Optimal. Leaf size=254 \[ -\frac {(d+e x)^{7/2}}{2 b (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {7 e (d+e x)^{5/2}}{4 b^2 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {35 e^2 (a+b x) (b d-a e)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{4 b^{9/2} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {35 e^2 (a+b x) \sqrt {d+e x} (b d-a e)}{4 b^4 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {35 e^2 (a+b x) (d+e x)^{3/2}}{12 b^3 \sqrt {a^2+2 a b x+b^2 x^2}} \]
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Rubi [A] time = 0.13, antiderivative size = 254, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {646, 47, 50, 63, 208} \begin {gather*} \frac {35 e^2 (a+b x) (d+e x)^{3/2}}{12 b^3 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {35 e^2 (a+b x) \sqrt {d+e x} (b d-a e)}{4 b^4 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {35 e^2 (a+b x) (b d-a e)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{4 b^{9/2} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {(d+e x)^{7/2}}{2 b (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {7 e (d+e x)^{5/2}}{4 b^2 \sqrt {a^2+2 a b x+b^2 x^2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 47
Rule 50
Rule 63
Rule 208
Rule 646
Rubi steps
\begin {align*} \int \frac {(d+e x)^{7/2}}{\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 {(d+e x)^{7/2}}{\left (a b+b^2 x\right )^3} \, dx}{\sqrt {a^2+2 a b x+b^2 x^2}}\\ &=-\frac {(d+e x)^{7/2}}{2 b (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {\left (7 e \left (a b+b^2 x\right )\right ) \int \frac {(d+e x)^{5/2}}{\left (a b+b^2 x\right )^2} \, dx}{4 \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=-\frac {7 e (d+e x)^{5/2}}{4 b^2 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {(d+e x)^{7/2}}{2 b (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {\left (35 e^2 \left (a b+b^2 x\right )\right ) \int \frac {(d+e x)^{3/2}}{a b+b^2 x} \, dx}{8 b^2 \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=\frac {35 e^2 (a+b x) (d+e x)^{3/2}}{12 b^3 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {7 e (d+e x)^{5/2}}{4 b^2 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {(d+e x)^{7/2}}{2 b (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {\left (35 e^2 \left (b^2 d-a b e\right ) \left (a b+b^2 x\right )\right ) \int \frac {\sqrt {d+e x}}{a b+b^2 x} \, dx}{8 b^4 \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=\frac {35 e^2 (b d-a e) (a+b x) \sqrt {d+e x}}{4 b^4 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {35 e^2 (a+b x) (d+e x)^{3/2}}{12 b^3 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {7 e (d+e x)^{5/2}}{4 b^2 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {(d+e x)^{7/2}}{2 b (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {\left (35 e^2 \left (b^2 d-a b e\right )^2 \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^6 \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=\frac {35 e^2 (b d-a e) (a+b x) \sqrt {d+e x}}{4 b^4 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {35 e^2 (a+b x) (d+e x)^{3/2}}{12 b^3 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {7 e (d+e x)^{5/2}}{4 b^2 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {(d+e x)^{7/2}}{2 b (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {\left (35 e \left (b^2 d-a b e\right )^2 \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^6 \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=\frac {35 e^2 (b d-a e) (a+b x) \sqrt {d+e x}}{4 b^4 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {35 e^2 (a+b x) (d+e x)^{3/2}}{12 b^3 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {7 e (d+e x)^{5/2}}{4 b^2 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {(d+e x)^{7/2}}{2 b (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {35 e^2 (b d-a e)^{3/2} (a+b x) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{4 b^{9/2} \sqrt {a^2+2 a b x+b^2 x^2}}\\ \end {align*}
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Mathematica [C] time = 0.04, size = 67, normalized size = 0.26 \begin {gather*} -\frac {2 e^2 (a+b x) (d+e x)^{9/2} \, _2F_1\left (3,\frac {9}{2};\frac {11}{2};\frac {b (d+e x)}{b d-a e}\right )}{9 \sqrt {(a+b x)^2} (b d-a e)^3} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 50.49, size = 285, normalized size = 1.12 \begin {gather*} \frac {(-a e-b e x) \left (-\frac {35 \left (-a^3 e^5+3 a^2 b d e^4-3 a b^2 d^2 e^3+b^3 d^3 e^2\right ) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x} \sqrt {a e-b d}}{b d-a e}\right )}{4 b^{9/2} (a e-b d)^{3/2}}-\frac {e^2 \sqrt {d+e x} \left (-105 a^3 e^3-175 a^2 b e^2 (d+e x)+315 a^2 b d e^2-315 a b^2 d^2 e-56 a b^2 e (d+e x)^2+350 a b^2 d e (d+e x)+105 b^3 d^3-175 b^3 d^2 (d+e x)+8 b^3 (d+e x)^3+56 b^3 d (d+e x)^2\right )}{12 b^4 (a e+b (d+e x)-b d)^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 = 0.42, size = 520, normalized size = 2.05 \begin {gather*} \left [-\frac {105 \, {\left (a^{2} b d e^{2} - a^{3} e^{3} + {\left (b^{3} d e^{2} - a b^{2} e^{3}\right )} x^{2} + 2 \, {\left (a b^{2} d e^{2} - a^{2} b e^{3}\right )} x\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 (8 \, b^{3} e^{3} x^{3} - 6 \, b^{3} d^{3} - 21 \, a b^{2} d^{2} e + 140 \, a^{2} b d e^{2} - 105 \, a^{3} e^{3} + 8 \, {\left (10 \, b^{3} d e^{2} - 7 \, a b^{2} e^{3}\right )} x^{2} - {\left (39 \, b^{3} d^{2} e - 238 \, a b^{2} d e^{2} + 175 \, a^{2} b e^{3}\right )} x\right )} \sqrt {e x + d}}{24 \, {\left (b^{6} x^{2} + 2 \, a b^{5} x + a^{2} b^{4}\right )}}, -\frac {105 \, {\left (a^{2} b d e^{2} - a^{3} e^{3} + {\left (b^{3} d e^{2} - a b^{2} e^{3}\right )} x^{2} + 2 \, {\left (a b^{2} d e^{2} - a^{2} b e^{3}\right )} x\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 (8 \, b^{3} e^{3} x^{3} - 6 \, b^{3} d^{3} - 21 \, a b^{2} d^{2} e + 140 \, a^{2} b d e^{2} - 105 \, a^{3} e^{3} + 8 \, {\left (10 \, b^{3} d e^{2} - 7 \, a b^{2} e^{3}\right )} x^{2} - {\left (39 \, b^{3} d^{2} e - 238 \, a b^{2} d e^{2} + 175 \, a^{2} b e^{3}\right )} x\right )} \sqrt {e x + d}}{12 \, {\left (b^{6} x^{2} + 2 \, a b^{5} x + a^{2} b^{4}\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.30, size = 337, normalized size = 1.33 \begin {gather*} \frac {35 \, {\left (b^{2} d^{2} e^{2} - 2 \, a b d e^{3} + a^{2} 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^{4} \mathrm {sgn}\left ({\left (x e + d\right )} b e - b d e + a e^{2}\right )} - \frac {13 \, {\left (x e + d\right )}^{\frac {3}{2}} b^{3} d^{2} e^{2} - 11 \, \sqrt {x e + d} b^{3} d^{3} e^{2} - 