Optimal. Leaf size=106 \[ \frac {\sqrt {a^2+2 a b x+b^2 x^2} (a+b x)^3 (A b-a B)}{4 (d+e x)^4 (b d-a e)^2}+\frac {\left (a^2+2 a b x+b^2 x^2\right )^{5/2} (B d-A e)}{5 (d+e x)^5 (b d-a e)^2} \]
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Rubi [A] time = 0.06, antiderivative size = 106, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {769, 646, 37} \begin {gather*} \frac {\sqrt {a^2+2 a b x+b^2 x^2} (a+b x)^3 (A b-a B)}{4 (d+e x)^4 (b d-a e)^2}+\frac {\left (a^2+2 a b x+b^2 x^2\right )^{5/2} (B d-A e)}{5 (d+e x)^5 (b d-a e)^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 37
Rule 646
Rule 769
Rubi steps
\begin {align*} \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^{3/2}}{(d+e x)^6} \, dx &=\frac {(B d-A e) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{5 (b d-a e)^2 (d+e x)^5}+\frac {(A b-a B) \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^{3/2}}{(d+e x)^5} \, dx}{b d-a e}\\ &=\frac {(B d-A e) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{5 (b d-a e)^2 (d+e x)^5}+\frac {\left ((A b-a B) \sqrt {a^2+2 a b x+b^2 x^2}\right ) \int \frac {\left (a b+b^2 x\right )^3}{(d+e x)^5} \, dx}{b^2 (b d-a e) \left (a b+b^2 x\right )}\\ &=\frac {(A b-a B) (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}{4 (b d-a e)^2 (d+e x)^4}+\frac {(B d-A e) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{5 (b d-a e)^2 (d+e x)^5}\\ \end {align*}
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Mathematica [B] time = 0.10, size = 229, normalized size = 2.16 \begin {gather*} -\frac {\sqrt {(a+b x)^2} \left (a^3 e^3 (4 A e+B (d+5 e x))+a^2 b e^2 \left (3 A e (d+5 e x)+2 B \left (d^2+5 d e x+10 e^2 x^2\right )\right )+a b^2 e \left (2 A e \left (d^2+5 d e x+10 e^2 x^2\right )+3 B \left (d^3+5 d^2 e x+10 d e^2 x^2+10 e^3 x^3\right )\right )+b^3 \left (A e \left (d^3+5 d^2 e x+10 d e^2 x^2+10 e^3 x^3\right )+4 B \left (d^4+5 d^3 e x+10 d^2 e^2 x^2+10 d e^3 x^3+5 e^4 x^4\right )\right )\right )}{20 e^5 (a+b x) (d+e x)^5} \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 = 304, normalized size = 2.87 \begin {gather*} -\frac {20 \, B b^{3} e^{4} x^{4} + 4 \, B b^{3} d^{4} + 4 \, A a^{3} e^{4} + {\left (3 \, B a b^{2} + A b^{3}\right )} d^{3} e + 2 \, {\left (B a^{2} b + A a b^{2}\right )} d^{2} e^{2} + {\left (B a^{3} + 3 \, A a^{2} b\right )} d e^{3} + 10 \, {\left (4 \, B b^{3} d e^{3} + {\left (3 \, B a b^{2} + A b^{3}\right )} e^{4}\right )} x^{3} + 10 \, {\left (4 \, B b^{3} d^{2} e^{2} + {\left (3 \, B a b^{2} + A b^{3}\right )} d e^{3} + 2 \, {\left (B a^{2} b + A a b^{2}\right )} e^{4}\right )} x^{2} + 5 \, {\left (4 \, B b^{3} d^{3} e + {\left (3 \, B a b^{2} + A b^{3}\right )} d^{2} e^{2} + 2 \, {\left (B a^{2} b + A a b^{2}\right )} d e^{3} + {\left (B a^{3} + 3 \, A a^{2} b\right )} e^{4}\right )} x}{20 \, {\left (e^{10} x^{5} + 5 \, d e^{9} x^{4} + 10 \, d^{2} e^{8} x^{3} + 10 \, d^{3} e^{7} x^{2} + 5 \, d^{4} e^{6} x + d^{5} e^{5}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.19, size = 425, normalized size = 4.01 \begin {gather*} -\frac {{\left (20 \, B b^{3} x^{4} e^{4} \mathrm {sgn}\left (b x + a\right ) + 40 \, B b^{3} d x^{3} e^{3} \mathrm {sgn}\left (b x + a\right ) + 40 \, B b^{3} d^{2} x^{2} e^{2} \mathrm {sgn}\left (b x + a\right ) + 20 \, B b^{3} d^{3} x e \mathrm {sgn}\left (b x + a\right ) + 4 \, B b^{3} d^{4} \mathrm {sgn}\left (b x + a\right ) + 30 \, B a b^{2} x^{3} e^{4} \mathrm {sgn}\left (b x + a\right ) + 10 \, A b^{3} x^{3} e^{4} \mathrm {sgn}\left (b x + a\right ) + 30 \, B a b^{2} d x^{2} e^{3} \mathrm {sgn}\left (b x + a\right ) + 10 \, A b^{3} d x^{2} e^{3} \mathrm {sgn}\left (b x + a\right ) + 