Optimal. Leaf size=250 \[ -\frac {(d+e x)^{7/2}}{b \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {7 e (a+b x) (d+e x)^{5/2}}{5 b^2 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {7 e (a+b x) (b d-a e)^{5/2} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{b^{9/2} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {7 e (a+b x) \sqrt {d+e x} (b d-a e)^2}{b^4 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {7 e (a+b x) (d+e x)^{3/2} (b d-a e)}{3 b^3 \sqrt {a^2+2 a b x+b^2 x^2}} \]
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Rubi [A] time = 0.17, antiderivative size = 250, 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 = {768, 646, 50, 63, 208} \[ -\frac {(d+e x)^{7/2}}{b \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {7 e (a+b x) (d+e x)^{5/2}}{5 b^2 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {7 e (a+b x) (d+e x)^{3/2} (b d-a e)}{3 b^3 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {7 e (a+b x) \sqrt {d+e x} (b d-a e)^2}{b^4 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {7 e (a+b x) (b d-a e)^{5/2} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{b^{9/2} \sqrt {a^2+2 a b x+b^2 x^2}} \]
Antiderivative was successfully verified.
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Rule 50
Rule 63
Rule 208
Rule 646
Rule 768
Rubi steps
\begin {align*} \int \frac {(a+b x) (d+e x)^{7/2}}{\left (a^2+2 a b x+b^2 x^2\right )^{3/2}} \, dx &=-\frac {(d+e x)^{7/2}}{b \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {(7 e) \int \frac {(d+e x)^{5/2}}{\sqrt {a^2+2 a b x+b^2 x^2}} \, dx}{2 b}\\ &=-\frac {(d+e x)^{7/2}}{b \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}}{a b+b^2 x} \, dx}{2 b \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=\frac {7 e (a+b x) (d+e x)^{5/2}}{5 b^2 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {(d+e x)^{7/2}}{b \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {\left (7 e \left (b^2 d-a b e\right ) \left (a b+b^2 x\right )\right ) \int \frac {(d+e x)^{3/2}}{a b+b^2 x} \, dx}{2 b^3 \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=\frac {7 e (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 {7 e (a+b x) (d+e x)^{5/2}}{5 b^2 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {(d+e x)^{7/2}}{b \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {\left (7 e \left (b^2 d-a b e\right )^2 \left (a b+b^2 x\right )\right ) \int \frac {\sqrt {d+e x}}{a b+b^2 x} \, dx}{2 b^5 \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=\frac {7 e (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 {7 e (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 {7 e (a+b x) (d+e x)^{5/2}}{5 b^2 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {(d+e x)^{7/2}}{b \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {\left (7 e \left (b^2 d-a b e\right )^3 \left (a b+b^2 x\right )\right ) \int \frac {1}{\left (a b+b^2 x\right ) \sqrt {d+e x}} \, dx}{2 b^7 \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=\frac {7 e (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 {7 e (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 {7 e (a+b x) (d+e x)^{5/2}}{5 b^2 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {(d+e x)^{7/2}}{b \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {\left (7 \left (b^2 d-a b e\right )^3 \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 )}{b^7 \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=\frac {7 e (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 {7 e (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 {7 e (a+b x) (d+e x)^{5/2}}{5 b^2 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {(d+e x)^{7/2}}{b \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {7 e (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 [C] time = 0.03, size = 66, normalized size = 0.26 \[ \frac {2 e (a+b x) (d+e x)^{9/2} \, _2F_1\left (2,\frac {9}{2};\frac {11}{2};-\frac {b (d+e x)}{a e-b d}\right )}{9 \sqrt {(a+b x)^2} (a e-b d)^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.83, size = 486, normalized size = 1.94 \[ \left [\frac {105 \, {\left (a b^{2} d^{2} e - 2 \, a^{2} b d e^{2} + a^{3} e^{3} + {\left (b^{3} d^{2} e - 2 \, a b^{2} d e^{2} + a^{2} b e^{3}\right )} x\right )} \sqrt {\frac {b d - a e}{b}} \log \left (\frac {b e x + 2 \, b d - a e - 2 \, \sqrt {e x + d} b \sqrt {\frac {b d - a e}{b}}}{b x + a}\right ) + 2 \, {\left (6 \, b^{3} e^{3} x^{3} - 15 \, b^{3} d^{3} + 161 \, a b^{2} d^{2} e - 245 \, a^{2} b d e^{2} + 105 \, a^{3} e^{3} + 