ({\left (x e + d\right )} b - b d + a e\right )}^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.08, size = 1296, normalized size = 3.61
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {B x + A}{{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} \sqrt {e x + d}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {A+B\,x}{\sqrt {d+e\,x}\,{\left (a^2+2\,a\,b\,x+b^2\,x^2\right )}^{5/2}} \,d x \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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