Optimal. Leaf size=348 \[ -\frac {A b-a B}{2 b (a+b x) \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{3/2} (b d-a e)}-\frac {5 e (a+b x) (3 a B e-7 A b e+4 b B d)}{4 \sqrt {a^2+2 a b x+b^2 x^2} \sqrt {d+e x} (b d-a e)^4}+\frac {-3 a B e+7 A b e-4 b B d}{4 b \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{3/2} (b d-a e)^2}-\frac {5 e (a+b x) (3 a B e-7 A b e+4 b B d)}{12 b \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{3/2} (b d-a e)^3}+\frac {5 \sqrt {b} e (a+b x) (3 a B e-7 A b e+4 b B d) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{4 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^{9/2}} \]
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Rubi [A] time = 0.32, antiderivative size = 348, 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 {A b-a B}{2 b (a+b x) \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{3/2} (b d-a e)}-\frac {5 e (a+b x) (3 a B e-7 A b e+4 b B d)}{4 \sqrt {a^2+2 a b x+b^2 x^2} \sqrt {d+e x} (b d-a e)^4}-\frac {3 a B e-7 A b e+4 b B d}{4 b \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{3/2} (b d-a e)^2}-\frac {5 e (a+b x) (3 a B e-7 A b e+4 b B d)}{12 b \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{3/2} (b d-a e)^3}+\frac {5 \sqrt {b} e (a+b x) (3 a B e-7 A b e+4 b B d) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{4 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^{9/2}} \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}{(d+e x)^{5/2} \left (a^2+2 a b x+b^2 x^2\right )^{3/2}} \, dx &=\frac {\left (b^2 \left (a b+b^2 x\right )\right ) \int \frac {A+B x}{\left (a b+b^2 x\right )^3 (d+e x)^{5/2}} \, dx}{\sqrt {a^2+2 a b x+b^2 x^2}}\\ &=-\frac {A b-a B}{2 b (b d-a e) (a+b x) (d+e x)^{3/2} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {\left ((4 b B d-7 A b e+3 a B e) \left (a b+b^2 x\right )\right ) \int \frac {1}{\left (a b+b^2 x\right )^2 (d+e x)^{5/2}} \, dx}{4 (b d-a e) \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=-\frac {4 b B d-7 A b e+3 a B e}{4 b (b d-a e)^2 (d+e x)^{3/2} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {A b-a B}{2 b (b d-a e) (a+b x) (d+e x)^{3/2} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {\left (5 e (4 b B d-7 A b e+3 a B e) \left (a b+b^2 x\right )\right ) \int \frac {1}{\left (a b+b^2 x\right ) (d+e x)^{5/2}} \, dx}{8 b (b d-a e)^2 \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=-\frac {4 b B d-7 A b e+3 a B e}{4 b (b d-a e)^2 (d+e x)^{3/2} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {A b-a B}{2 b (b d-a e) (a+b x) (d+e x)^{3/2} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {5 e (4 b B d-7 A b e+3 a B e) (a+b x)}{12 b (b d-a e)^3 (d+e x)^{3/2} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {\left (5 e (4 b B d-7 A b e+3 a B e) \left (a b+b^2 x\right )\right ) \int \frac {1}{\left (a b+b^2 x\right ) (d+e x)^{3/2}} \, dx}{8 (b d-a e)^3 \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=-\frac {4 b