26 \, {\left (x e + d\right )}^{\frac {3}{2}} a b^{2} d e^{3} + 33 \, \sqrt {x e + d} a b^{2} d^{2} e^{3} + 13 \, {\left (x e + d\right )}^{\frac {3}{2}} a^{2} b e^{4} - 33 \, \sqrt {x e + d} a^{2} b d e^{4} + 11 \, \sqrt {x e + d} a^{3} e^{5}}{4 \, {\left ({\left (x e + d\right )} b - b d + a e\right )}^{2} b^{4} \mathrm {sgn}\left ({\left (x e + d\right )} b e - b d e + a e^{2}\right )} + \frac {2 \, {\left ({\left (x e + d\right )}^{\frac {3}{2}} b^{6} e^{2} + 9 \, \sqrt {x e + d} b^{6} d e^{2} - 9 \, \sqrt {x e + d} a b^{5} e^{3}\right )}}{3 \, b^{9} \mathrm {sgn}\left ({\left (x e + d\right )} b e - b d e + a e^{2}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.06, size = 714, normalized size = 2.81 \begin {gather*} \frac {\left (105 a^{2} b^{2} e^{4} x^{2} \arctan \left (\frac {\sqrt {e x +d}\, b}{\sqrt {\left (a e -b d \right ) b}}\right )-210 a \,b^{3} d \,e^{3} x^{2} \arctan \left (\frac {\sqrt {e x +d}\, b}{\sqrt {\left (a e -b d \right ) b}}\right )+105 b^{4} d^{2} e^{2} x^{2} \arctan \left (\frac {\sqrt {e x +d}\, b}{\sqrt {\left (a e -b d \right ) b}}\right )+210 a^{3} b \,e^{4} x \arctan \left (\frac {\sqrt {e x +d}\, b}{\sqrt {\left (a e -b d \right ) b}}\right )-420 a^{2} b^{2} d \,e^{3} x \arctan \left (\frac {\sqrt {e x +d}\, b}{\sqrt {\left (a e -b d \right ) b}}\right )+210 a \,b^{3} d^{2} e^{2} x \arctan \left (\frac {\sqrt {e x +d}\, b}{\sqrt {\left (a e -b d \right ) b}}\right )+105 a^{4} e^{4} \arctan \left (\frac {\sqrt {e x +d}\, b}{\sqrt {\left (a e -b d \right ) b}}\right )-210 a^{3} b d \,e^{3} \arctan \left (\frac {\sqrt {e x +d}\, b}{\sqrt {\left (a e -b d \right ) b}}\right )+105 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 )-72 \sqrt {e x +d}\, \sqrt {\left (a e -b d \right ) b}\, a \,b^{2} e^{3} x^{2}+72 \sqrt {e x +d}\, \sqrt {\left (a e -b d \right ) b}\, b^{3} d \,e^{2} x^{2}-144 \sqrt {e x +d}\, \sqrt {\left (a e -b d \right ) b}\, a^{2} b \,e^{3} x +144 \sqrt {e x +d}\, \sqrt {\left (a e -b d \right ) b}\, a \,b^{2} d \,e^{2} x +8 \left (e x +d \right )^{\frac {3}{2}} \sqrt {\left (a e -b d \right ) b}\, b^{3} e^{2} x^{2}-105 \sqrt {e x +d}\, \sqrt {\left (a e -b d \right ) b}\, a^{3} e^{3}+171 \sqrt {e x +d}\, \sqrt {\left (a e -b d \right ) b}\, a^{2} b d \,e^{2}-99 \sqrt {e x +d}\, \sqrt {\left (a e -b d \right ) b}\, a \,b^{2} d^{2} e +16 \left (e x +d \right )^{\frac {3}{2}} \sqrt {\left (a e -b d \right ) b}\, a \,b^{2} e^{2} x +33 \sqrt {e x +d}\, \sqrt {\left (a e -b d \right ) b}\, b^{3} d^{3}-31 \left (e x +d \right )^{\frac {3}{2}} \sqrt {\left (a e -b d \right ) b}\, a^{2} b \,e^{2}+78 \left (e x +d \right )^{\frac {3}{2}} \sqrt {\left (a e -b d \right ) b}\, a \,b^{2} d e -39 \left (e x +d \right )^{\frac {3}{2}} \sqrt {\left (a e -b d \right ) b}\, b^{3} d^{2}\right ) \left (b x +a \right )}{12 \sqrt {\left (a e -b d \right ) b}\, \left (\left (b x +a \right )^{2}\right )^{\frac {3}{2}} b^{4}} \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 (e x + d\right )}^{\frac {7}{2}}}{{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{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 (d+e\,x\right )}^{7/2}}{{\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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