Optimal. Leaf size=152 \[ -\frac {35 e^4 \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{64 b^{9/2} \sqrt {b d-a e}}-\frac {35 e^3 \sqrt {d+e x}}{64 b^4 (a+b x)}-\frac {35 e^2 (d+e x)^{3/2}}{96 b^3 (a+b x)^2}-\frac {7 e (d+e x)^{5/2}}{24 b^2 (a+b x)^3}-\frac {(d+e x)^{7/2}}{4 b (a+b x)^4} \]
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Rubi [A] time = 0.07, antiderivative size = 152, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 4, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.121, Rules used = {27, 47, 63, 208} \begin {gather*} -\frac {35 e^3 \sqrt {d+e x}}{64 b^4 (a+b x)}-\frac {35 e^2 (d+e x)^{3/2}}{96 b^3 (a+b x)^2}-\frac {35 e^4 \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{64 b^{9/2} \sqrt {b d-a e}}-\frac {7 e (d+e x)^{5/2}}{24 b^2 (a+b x)^3}-\frac {(d+e x)^{7/2}}{4 b (a+b x)^4} \end {gather*}
Antiderivative was successfully verified.
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Rule 27
Rule 47
Rule 63
Rule 208
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} \, dx &=\int \frac {(d+e x)^{7/2}}{(a+b x)^5} \, dx\\ &=-\frac {(d+e x)^{7/2}}{4 b (a+b x)^4}+\frac {(7 e) \int \frac {(d+e x)^{5/2}}{(a+b x)^4} \, dx}{8 b}\\ &=-\frac {7 e (d+e x)^{5/2}}{24 b^2 (a+b x)^3}-\frac {(d+e x)^{7/2}}{4 b (a+b x)^4}+\frac {\left (35 e^2\right ) \int \frac {(d+e x)^{3/2}}{(a+b x)^3} \, dx}{48 b^2}\\ &=-\frac {35 e^2 (d+e x)^{3/2}}{96 b^3 (a+b x)^2}-\frac {7 e (d+e x)^{5/2}}{24 b^2 (a+b x)^3}-\frac {(d+e x)^{7/2}}{4 b (a+b x)^4}+\frac {\left (35 e^3\right ) \int \frac {\sqrt {d+e x}}{(a+b x)^2} \, dx}{64 b^3}\\ &=-\frac {35 e^3 \sqrt {d+e x}}{64 b^4 (a+b x)}-\frac {35 e^2 (d+e x)^{3/2}}{96 b^3 (a+b x)^2}-\frac {7 e (d+e x)^{5/2}}{24 b^2 (a+b x)^3}-\frac {(d+e x)^{7/2}}{4 b (a+b x)^4}+\frac {\left (35 e^4\right ) \int \frac {1}{(a+b x) \sqrt {d+e x}} \, dx}{128 b^4}\\ &=-\frac {35 e^3 \sqrt {d+e x}}{64 b^4 (a+b x)}-\frac {35 e^2 (d+e x)^{3/2}}{96 b^3 (a+b x)^2}-\frac {7 e (d+e x)^{5/2}}{24 b^2 (a+b x)^3}-\frac {(d+e x)^{7/2}}{4 b (a+b x)^4}+\frac {\left (35 e^3\right ) \operatorname {Subst}\left (\int \frac {1}{a-\frac {b d}{e}+\frac {b x^2}{e}} \, dx,x,\sqrt {d+e x}\right )}{64 b^4}\\ &=-\frac {35 e^3 \sqrt {d+e x}}{64 b^4 (a+b x)}-\frac {35 e^2 (d+e x)^{3/2}}{96 b^3 (a+b x)^2}-\frac {7 e (d+e x)^{5/2}}{24 b^2 (a+b x)^3}-\frac {(d+e x)^{7/2}}{4 b (a+b x)^4}-\frac {35 e^4 \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{64 b^{9/2} \sqrt {b d-a e}}\\ \end {align*}
