Optimal. Leaf size=289 \[ \frac {2 (a+b x) (d+e x)^{5/2} (A b-a B)}{5 b^2 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {2 (a+b x) (A b-a B) (b d-a e)^{5/2} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{b^{9/2} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {2 (a+b x) \sqrt {d+e x} (A b-a B) (b d-a e)^2}{b^4 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {2 (a+b x) (d+e x)^{3/2} (A b-a B) (b d-a e)}{3 b^3 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {2 B (a+b x) (d+e x)^{7/2}}{7 b e \sqrt {a^2+2 a b x+b^2 x^2}} \]
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Rubi [A] time = 0.22, antiderivative size = 289, 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, 80, 50, 63, 208} \begin {gather*} \frac {2 (a+b x) (d+e x)^{5/2} (A b-a B)}{5 b^2 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {2 (a+b x) (d+e x)^{3/2} (A b-a B) (b d-a e)}{3 b^3 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {2 (a+b x) \sqrt {d+e x} (A b-a B) (b d-a e)^2}{b^4 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {2 (a+b x) (A b-a B) (b d-a e)^{5/2} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{b^{9/2} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {2 B (a+b x) (d+e x)^{7/2}}{7 b e \sqrt {a^2+2 a b x+b^2 x^2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 50
Rule 63
Rule 80
Rule 208
Rule 770
Rubi steps
\begin {align*} \int \frac {(A+B x) (d+e x)^{5/2}}{\sqrt {a^2+2 a b x+b^2 x^2}} \, dx &=\frac {\left (a b+b^2 x\right ) \int \frac {(A+B x) (d+e x)^{5/2}}{a b+b^2 x} \, dx}{\sqrt {a^2+2 a b x+b^2 x^2}}\\ &=\frac {2 B (a+b x) (d+e x)^{7/2}}{7 b e \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {\left (2 \left (\frac {7}{2} A b^2 e-\frac {7}{2} a b B e\right ) \left (a b+b^2 x\right )\right ) \int \frac {(d+e x)^{5/2}}{a b+b^2 x} \, dx}{7 b^2 e \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=\frac {2 (A b-a B) (a+b x) (d+e x)^{5/2}}{5 b^2 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {2 B (a+b x) (d+e x)^{7/2}}{7 b e \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {\left (2 \left (b^2 d-a b e\right ) \left (\frac {7}{2} A b^2 e-\frac {7}{2} a b B e\right ) \left (a b+b^2 x\right )\right ) \int \frac {(d+e x)^{3/2}}{a b+b^2 x} \, dx}{7 b^4 e \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=\frac {2 (A b-a B) (b d-a e) (a+b x) (d+e x)^{3/2}}{3 b^3 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {2 (A b-a B) (a+b x) (d+e x)^{5/2}}{5 b^2 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {2 B (a+b x) (d+e x)^{7/2}}{7 b e \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {\left (2 \left (b^2 d-a b e\right )^2 \left (\frac {7}{2} A b^2 e-\frac {7}{2} a b B e\right ) \left (a b+b^2 x\right )\right ) \int \frac {\sqrt {d+e x}}{a b+b^2 x} \, dx}{7 b^6 e \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=\frac {2 (A b-a B) (b d-a e)^2 (a+b x) \sqrt {d+e x}}{b^4 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {2 (A b-a B) (b d-a e) (a+b x) (d+e x)^{3/2}}{3 b^3 