B d-7 A b e+3 a B e}{4 b (b d-a e)^2 (d+e x)^{3/2} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {A b-a B}{2 b (b d-a e) (a+b x) (d+e x)^{3/2} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {5 e (4 b B d-7 A b e+3 a B e) (a+b x)}{12 b (b d-a e)^3 (d+e x)^{3/2} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {5 e (4 b B d-7 A b e+3 a B e) (a+b x)}{4 (b d-a e)^4 \sqrt {d+e x} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {\left (5 b e (4 b B d-7 A b e+3 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}{8 (b d-a e)^4 \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=-\frac {4 b B d-7 A b e+3 a B e}{4 b (b d-a e)^2 (d+e x)^{3/2} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {A b-a B}{2 b (b d-a e) (a+b x) (d+e x)^{3/2} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {5 e (4 b B d-7 A b e+3 a B e) (a+b x)}{12 b (b d-a e)^3 (d+e x)^{3/2} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {5 e (4 b B d-7 A b e+3 a B e) (a+b x)}{4 (b d-a e)^4 \sqrt {d+e x} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {\left (5 b (4 b B d-7 A b e+3 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 )}{4 (b d-a e)^4 \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=-\frac {4 b B d-7 A b e+3 a B e}{4 b (b d-a e)^2 (d+e x)^{3/2} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {A b-a B}{2 b (b d-a e) (a+b x) (d+e x)^{3/2} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {5 e (4 b B d-7 A b e+3 a B e) (a+b x)}{12 b (b d-a e)^3 (d+e x)^{3/2} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {5 e (4 b B d-7 A b e+3 a B e) (a+b x)}{4 (b d-a e)^4 \sqrt {d+e x} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {5 \sqrt {b} e (4 b B d-7 A b e+3 a B e) (a+b x) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{4 (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 = 0.07, size = 111, normalized size = 0.32 \begin {gather*} \frac {(a+b x) \left (\frac {e (a+b x)^2 (-3 a B e+7 A b e-4 b B d) \, _2F_1\left (-\frac {3}{2},2;-\frac {1}{2};\frac {b (d+e x)}{b d-a e}\right )}{(b d-a e)^2}+3 a B-3 A b\right )}{6 b \left ((a+b x)^2\right )^{3/2} (d+e x)^{3/2} (b d-a e)} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 63.32, size = 490, normalized size = 1.41 \begin {gather*} \frac {(-a e-b e x) \left (\frac {e \left (8 a^3 A e^4+24 a^3 B e^3 (d+e x)-8 a^3 B d e^3-56 a^2 A b e^3 (d+e x)-24 a^2 A b d e^3+24 a^2 b B d^2 e^2-16 a^2 b B d e^2 (d+e x)+75 a^2 b B e^2 (d+e x)^2+24 a A b^2 d^2 e^2+112 a A b^2 d e^2 (d+e x)-175 a A b^2 e^2 (d+e x)^2-24 a b^2 B d^3 e-40 a b^2 B d^2 e (d+e x)+25 a b^2 B d e (d+e x)^2+45 a b^2 B e (d+e x)^3-8 A b^3 d^3 e-56 A b^3 d^2 e (d+e x)+175 A b^3 d e (d+e x)^2-105 A b^3 e (d+e x)^3+8 b^3 B d^4+32 b^3 B d^3 (d+e x)-100 b^3 B d^2 (d+e x)^2+60 b^3 B d (d+e x)^3\right )}{12 (d+e x)^{3/2} (b d-a e)^4 (-a e-b (d+e x)+b d)^2}-\frac {5 \left (3 a \sqrt {b} B e^2-7 A b^{3/2} e^2+4 b^{3/2} 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 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 = 0.55, size = 1776, normalized size = 5.10
