Optimal. Leaf size=98 \[ \frac {b \sqrt {a^2+2 a b x+b^2 x^2} (a+b x)^6}{56 (d+e x)^7 (b d-a e)^2}+\frac {\sqrt {a^2+2 a b x+b^2 x^2} (a+b x)^6}{8 (d+e x)^8 (b d-a e)} \]
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Rubi [A] time = 0.05, antiderivative size = 98, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.121, Rules used = {770, 21, 45, 37} \begin {gather*} \frac {b \sqrt {a^2+2 a b x+b^2 x^2} (a+b x)^6}{56 (d+e x)^7 (b d-a e)^2}+\frac {\sqrt {a^2+2 a b x+b^2 x^2} (a+b x)^6}{8 (d+e x)^8 (b d-a e)} \end {gather*}
Antiderivative was successfully verified.
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Rule 21
Rule 37
Rule 45
Rule 770
Rubi steps
\begin {align*} \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^9} \, dx &=\frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \frac {(a+b x) \left (a b+b^2 x\right )^5}{(d+e x)^9} \, dx}{b^4 \left (a b+b^2 x\right )}\\ &=\frac {\left (b \sqrt {a^2+2 a b x+b^2 x^2}\right ) \int \frac {(a+b x)^6}{(d+e x)^9} \, dx}{a b+b^2 x}\\ &=\frac {(a+b x)^6 \sqrt {a^2+2 a b x+b^2 x^2}}{8 (b d-a e) (d+e x)^8}+\frac {\left (b^2 \sqrt {a^2+2 a b x+b^2 x^2}\right ) \int \frac {(a+b x)^6}{(d+e x)^8} \, dx}{8 (b d-a e) \left (a b+b^2 x\right )}\\ &=\frac {(a+b x)^6 \sqrt {a^2+2 a b x+b^2 x^2}}{8 (b d-a e) (d+e x)^8}+\frac {b (a+b x)^6 \sqrt {a^2+2 a b x+b^2 x^2}}{56 (b d-a e)^2 (d+e x)^7}\\ \end {align*}
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Mathematica [B] time = 0.11, size = 295, normalized size = 3.01 \begin {gather*} -\frac {\sqrt {(a+b x)^2} \left (7 a^6 e^6+6 a^5 b e^5 (d+8 e x)+5 a^4 b^2 e^4 \left (d^2+8 d e x+28 e^2 x^2\right )+4 a^3 b^3 e^3 \left (d^3+8 d^2 e x+28 d e^2 x^2+56 e^3 x^3\right )+3 a^2 b^4 e^2 \left (d^4+8 d^3 e x+28 d^2 e^2 x^2+56 d e^3 x^3+70 e^4 x^4\right )+2 a b^5 e \left (d^5+8 d^4 e x+28 d^3 e^2 x^2+56 d^2 e^3 x^3+70 d e^4 x^4+56 e^5 x^5\right )+b^6 \left (d^6+8 d^5 e x+28 d^4 e^2 x^2+56 d^3 e^3 x^3+70 d^2 e^4 x^4+56 d e^5 x^5+28 e^6 x^6\right )\right )}{56 e^7 (a+b x) (d+e x)^8} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 180.02, size = 0, normalized size = 0.00 \begin {gather*} \text {\$Aborted} \end {gather*}
Verification is not applicable to the result.
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fricas [B] time = 0.41, size = 430, normalized size = 4.39 \begin {gather*} -\frac {28 \, b^{6} e^{6} x^{6} + b^{6} d^{6} + 2 \, a b^{5} d^{5} e + 3 \, a^{2} b^{4} d^{4} e^{2} + 4 \, a^{3} b^{3} d^{3} e^{3} + 5 \, a^{4} b^{2} d^{2} e^{4} + 6 \, a^{5} b d e^{5} + 7 \, a^{6} e^{6} + 56 \, {\left (b^{6} d e^{5} + 2 \, a b^{5} e^{6}\right )} x^{5} + 70 \, {\left (b^{6} d^{2} e^{4} + 2 \, a b^{5} d e^{5} + 3 \, a^{2} b^{4} e^{6}\right )} x^{4} + 56 \, {\left (b^{6} d^{3} e^{3} + 2 \, a b^{5} d^{2} e^{4} + 3 \, a^{2} b^{4} d e^{5} + 4 \, a^{3} b^{3} e^{6}\right )} x^{3} + 28 \, {\left (b^{6} d^{4} e^{2} + 2 \, a b^{5} d^{3} e^{3} + 3 \, a^{2} b^{4} d^{2} e^{4} + 4 \, a^{3} b^{3} d e^{5} + 5 \, a^{4} b^{2} e^{6}\right )} x^{2} + 8 \, {\left (b^{6} d^{5} e + 2 \, a b^{5} d^{4} e^{2} + 3 \, a^{2} b^{4} d^{3} e^{3} + 4 \, a^{3} b^{3} d^{2} e^{4} + 5 \, a^{4} b^{2} d