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Mathematica [A] time = 0.21, size = 152, normalized size = 1.00 \begin {gather*} \frac {35 e^4 \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {a e-b d}}\right )}{64 b^{9/2} \sqrt {a e-b d}}-\frac {35 e^3 \sqrt {d+e x}}{64 b^4 (a+b x)}-\frac {35 e^2 (d+e x)^{3/2}}{96 b^3 (a+b x)^2}-\frac {7 e (d+e x)^{5/2}}{24 b^2 (a+b x)^3}-\frac {(d+e x)^{7/2}}{4 b (a+b x)^4} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 1.14, size = 215, normalized size = 1.41 \begin {gather*} -\frac {e^4 \sqrt {d+e x} \left (105 a^3 e^3+385 a^2 b e^2 (d+e x)-315 a^2 b d e^2+315 a b^2 d^2 e+511 a b^2 e (d+e x)^2-770 a b^2 d e (d+e x)-105 b^3 d^3+385 b^3 d^2 (d+e x)+279 b^3 (d+e x)^3-511 b^3 d (d+e x)^2\right )}{192 b^4 (a e+b (d+e x)-b d)^4}-\frac {35 e^4 \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x} \sqrt {a e-b d}}{b d-a e}\right )}{64 b^{9/2} \sqrt {a e-b d}} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 0.45, size = 765, normalized size = 5.03 \begin {gather*} \left [\frac {105 \, {\left (b^{4} e^{4} x^{4} + 4 \, a b^{3} e^{4} x^{3} + 6 \, a^{2} b^{2} e^{4} x^{2} + 4 \, a^{3} b e^{4} x + a^{4} e^{4}\right )} \sqrt {b^{2} d - a b e} \log \left (\frac {b e x + 2 \, b d - a e - 2 \, \sqrt {b^{2} d - a b e} \sqrt {e x + d}}{b x + a}\right ) - 2 \, {\left (48 \, b^{5} d^{4} + 8 \, a b^{4} d^{3} e + 14 \, a^{2} b^{3} d^{2} e^{2} + 35 \, a^{3} b^{2} d e^{3} - 105 \, a^{4} b e^{4} + 279 \, {\left (b^{5} d e^{3} - a b^{4} e^{4}\right )} x^{3} + {\left (326 \, b^{5} d^{2} e^{2} + 185 \, a b^{4} d e^{3} - 511 \, a^{2} b^{3} e^{4}\right )} x^{2} + {\left (200 \, b^{5} d^{3} e + 52 \, a b^{4} d^{2} e^{2} + 133 \, a^{2} b^{3} d e^{3} - 385 \, a^{3} b^{2} e^{4}\right )} x\right )} \sqrt {e x + d}}{384 \, {\left (a^{4} b^{6} d - a^{5} b^{5} e + {\left (b^{10} d - a b^{9} e\right )} x^{4} + 4 \, {\left (a b^{9} d - a^{2} b^{8} e\right )} x^{3} + 6 \, {\left (a^{2} b^{8} d - a^{3} b^{7} e\right )} x^{2} + 4 \, {\left (a^{3} b^{7} d - a^{4} b^{6} e\right )} x\right )}}, \frac {105 \, {\left (b^{4} e^{4} x^{4} + 4 \, a b^{3} e^{4} x^{3} + 6 \, a^{2} b^{2} e^{4} x^{2} + 4 \, a^{3} b e^{4} x + a^{4} e^{4}\right )} \sqrt {-b^{2} d + a b e} \arctan \left (\frac {\sqrt {-b^{2} d + a b e} \sqrt {e x + d}}{b e x + b d}\right ) - {\left (48 \, b^{5} d^{4} + 8 \, a b^{4} d^{3} e + 14 \, a^{2} b^{3} d^{2} e^{2} + 35 \, a^{3} b^{2} d e^{3} - 105 \, a^{4} b e^{4} + 279 \, {\left (b^{5} d e^{3} - a b^{4} e^{4}\right )} x^{3} + {\left (326 \, b^{5} d^{2} e^{2} + 185 \, a b^{4} d e^{3} - 511 \, a^{2} b^{3} e^{4}\right )} x^{2} + {\left (200 \, b^{5} d^{3} e + 52 \, a b^{4} d^{2} e^{2} + 133 \, a^{2} b^{3} d e^{3} - 385 \, a^{3} b^{2} e^{4}\right )} x\right )} \sqrt {e x + d}}{192 \, {\left (a^{4} b^{6} d - a^{5} b^{5} e + {\left (b^{10} d - a b^{9} e\right )} x^{4} + 4 \, {\left (a b^{9} d - a^{2} b^{8} e\right )} x^{3} + 6 \, {\left (a^{2} b^{8} d - a^{3} b^{7} e\right )} x^{2} + 4 \, {\left (a^{3} b^{7} d - a^{4} b^{6} e\right )} x\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.22, size = 239, normalized size = 1.57 \begin {gather*} \frac {35 \, \arctan \left (\frac {\sqrt {x e + d} b}{\sqrt {-b^{2} d + a b e}}\right ) e^{4}}{64 \, \sqrt {-b^{2} d + a b e} b^{4}} - \frac {279 \, {\left (x