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.49, size = 785, normalized size = 2.26 \begin {gather*} -\frac {5 \, {\left (4 \, B b^{2} d e^{2} + 3 \, B a b e^{3} - 7 \, A b^{2} e^{3}\right )} \arctan \left (\frac {\sqrt {x e + d} b}{\sqrt {-b^{2} d + a b e}}\right )}{4 \, {\left (b^{4} d^{4} e \mathrm {sgn}\left ({\left (x e + d\right )} b e - b d e + a e^{2}\right ) - 4 \, a b^{3} d^{3} e^{2} \mathrm {sgn}\left ({\left (x e + d\right )} b e - b d e + a e^{2}\right ) + 6 \, a^{2} b^{2} d^{2} e^{3} \mathrm {sgn}\left ({\left (x e + d\right )} b e - b d e + a e^{2}\right ) - 4 \, a^{3} b d e^{4} \mathrm {sgn}\left ({\left (x e + d\right )} b e - b d e + a e^{2}\right ) + a^{4} e^{5} \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 {2 \, {\left (6 \, {\left (x e + d\right )} B b d e^{2} + B b d^{2} e^{2} + 3 \, {\left (x e + d\right )} B a e^{3} - 9 \, {\left (x e + d\right )} A b e^{3} - B a d e^{3} - A b d e^{3} + A a e^{4}\right )}}{3 \, {\left (b^{4} d^{4} e \mathrm {sgn}\left ({\left (x e + d\right )} b e - b d e + a e^{2}\right ) - 4 \, a b^{3} d^{3} e^{2} \mathrm {sgn}\left ({\left (x e + d\right )} b e - b d e + a e^{2}\right ) + 6 \, a^{2} b^{2} d^{2} e^{3} \mathrm {sgn}\left ({\left (x e + d\right )} b e - b d e + a e^{2}\right ) - 4 \, a^{3} b d e^{4} \mathrm {sgn}\left ({\left (x e + d\right )} b e - b d e + a e^{2}\right ) + a^{4} e^{5} \mathrm {sgn}\left ({\left (x e + d\right )} b e - b d e + a e^{2}\right )\right )} {\left (x e + d\right )}^{\frac {3}{2}}} - \frac {4 \, {\left (x e + d\right )}^{\frac {3}{2}} B b^{3} d e^{2} - 4 \, \sqrt {x e + d} B b^{3} d^{2} e^{2} + 7 \, {\left (x e + d\right )}^{\frac {3}{2}} B a b^{2} e^{3} - 11 \, {\left (x e + d\right )}^{\frac {3}{2}} A b^{3} e^{3} - 5 \, \sqrt {x e + d} B a b^{2} d e^{3} + 13 \, \sqrt {x e + d} A b^{3} d e^{3} + 9 \, \sqrt {x e + d} B a^{2} b e^{4} - 13 \, \sqrt {x e + d} A a b^{2} e^{4}}{4 \, {\left (b^{4} d^{4} e \mathrm {sgn}\left ({\left (x e + d\right )} b e - b d e + a e^{2}\right ) - 4 \, a b^{3} d^{3} e^{2} \mathrm {sgn}\left ({\left (x e + d\right )} b e - b d e + a e^{2}\right ) + 6 \, a^{2} b^{2} d^{2} e^{3} \mathrm {sgn}\left ({\left (x e + d\right )} b e - b d e + a e^{2}\right ) - 4 \, a^{3} b d e^{4} \mathrm {sgn}\left ({\left (x e + d\right )} b e - b d e + a e^{2}\right ) + a^{4} e^{5} \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 )}^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.13, size = 928, normalized size = 2.67 \begin {gather*} \frac {\left (105 \left (e x +d \right )^{\frac {3}{2}} A \,b^{4} e^{2} x^{2} \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}\, A \,b^{3} e^{3} x^{3}-45 \left (e x +d \right )^{\frac {3}{2}} B a \,b^{3} e^{2} x^{2} \arctan \left (\frac {\sqrt {e x +d}\, b}{\sqrt {\left (a e -b d \right ) b}}\right )-45 \sqrt {\left (a e -b d \right ) b}\, B a \,b^{2} e^{3} x^{3}-60 \left (e x +d \right )^{\frac {3}{2}} B \,b^{4} d e \,x^{2} \arctan \left (\frac {\sqrt {e x +d}\, b}{\sqrt {\left (a e -b d \right ) b}}\right )-60 \sqrt {\left (a e -b d \right ) b}\, B \,b^{3} d \,e^{2} x^{3}+210 \left (e x +d \right )^{\frac {3}{2}} A a \,b^{3} e^{2} x \arctan \left (\frac {\sqrt {e x +d}\, b}{\sqrt {\left (a e -b d \right ) b}}\right )+175 \sqrt {\left (a e -b d \right ) b}\, A a \,b^{2} e^{3} x^{2}+140 \sqrt {\left (a e -b d \right ) b}\, A \,b^{3} d \,e^{2} x^{2}-90 \left (e x +d \right )^{\frac {3}{2}} B \,a^{2} b^{2} e^{2} x \arctan \left (\frac {\sqrt {e x +d}\, b}{\sqrt {\left (a e -b d \right ) b}}\right )-75 \sqrt {\left (a e -b d \right ) b}\, B \,a^{2} b \,e^{3} x^{2}-120 \left (e x +d \right )^{\frac {3}{2}} B a \,b^{3} d e x \arctan \left (\frac {\sqrt {e x +d}\, b}{\sqrt {\left (a e -b d \right ) b}}\right )-160 \sqrt {\left (a e -b d \right ) b}\, B a \,b^{2} d \,e^{2} x^{2}-80 \sqrt {\left (a e -b d \right ) b}\, B \,b^{3} d^{2} e \,x^{2}+105 \left (e x +d \right )^{\frac {3}{2}} A \,a^{2} b^{2} e^{2} \arctan \left (\frac {\sqrt {e x +d}\, b}{\sqrt {\left (a e -b d \right ) b}}\right )+56 \sqrt {\left (a e -b d \right ) b}\, A \,a^{2} b \,e^{3} x +238 \sqrt {\left (a e -b d \right ) b}\, A a \,b^{2} d \,e^{2} x +21 \sqrt {\left (a e -b d \right ) b}\, A \,b^{3} d^{2} e x -45 \left (e x +d \right )^{\frac {3}{2}} B \,a^{3} b \,e^{2} \arctan \left (\frac {\sqrt {e x +d}\, b}{\sqrt {\left (a e -b d \right ) b}}\right )-24 \sqrt {\left (a e -b d \right ) b}\, B \,a^{3} e^{3} x -60 \left (e x +d \right )^{\frac {3}{2}} B \,a^{2} b^{2} d e \arctan \left (\frac {\sqrt {e x +d}\, b}{\sqrt {\left (a e -b d \right ) b}}\right )-134 \sqrt {\left (a e -b d \right ) b}\, B \,a^{2} b d \,e^{2} x -145 \sqrt {\left (a e -b d \right ) b}\, B a \,b^{2} d^{2} e x -12 \sqrt {\left (a e -b d \right ) b}\, B \,b^{3} d^{3} x -8 \sqrt {\left (a e -b d \right ) b}\, A \,a^{3} e^{3}+80 \sqrt {\left (a e -b d \right ) b}\, A \,a^{2} b d \,e^{2}+39 \sqrt {\left (a e -b d \right ) b}\, A a \,b^{2} d^{2} e -6 \sqrt {\left (a e -b d \right ) b}\, A \,b^{3} d^{3}-16 \sqrt {\left (a e -b d \right ) b}\, B \,a^{3} d \,e^{2}-83 \sqrt {\left (a e -b d \right ) b}\, B \,a^{2} b \,d^{2} e -6 \sqrt {\left (a e -b d \right ) b}\, B a \,b^{2} d^{3}\right ) \left (b x +a \right )}{12 \left (e x +d \right )^{\frac {3}{2}} \sqrt {\left (a e -b d \right ) b}\, \left (a e -b d \right )^{4} \left (\left (b x +a \right )^{2}\right )^{\frac {3}{2}}} \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 {B x + A}{{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} {\left (e x + d\right )}^{\frac {5}{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 {A+B\,x}{{\left (d+e\,x\right )}^{5/2}\,{\left (a^2+2\,a\,b\,x+b^2\,x^2\right )}^{3/2}} \,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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