e^{5} + 6 \, a^{5} b e^{6}\right )} x}{56 \, {\left (e^{15} x^{8} + 8 \, d e^{14} x^{7} + 28 \, d^{2} e^{13} x^{6} + 56 \, d^{3} e^{12} x^{5} + 70 \, d^{4} e^{11} x^{4} + 56 \, d^{5} e^{10} x^{3} + 28 \, d^{6} e^{9} x^{2} + 8 \, d^{7} e^{8} x + d^{8} e^{7}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.24, size = 520, normalized size = 5.31 \begin {gather*} -\frac {{\left (28 \, b^{6} x^{6} e^{6} \mathrm {sgn}\left (b x + a\right ) + 56 \, b^{6} d x^{5} e^{5} \mathrm {sgn}\left (b x + a\right ) + 70 \, b^{6} d^{2} x^{4} e^{4} \mathrm {sgn}\left (b x + a\right ) + 56 \, b^{6} d^{3} x^{3} e^{3} \mathrm {sgn}\left (b x + a\right ) + 28 \, b^{6} d^{4} x^{2} e^{2} \mathrm {sgn}\left (b x + a\right ) + 8 \, b^{6} d^{5} x e \mathrm {sgn}\left (b x + a\right ) + b^{6} d^{6} \mathrm {sgn}\left (b x + a\right ) + 112 \, a b^{5} x^{5} e^{6} \mathrm {sgn}\left (b x + a\right ) + 140 \, a b^{5} d x^{4} e^{5} \mathrm {sgn}\left (b x + a\right ) + 112 \, a b^{5} d^{2} x^{3} e^{4} \mathrm {sgn}\left (b x + a\right ) + 56 \, a b^{5} d^{3} x^{2} e^{3} \mathrm {sgn}\left (b x + a\right ) + 16 \, a b^{5} d^{4} x e^{2} \mathrm {sgn}\left (b x + a\right ) + 2 \, a b^{5} d^{5} e \mathrm {sgn}\left (b x + a\right ) + 210 \, a^{2} b^{4} x^{4} e^{6} \mathrm {sgn}\left (b x + a\right ) + 168 \, a^{2} b^{4} d x^{3} e^{5} \mathrm {sgn}\left (b x + a\right ) + 84 \, a^{2} b^{4} d^{2} x^{2} e^{4} \mathrm {sgn}\left (b x + a\right ) + 24 \, a^{2} b^{4} d^{3} x e^{3} \mathrm {sgn}\left (b x + a\right ) + 3 \, a^{2} b^{4} d^{4} e^{2} \mathrm {sgn}\left (b x + a\right ) + 224 \, a^{3} b^{3} x^{3} e^{6} \mathrm {sgn}\left (b x + a\right ) + 112 \, a^{3} b^{3} d x^{2} e^{5} \mathrm {sgn}\left (b x + a\right ) + 32 \, a^{3} b^{3} d^{2} x e^{4} \mathrm {sgn}\left (b x + a\right ) + 4 \, a^{3} b^{3} d^{3} e^{3} \mathrm {sgn}\left (b x + a\right ) + 140 \, a^{4} b^{2} x^{2} e^{6} \mathrm {sgn}\left (b x + a\right ) + 40 \, a^{4} b^{2} d x e^{5} \mathrm {sgn}\left (b x + a\right ) + 5 \, a^{4} b^{2} d^{2} e^{4} \mathrm {sgn}\left (b x + a\right ) + 48 \, a^{5} b x e^{6} \mathrm {sgn}\left (b x + a\right ) + 6 \, a^{5} b d e^{5} \mathrm {sgn}\left (b x + a\right ) + 7 \, a^{6} e^{6} \mathrm {sgn}\left (b x + a\right )\right )} e^{\left (-7\right )}}{56 \, {\left (x e + d\right )}^{8}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.05, size = 392, normalized size = 4.00 \begin {gather*} -\frac {\left (28 b^{6} e^{6} x^{6}+112 a \,b^{5} e^{6} x^{5}+56 b^{6} d \,e^{5} x^{5}+210 a^{2} b^{4} e^{6} x^{4}+140 a \,b^{5} d \,e^{5} x^{4}+70 b^{6} d^{2} e^{4} x^{4}+224 a^{3} b^{3} e^{6} x^{3}+168 a^{2} b^{4} d \,e^{5} x^{3}+112 a \,b^{5} d^{2} e^{4} x^{3}+56 b^{6} d^{3} e^{3} x^{3}+140 a^{4} b^{2} e^{6} x^{2}+112 a^{3} b^{3} d \,e^{5} x^{2}+84 a^{2} b^{4} d^{2} e^{4} x^{2}+56 a \,b^{5} d^{3} e^{3} x^{2}+28 b^{6} d^{4} e^{2} x^{2}+48 a^{5} b \,e^{6} x +40 a^{4} b^{2} d \,e^{5} x +32 a^{3} b^{3} d^{2} e^{4} x +24 a^{2} b^{4} d^{3} e^{3} x +16 a \,b^{5} d^{4} e^{2} x +8 b^{6} d^{5} e x +7 a^{6} e^{6}+6 a^{5} b d \,e^{5}+5 a^{4} b^{2} d^{2} e^{4}+4 a^{3} b^{3} d^{3} e^{3}+3 a^{2} b^{4} d^{4} e^{2}+2 