15 \, B a b^{2} d^{2} x e^{2} \mathrm {sgn}\left (b x + a\right ) + 5 \, A b^{3} d^{2} x e^{2} \mathrm {sgn}\left (b x + a\right ) + 3 \, B a b^{2} d^{3} e \mathrm {sgn}\left (b x + a\right ) + A b^{3} d^{3} e \mathrm {sgn}\left (b x + a\right ) + 20 \, B a^{2} b x^{2} e^{4} \mathrm {sgn}\left (b x + a\right ) + 20 \, A a b^{2} x^{2} e^{4} \mathrm {sgn}\left (b x + a\right ) + 10 \, B a^{2} b d x e^{3} \mathrm {sgn}\left (b x + a\right ) + 10 \, A a b^{2} d x e^{3} \mathrm {sgn}\left (b x + a\right ) + 2 \, B a^{2} b d^{2} e^{2} \mathrm {sgn}\left (b x + a\right ) + 2 \, A a b^{2} d^{2} e^{2} \mathrm {sgn}\left (b x + a\right ) + 5 \, B a^{3} x e^{4} \mathrm {sgn}\left (b x + a\right ) + 15 \, A a^{2} b x e^{4} \mathrm {sgn}\left (b x + a\right ) + B a^{3} d e^{3} \mathrm {sgn}\left (b x + a\right ) + 3 \, A a^{2} b d e^{3} \mathrm {sgn}\left (b x + a\right ) + 4 \, A a^{3} e^{4} \mathrm {sgn}\left (b x + a\right )\right )} e^{\left (-5\right )}}{20 \, {\left (x e + d\right )}^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.06, size = 315, normalized size = 2.97 \begin {gather*} -\frac {\left (20 B \,b^{3} e^{4} x^{4}+10 A \,b^{3} e^{4} x^{3}+30 B a \,b^{2} e^{4} x^{3}+40 B \,b^{3} d \,e^{3} x^{3}+20 A a \,b^{2} e^{4} x^{2}+10 A \,b^{3} d \,e^{3} x^{2}+20 B \,a^{2} b \,e^{4} x^{2}+30 B a \,b^{2} d \,e^{3} x^{2}+40 B \,b^{3} d^{2} e^{2} x^{2}+15 A \,a^{2} b \,e^{4} x +10 A a \,b^{2} d \,e^{3} x +5 A \,b^{3} d^{2} e^{2} x +5 B \,a^{3} e^{4} x +10 B \,a^{2} b d \,e^{3} x +15 B a \,b^{2} d^{2} e^{2} x +20 B \,b^{3} d^{3} e x +4 A \,a^{3} e^{4}+3 A \,a^{2} b d \,e^{3}+2 A a \,b^{2} d^{2} e^{2}+A \,b^{3} d^{3} e +B \,a^{3} d \,e^{3}+2 B \,a^{2} b \,d^{2} e^{2}+3 B a \,b^{2} d^{3} e +4 B \,b^{3} d^{4}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {3}{2}}}{20 \left (e x +d \right )^{5} \left (b x +a \right )^{3} e^{5}} \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 = 2.20, size = 577, normalized size = 5.44 \begin {gather*} -\frac {\left (\frac {A\,b^3\,e-3\,B\,b^3\,d+3\,B\,a\,b^2\,e}{2\,e^5}-\frac {B\,b^3\,d}{2\,e^5}\right )\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{\left (a+b\,x\right )\,{\left (d+e\,x\right )}^2}-\frac {\left (\frac {A\,a^3}{5\,e}-\frac {d\,\left (\frac {B\,a^3+3\,A\,b\,a^2}{5\,e}+\frac {d\,\left (\frac {d\,\left (\frac {A\,b^3+3\,B\,a\,b^2}{5\,e}-\frac {B\,b^3\,d}{5\,e^2}\right )}{e}-\frac {3\,a\,b\,\left (A\,b+B\,a\right )}{5\,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 )}^5}-\frac {\left (\frac {B\,a^3\,e^3-3\,B\,a^2\,b\,d\,e^2+3\,A\,a^2\,b\,e^3+3\,B\,a\,b^2\,d^2\,e-3\,A\,a\,b^2\,d\,e^2-B\,b^3\,d^3+A\,b^3\,d^2\,e}{4\,e^5}-\frac {d\,\left (\frac {3\,B\,a^2\,b\,e^3-3\,B\,a\,b^2\,d\,e^2+3\,A\,a\,b^2\,e^3+B\,b^3\,d^2\,e-A\,b^3\,d\,e^2}{4\,e^5}-\frac {d\,\left (\frac {b^2\,\left (A\,b\,e+3\,B\,a\,e-B\,b\,d\right )}{4\,e^3}-\frac {B\,b^3\,d}{4\,e^3}\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 )}^4}-\frac {\left (\frac {3\,B\,a^2\,b\,e^2-6\,B\,a\,b^2\,d\,e+3\,A\,a\,b^2\,e^2+3\,B\,b^3\,d^2-2\,A\,b^3\,d\,e}{3\,e^5}-\frac {d\,\left (\frac {b^2\,\left (A\,b\,e+3\,B\,a\,e-2\,B\,b\,d\right )}{3\,e^4}-\frac {B\,b^3\,d}{3\,e^4}\right )}{e}\right )\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{\left (a+b\,x\right )\,{\left (d+e\,x\right )}^3}-\frac {B\,b^3\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{e^5\,\left (a+b\,x\right )\,\left (d+e\,x\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (A + B x\right ) \left (\left (a + b x\right )^{2}\right )^{\frac {3}{2}}}{\left (d + e x\right )^{6}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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