2 \, {\left (16 \, b^{3} d e^{2} - 7 \, a b^{2} e^{3}\right )} x^{2} + 2 \, {\left (58 \, b^{3} d^{2} e - 84 \, a b^{2} d e^{2} + 35 \, a^{2} b e^{3}\right )} x\right )} \sqrt {e x + d}}{30 \, {\left (b^{5} x + a b^{4}\right )}}, -\frac {105 \, {\left (a b^{2} d^{2} e - 2 \, a^{2} b d e^{2} + a^{3} e^{3} + {\left (b^{3} d^{2} e - 2 \, a b^{2} d e^{2} + a^{2} b e^{3}\right )} x\right )} \sqrt {-\frac {b d - a e}{b}} \arctan \left (-\frac {\sqrt {e x + d} b \sqrt {-\frac {b d - a e}{b}}}{b d - a e}\right ) - {\left (6 \, b^{3} e^{3} x^{3} - 15 \, b^{3} d^{3} + 161 \, a b^{2} d^{2} e - 245 \, a^{2} b d e^{2} + 105 \, a^{3} e^{3} + 2 \, {\left (16 \, b^{3} d e^{2} - 7 \, a b^{2} e^{3}\right )} x^{2} + 2 \, {\left (58 \, b^{3} d^{2} e - 84 \, a b^{2} d e^{2} + 35 \, a^{2} b e^{3}\right )} x\right )} \sqrt {e x + d}}{15 \, {\left (b^{5} x + a b^{4}\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.30, size = 359, normalized size = 1.44 \[ \frac {7 \, {\left (b^{3} d^{3} e^{2} - 3 \, a b^{2} d^{2} e^{3} + 3 \, a^{2} b d e^{4} - a^{3} e^{5}\right )} \arctan \left (\frac {\sqrt {x e + d} b}{\sqrt {-b^{2} d + a b e}}\right ) e^{\left (-1\right )}}{\sqrt {-b^{2} d + a b e} b^{4} \mathrm {sgn}\left ({\left (x e + d\right )} b e - b d e + a e^{2}\right )} - \frac {{\left (\sqrt {x e + d} b^{3} d^{3} e^{2} - 3 \, \sqrt {x e + d} a b^{2} d^{2} e^{3} + 3 \, \sqrt {x e + d} a^{2} b d e^{4} - \sqrt {x e + d} a^{3} e^{5}\right )} e^{\left (-1\right )}}{{\left ({\left (x e + d\right )} b - b d + a e\right )} b^{4} \mathrm {sgn}\left ({\left (x e + d\right )} b e - b d e + a e^{2}\right )} + \frac {2 \, {\left (3 \, {\left (x e + d\right )}^{\frac {5}{2}} b^{8} e^{6} + 10 \, {\left (x e + d\right )}^{\frac {3}{2}} b^{8} d e^{6} + 45 \, \sqrt {x e + d} b^{8} d^{2} e^{6} - 10 \, {\left (x e + d\right )}^{\frac {3}{2}} a b^{7} e^{7} - 90 \, \sqrt {x e + d} a b^{7} d e^{7} + 45 \, \sqrt {x e + d} a^{2} b^{6} e^{8}\right )} e^{\left (-5\right )}}{15 \, b^{10} \mathrm {sgn}\left ({\left (x e + d\right )} b e - b d e + a e^{2}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.08, size = 662, normalized size = 2.65 \[ \frac {\left (-105 a^{3} b \,e^{4} x \arctan \left (\frac {\sqrt {e x +d}\, b}{\sqrt {\left (a e -b d \right ) b}}\right )+315 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 )-315 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 b^{4} d^{3} e 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 )+315 a^{3} b d \,e^{3} \arctan \left (\frac {\sqrt {e x +d}\, b}{\sqrt {\left (a e -b d \right ) b}}\right )-315 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 a \,b^{3} d^{3} e \arctan \left (\frac {\sqrt {e x +d}\, b}{\sqrt {\left (a e -b d \right ) b}}\right )+90 \sqrt {\left (a e -b d \right ) b}\, \sqrt {e x +d}\, a^{2} b \,e^{3} x -180 \sqrt {\left (a e -b d \right ) b}\, \sqrt {e x +d}\, a \,b^{2} d \,e^{2} x +90 \sqrt {\left (a e -b d \right ) b}\, \sqrt {e x +d}\, b^{3} d^{2} e x +105 \sqrt {e x +d}\, \sqrt {\left (a e -b d \right ) b}\, a^{3} e^{3}-225 \sqrt {e x +d}\, \sqrt {\left (a e -b d \right ) b}\, a^{2} b d \,e^{2}+135 \sqrt {e x +d}\, \sqrt {\left (a e -b d \right ) b}\, a \,b^{2} d^{2} e -20 \sqrt {\left (a e -b d \right ) b}\, \left (e x +d \right )^{\frac {3}{2}} a \,b^{2} e^{2} x -15 \sqrt {e x +d}\, \sqrt {\left (a e -b d \right ) b}\, b^{3} d^{3}+20 \sqrt {\left (a e -b d \right ) b}\, \left (e x +d \right )^{\frac {3}{2}} b^{3} d e x -20 \left (e x +d \right )^{\frac {3}{2}} \sqrt {\left (a e -b d \right ) b}\, a^{2} b \,e^{2}+20 \left (e x +d \right )^{\frac {3}{2}} \sqrt {\left (a e -b d \right ) b}\, a \,b^{2} d e +6 \sqrt {\left (a e -b d \right ) b}\, \left (e x +d \right )^{\frac {5}{2}} b^{3} e x +6 \left (e x +d \right )^{\frac {5}{2}} \sqrt {\left (a e -b d \right ) b}\, a \,b^{2} e \right ) \left (b x +a \right )^{2}}{15 \sqrt {\left (a e -b d \right ) b}\, \left (\left (b x +a \right )^{2}\right )^{\frac {3}{2}} b^{4}} \]
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 \[ \int \frac {{\left (b x + a\right )} {\left (e x + d\right )}^{\frac {7}{2}}}{{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}}}\,{d x} \]
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 \[ \int \frac {\left (a+b\,x\right )\,{\left (d+e\,x\right )}^{7/2}}{{\left (a^2+2\,a\,b\,x+b^2\,x^2\right )}^{3/2}} \,d x \]
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 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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