a \,b^{5} d^{5} e +b^{6} d^{6}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}}}{56 \left (e x +d \right )^{8} \left (b x +a \right )^{5} e^{7}} \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 = 2.30, size = 1010, normalized size = 10.31 \begin {gather*} \frac {\left (\frac {-6\,a^5\,b\,e^5+15\,a^4\,b^2\,d\,e^4-20\,a^3\,b^3\,d^2\,e^3+15\,a^2\,b^4\,d^3\,e^2-6\,a\,b^5\,d^4\,e+b^6\,d^5}{7\,e^7}+\frac {d\,\left (\frac {15\,a^4\,b^2\,e^5-20\,a^3\,b^3\,d\,e^4+15\,a^2\,b^4\,d^2\,e^3-6\,a\,b^5\,d^3\,e^2+b^6\,d^4\,e}{7\,e^7}-\frac {d\,\left (\frac {20\,a^3\,b^3\,e^5-15\,a^2\,b^4\,d\,e^4+6\,a\,b^5\,d^2\,e^3-b^6\,d^3\,e^2}{7\,e^7}-\frac {d\,\left (\frac {d\,\left (\frac {b^6\,d}{7\,e^3}-\frac {b^5\,\left (6\,a\,e-b\,d\right )}{7\,e^3}\right )}{e}+\frac {b^4\,\left (15\,a^2\,e^2-6\,a\,b\,d\,e+b^2\,d^2\right )}{7\,e^4}\right )}{e}\right )}{e}\right )}{e}\right )\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{\left (a+b\,x\right )\,{\left (d+e\,x\right )}^7}-\frac {\left (\frac {15\,a^2\,b^4\,e^2-24\,a\,b^5\,d\,e+10\,b^6\,d^2}{4\,e^7}+\frac {d\,\left (\frac {b^6\,d}{4\,e^6}-\frac {b^5\,\left (3\,a\,e-2\,b\,d\right )}{2\,e^6}\right )}{e}\right )\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{\left (a+b\,x\right )\,{\left (d+e\,x\right )}^4}-\frac {\left (\frac {a^6}{8\,e}-\frac {d\,\left (\frac {d\,\left (\frac {d\,\left (\frac {d\,\left (\frac {d\,\left (\frac {3\,a\,b^5}{4\,e}-\frac {b^6\,d}{8\,e^2}\right )}{e}-\frac {15\,a^2\,b^4}{8\,e}\right )}{e}+\frac {5\,a^3\,b^3}{2\,e}\right )}{e}-\frac {15\,a^4\,b^2}{8\,e}\right )}{e}+\frac {3\,a^5\,b}{4\,e}\right )}{e}\right )\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{\left (a+b\,x\right )\,{\left (d+e\,x\right )}^8}-\frac {\left (\frac {15\,a^4\,b^2\,e^4-40\,a^3\,b^3\,d\,e^3+45\,a^2\,b^4\,d^2\,e^2-24\,a\,b^5\,d^3\,e+5\,b^6\,d^4}{6\,e^7}+\frac {d\,\left (\frac {-20\,a^3\,b^3\,e^4+30\,a^2\,b^4\,d\,e^3-18\,a\,b^5\,d^2\,e^2+4\,b^6\,d^3\,e}{6\,e^7}+\frac {d\,\left (\frac {d\,\left (\frac {b^6\,d}{6\,e^4}-\frac {b^5\,\left (3\,a\,e-b\,d\right )}{3\,e^4}\right )}{e}+\frac {b^4\,\left (5\,a^2\,e^2-4\,a\,b\,d\,e+b^2\,d^2\right )}{2\,e^5}\right )}{e}\right )}{e}\right )\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{\left (a+b\,x\right )\,{\left (d+e\,x\right )}^6}+\frac {\left (\frac {5\,b^6\,d-6\,a\,b^5\,e}{3\,e^7}+\frac {b^6\,d}{3\,e^7}\right )\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{\left (a+b\,x\right )\,{\left (d+e\,x\right )}^3}+\frac {\left (\frac {-20\,a^3\,b^3\,e^3+45\,a^2\,b^4\,d\,e^2-36\,a\,b^5\,d^2\,e+10\,b^6\,d^3}{5\,e^7}+\frac {d\,\left (\frac {d\,\left (\frac {b^6\,d}{5\,e^5}-\frac {3\,b^5\,\left (2\,a\,e-b\,d\right )}{5\,e^5}\right )}{e}+\frac {3\,b^4\,\left (5\,a^2\,e^2-6\,a\,b\,d\,e+2\,b^2\,d^2\right )}{5\,e^6}\right )}{e}\right )\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{\left (a+b\,x\right )\,{\left (d+e\,x\right )}^5}-\frac {b^6\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{2\,e^7\,\left (a+b\,x\right )\,{\left (d+e\,x\right )}^2} \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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