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {2 (A b-a B) (a+b x) (d+e x)^{5/2}}{5 b^2 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {2 B (a+b x) (d+e x)^{7/2}}{7 b e \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {\left (2 \left (b^2 d-a b e\right )^3 \left (\frac {7}{2} A b^2 e-\frac {7}{2} a b B e\right ) \left (a b+b^2 x\right )\right ) \int \frac {1}{\left (a b+b^2 x\right ) \sqrt {d+e x}} \, dx}{7 b^8 e \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=\frac {2 (A b-a B) (b d-a e)^2 (a+b x) \sqrt {d+e x}}{b^4 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {2 (A b-a B) (b d-a e) (a+b x) (d+e x)^{3/2}}{3 b^3 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {2 (A b-a B) (a+b x) (d+e x)^{5/2}}{5 b^2 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {2 B (a+b x) (d+e x)^{7/2}}{7 b e \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {\left (4 \left (b^2 d-a b e\right )^3 \left (\frac {7}{2} A b^2 e-\frac {7}{2} a b B e\right ) \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 )}{7 b^8 e^2 \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=\frac {2 (A b-a B) (b d-a e)^2 (a+b x) \sqrt {d+e x}}{b^4 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {2 (A b-a B) (b d-a e) (a+b x) (d+e x)^{3/2}}{3 b^3 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {2 (A b-a B) (a+b x) (d+e x)^{5/2}}{5 b^2 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {2 B (a+b x) (d+e x)^{7/2}}{7 b e \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {2 (A b-a B) (b d-a e)^{5/2} (a+b x) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{b^{9/2} \sqrt {a^2+2 a b x+b^2 x^2}}\\ \end {align*}
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Mathematica [A] time = 0.30, size = 154, normalized size = 0.53 \begin {gather*} \frac {2 (a+b x) \left (\frac {7 e (A b-a B) \left (5 (b d-a e) \left (\sqrt {b} \sqrt {d+e x} (-3 a e+4 b d+b e x)-3 (b d-a e)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )\right )+3 b^{5/2} (d+e x)^{5/2}\right )}{15 b^{7/2}}+B (d+e x)^{7/2}\right )}{7 b e \sqrt {(a+b x)^2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 51.23, size = 346, normalized size = 1.20 \begin {gather*} \frac {(-a e-b e x) \left (-\frac {2 \left (-105 a^3 B e^3 \sqrt {d+e x}+105 a^2 A b e^3 \sqrt {d+e x}+35 a^2 b B e^2 (d+e x)^{3/2}+210 a^2 b B d e^2 \sqrt {d+e x}-35 a A b^2 e^2 (d+e x)^{3/2}-210 a A b^2 d e^2 \sqrt {d+e x}-105 a b^2 B d^2 e \sqrt {d+e x}-21 a b^2 B e (d+e x)^{5/2}-35 a b^2 B d e (d+e x)^{3/2}+105 A b^3 d^2 e \sqrt {d+e x}+21 A b^3 e (d+e x)^{5/2}+35 A b^3 d e (d+e x)^{3/2}+15 b^3 B (d+e x)^{7/2}\right )}{105 b^4 e}-\frac {2 (A b-a B) (a e-b d)^{5/2} \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x} \sqrt {a e-b d}}{b d-a e}\right )}{b^{9/2}}\right )}{e \sqrt {\frac {(a e+b e x)^2}{e^2}}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.46, size = 591, normalized size = 2.04 \begin {gather*} \left [\frac {105 \, {\left ({\left (B a b^{2} - A b^{3}\right )} d^{2} e - 2 \, {\left (B a^{2} b - A a b^{2}\right )} d e^{2} + {\left (B a^{3} - A a^{2} b\right )} e^{3}\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 (15 \, B b^{3} e^{3} x^{3} + 15 \, B b^{3} d^{3} - 161 \, {\left (B a b^{2} - A b^{3}\right )} d^{2} e + 245 \, {\left (B a^{2} b - A a b^{2}\right )} d e^{2} - 105 \, {\left (B a^{3} - A a^{2} b\right )} e^{3} + 3 \, {\left (15 \, B b^{3} d e^{2} - 7 \, {\left (B a b^{2} - A b^{3}\right )} e^{3}\right )} x^{2} + {\left (45 \, B b^{3} d^{2} e - 77 \, {\left (B a b^{2} - A b^{3}\right )} d e^{2} + 35 \, {\left (B a^{2} b - A a b^{2}\right )} e^{3}\right )} x\right )} \sqrt {e x + d}}{105 \, b^{4} e}, \frac {2 \, {\left (105 \, {\left ({\left (B a b^{2} - A b^{3}\right )} d^{2} e - 2 \, {\left (B a^{2} b - A a b^{2}\right )} d e^{2} + {\left (B a^{3} - A a^{2} b\right )} e^{3}\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 (15 \, B b^{3} e^{3} x^{3} + 15 \, B b^{3} d^{3} - 161 \, {\left (B a b^{2} - A b^{3}\right )} d^{2} e + 245 \, {\left (B a^{2} b - A a b^{2}\right )} d e^{2} - 105 \, {\left (B a^{3} - A a^{2} b\right )} e^{3} + 3 \, {\left (15 \, B b^{3} d e^{2} - 7 \, {\left (B a b^{2} - A b^{3}\right )} e^{3}\right )} x^{2} + {\left (45 \, B b^{3} d^{2} e - 77 \, {\left (B a b^{2} - A b^{3}\right )} d e^{2} + 35 \, {\left (B a^{2} b - A a b^{2}\right )} e^{3}\right )} x\right )} \sqrt {e x + d}\right )}}{105 \, b^{4} e}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.23, size = 497, normalized size = 1.72 \begin {gather*} -\frac {2 \, {\left (B a b^{3} d^{3} \mathrm {sgn}\left (b x + a\right ) - A b^{4} d^{3} \mathrm {sgn}\left (b x + a\right ) - 3 \, B a^{2} b^{2} d^{2} e \mathrm {sgn}\left (b x + a\right ) + 3 \, A a b^{3} d^{2} e \mathrm {sgn}\left (b x + a\right ) + 3 \, B a^{3} b d e^{2} \mathrm {sgn}\left (b x + a\right ) - 3 \, A a^{2} b^{2} d e^{2} \mathrm {sgn}\left (b x + a\right ) - B a^{4} e^{3} \mathrm {sgn}\left (b x + a\right ) + A a^{3} b e^{3} \mathrm {sgn}\left (b x + a\right )\right )} \arctan \left (\frac {\sqrt {x e + d} b}{\sqrt {-b^{2} d + a b e}}\right )}{\sqrt {-b^{2} d + a b e} b^{4}} + \frac {2 \, {\left (15 \, {\left (x e + d\right )}^{\frac {7}{2}} B b^{6} e^{6} \mathrm {sgn}\left (b x + a\right ) - 21 \, {\left (x e + d\right )}^{\frac {5}{2}} B a b^{5} e^{7} \mathrm {sgn}\left (b x + a\right ) + 21 \, {\left (x e + d\right )}^{\frac {5}{2}} A b^{6} e^{7} \mathrm {sgn}\left (b x + a\right ) - 35 \, {\left (x e + d\right )}^{\frac {3}{2}} B a b^{5} d e^{7} \mathrm {sgn}\left (b x + a\right ) + 35 \, {\left (x e + d\right )}^{\frac {3}{2}} A b^{6} d e^{7} \mathrm {sgn}\left (b x + a\right ) - 105 \, \sqrt {x e + d} B a b^{5} d^{2} e^{7} \mathrm {sgn}\left (b x + a\right ) + 105 \, \sqrt {x e + d} A b^{6} d^{2} e^{7} \mathrm {sgn}\left (b x + a\right ) + 35 \, {\left (x e + d\right )}^{\frac {3}{2}} B a^{2} b^{4} e^{8} \mathrm {sgn}\left (b x + a\right ) - 35 \, {\left (x e + d\right )}^{\frac {3}{2}} A a b^{5} e^{8} \mathrm {sgn}\left (b x + a\right ) + 210 \, \sqrt {x e + d} B a^{2} b^{4} d