e + d\right )}^{\frac {7}{2}} b^{3} e^{4} - 511 \, {\left (x e + d\right )}^{\frac {5}{2}} b^{3} d e^{4} + 385 \, {\left (x e + d\right )}^{\frac {3}{2}} b^{3} d^{2} e^{4} - 105 \, \sqrt {x e + d} b^{3} d^{3} e^{4} + 511 \, {\left (x e + d\right )}^{\frac {5}{2}} a b^{2} e^{5} - 770 \, {\left (x e + d\right )}^{\frac {3}{2}} a b^{2} d e^{5} + 315 \, \sqrt {x e + d} a b^{2} d^{2} e^{5} + 385 \, {\left (x e + d\right )}^{\frac {3}{2}} a^{2} b e^{6} - 315 \, \sqrt {x e + d} a^{2} b d e^{6} + 105 \, \sqrt {x e + d} a^{3} e^{7}}{192 \, {\left ({\left (x e + d\right )} b - b d + a e\right )}^{4} b^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.07, size = 318, normalized size = 2.09 \begin {gather*} -\frac {35 \sqrt {e x +d}\, a^{3} e^{7}}{64 \left (b e x +a e \right )^{4} b^{4}}+\frac {105 \sqrt {e x +d}\, a^{2} d \,e^{6}}{64 \left (b e x +a e \right )^{4} b^{3}}-\frac {105 \sqrt {e x +d}\, a \,d^{2} e^{5}}{64 \left (b e x +a e \right )^{4} b^{2}}+\frac {35 \sqrt {e x +d}\, d^{3} e^{4}}{64 \left (b e x +a e \right )^{4} b}-\frac {385 \left (e x +d \right )^{\frac {3}{2}} a^{2} e^{6}}{192 \left (b e x +a e \right )^{4} b^{3}}+\frac {385 \left (e x +d \right )^{\frac {3}{2}} a d \,e^{5}}{96 \left (b e x +a e \right )^{4} b^{2}}-\frac {385 \left (e x +d \right )^{\frac {3}{2}} d^{2} e^{4}}{192 \left (b e x +a e \right )^{4} b}-\frac {511 \left (e x +d \right )^{\frac {5}{2}} a \,e^{5}}{192 \left (b e x +a e \right )^{4} b^{2}}+\frac {511 \left (e x +d \right )^{\frac {5}{2}} d \,e^{4}}{192 \left (b e x +a e \right )^{4} b}-\frac {93 \left (e x +d \right )^{\frac {7}{2}} e^{4}}{64 \left (b e x +a e \right )^{4} b}+\frac {35 e^{4} \arctan \left (\frac {\sqrt {e x +d}\, b}{\sqrt {\left (a e -b d \right ) b}}\right )}{64 \sqrt {\left (a e -b d \right ) b}\, b^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.18, size = 337, normalized size = 2.22 \begin {gather*} \frac {35\,e^4\,\mathrm {atan}\left (\frac {\sqrt {b}\,\sqrt {d+e\,x}}{\sqrt {a\,e-b\,d}}\right )}{64\,b^{9/2}\,\sqrt {a\,e-b\,d}}-\frac {\frac {93\,e^4\,{\left (d+e\,x\right )}^{7/2}}{64\,b}+\frac {385\,e^4\,{\left (d+e\,x\right )}^{3/2}\,\left (a^2\,e^2-2\,a\,b\,d\,e+b^2\,d^2\right )}{192\,b^3}+\frac {35\,e^4\,\sqrt {d+e\,x}\,\left (a^3\,e^3-3\,a^2\,b\,d\,e^2+3\,a\,b^2\,d^2\,e-b^3\,d^3\right )}{64\,b^4}+\frac {511\,e^4\,\left (a\,e-b\,d\right )\,{\left (d+e\,x\right )}^{5/2}}{192\,b^2}}{b^4\,{\left (d+e\,x\right )}^4-\left (4\,b^4\,d-4\,a\,b^3\,e\right )\,{\left (d+e\,x\right )}^3-\left (d+e\,x\right )\,\left (-4\,a^3\,b\,e^3+12\,a^2\,b^2\,d\,e^2-12\,a\,b^3\,d^2\,e+4\,b^4\,d^3\right )+a^4\,e^4+b^4\,d^4+{\left (d+e\,x\right )}^2\,\left (6\,a^2\,b^2\,e^2-12\,a\,b^3\,d\,e+6\,b^4\,d^2\right )+6\,a^2\,b^2\,d^2\,e^2-4\,a\,b^3\,d^3\,e-4\,a^3\,b\,d\,e^3} \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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