e^{8} \mathrm {sgn}\left (b x + a\right ) - 210 \, \sqrt {x e + d} A a b^{5} d e^{8} \mathrm {sgn}\left (b x + a\right ) - 105 \, \sqrt {x e + d} B a^{3} b^{3} e^{9} \mathrm {sgn}\left (b x + a\right ) + 105 \, \sqrt {x e + d} A a^{2} b^{4} e^{9} \mathrm {sgn}\left (b x + a\right )\right )} e^{\left (-7\right )}}{105 \, b^{7}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.06, size = 671, normalized size = 2.32 \begin {gather*} \frac {2 \left (b x +a \right ) \left (-105 A \,a^{3} b \,e^{4} \arctan \left (\frac {\sqrt {e x +d}\, b}{\sqrt {\left (a e -b d \right ) b}}\right )+315 A \,a^{2} b^{2} d \,e^{3} \arctan \left (\frac {\sqrt {e x +d}\, b}{\sqrt {\left (a e -b d \right ) b}}\right )-315 A a \,b^{3} d^{2} e^{2} \arctan \left (\frac {\sqrt {e x +d}\, b}{\sqrt {\left (a e -b d \right ) b}}\right )+105 A \,b^{4} d^{3} e \arctan \left (\frac {\sqrt {e x +d}\, b}{\sqrt {\left (a e -b d \right ) b}}\right )+105 B \,a^{4} e^{4} \arctan \left (\frac {\sqrt {e x +d}\, b}{\sqrt {\left (a e -b d \right ) b}}\right )-315 B \,a^{3} b d \,e^{3} \arctan \left (\frac {\sqrt {e x +d}\, b}{\sqrt {\left (a e -b d \right ) b}}\right )+315 B \,a^{2} b^{2} d^{2} e^{2} \arctan \left (\frac {\sqrt {e x +d}\, b}{\sqrt {\left (a e -b d \right ) b}}\right )-105 B a \,b^{3} d^{3} e \arctan \left (\frac {\sqrt {e x +d}\, b}{\sqrt {\left (a e -b d \right ) b}}\right )+105 \sqrt {\left (a e -b d \right ) b}\, \sqrt {e x +d}\, A \,a^{2} b \,e^{3}-210 \sqrt {\left (a e -b d \right ) b}\, \sqrt {e x +d}\, A a \,b^{2} d \,e^{2}+105 \sqrt {\left (a e -b d \right ) b}\, \sqrt {e x +d}\, A \,b^{3} d^{2} e -105 \sqrt {\left (a e -b d \right ) b}\, \sqrt {e x +d}\, B \,a^{3} e^{3}+210 \sqrt {\left (a e -b d \right ) b}\, \sqrt {e x +d}\, B \,a^{2} b d \,e^{2}-105 \sqrt {\left (a e -b d \right ) b}\, \sqrt {e x +d}\, B a \,b^{2} d^{2} e -35 \sqrt {\left (a e -b d \right ) b}\, \left (e x +d \right )^{\frac {3}{2}} A a \,b^{2} e^{2}+35 \sqrt {\left (a e -b d \right ) b}\, \left (e x +d \right )^{\frac {3}{2}} A \,b^{3} d e +35 \sqrt {\left (a e -b d \right ) b}\, \left (e x +d \right )^{\frac {3}{2}} B \,a^{2} b \,e^{2}-35 \sqrt {\left (a e -b d \right ) b}\, \left (e x +d \right )^{\frac {3}{2}} B a \,b^{2} d e +21 \sqrt {\left (a e -b d \right ) b}\, \left (e x +d \right )^{\frac {5}{2}} A \,b^{3} e -21 \sqrt {\left (a e -b d \right ) b}\, \left (e x +d \right )^{\frac {5}{2}} B a \,b^{2} e +15 \sqrt {\left (a e -b d \right ) b}\, \left (e x +d \right )^{\frac {7}{2}} B \,b^{3}\right )}{105 \sqrt {\left (b x +a \right )^{2}}\, \sqrt {\left (a e -b d \right ) b}\, b^{4} e} \end {gather*}
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 {{\left (B x + A\right )} {\left (e x + d\right )}^{\frac {5}{2}}}{\sqrt {{\left (b x + a\right )}^{2}}}\,{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 {\left (A+B\,x\right )\,{\left (d+e\,x\right )}^{5/2}}{\sqrt {{\left (a+b\,x\right )}^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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