Optimal. Leaf size=193 \[ \frac {b \sqrt {a^2+2 a b x+b^2 x^2} (a+b x)^3 (-3 a B e+A b e+2 b B d)}{60 e (d+e x)^4 (b d-a e)^3}+\frac {\sqrt {a^2+2 a b x+b^2 x^2} (a+b x)^3 (-3 a B e+A b e+2 b B d)}{15 e (d+e x)^5 (b d-a e)^2}-\frac {\sqrt {a^2+2 a b x+b^2 x^2} (a+b x)^3 (B d-A e)}{6 e (d+e x)^6 (b d-a e)} \]
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Rubi [A] time = 0.15, antiderivative size = 193, 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, 78, 45, 37} \begin {gather*} \frac {b \sqrt {a^2+2 a b x+b^2 x^2} (a+b x)^3 (-3 a B e+A b e+2 b B d)}{60 e (d+e x)^4 (b d-a e)^3}+\frac {\sqrt {a^2+2 a b x+b^2 x^2} (a+b x)^3 (-3 a B e+A b e+2 b B d)}{15 e (d+e x)^5 (b d-a e)^2}-\frac {\sqrt {a^2+2 a b x+b^2 x^2} (a+b x)^3 (B d-A e)}{6 e (d+e x)^6 (b d-a e)} \end {gather*}
Antiderivative was successfully verified.
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Rule 37
Rule 45
Rule 78
Rule 770
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)^7} \, dx &=\frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \frac {\left (a b+b^2 x\right )^3 (A+B x)}{(d+e x)^7} \, dx}{b^2 \left (a b+b^2 x\right )}\\ &=-\frac {(B d-A e) (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}{6 e (b d-a e) (d+e x)^6}+\frac {\left ((2 b B d+A b e-3 a B e) \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)^6} \, dx}{3 b^2 e (b d-a e) \left (a b+b^2 x\right )}\\ &=-\frac {(B d-A e) (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}{6 e (b d-a e) (d+e x)^6}+\frac {(2 b B d+A b e-3 a B e) (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}{15 e (b d-a e)^2 (d+e x)^5}+\frac {\left ((2 b B d+A b e-3 a B e) \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}{15 b e (b d-a e)^2 \left (a b+b^2 x\right )}\\ &=-\frac {(B d-A e) (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}{6 e (b d-a e) (d+e x)^6}+\frac {(2 b B d+A b e-3 a B e) (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}{15 e (b d-a e)^2 (d+e x)^5}+\frac {b (2 b B d+A b e-3 a B e) (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}{60 e (b d-a e)^3 (d+e x)^4}\\ \end {align*}
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Mathematica [A] time = 0.10, size = 229, normalized size = 1.19 \begin {gather*} -\frac {\sqrt {(a+b x)^2} \left (2 a^3 e^3 (5 A e+B (d+6 e x))+3 a^2 b e^2 \left (2 A e (d+6 e x)+B \left (d^2+6 d e x+15 e^2 x^2\right )\right )+3 a b^2 e \left (A e \left (d^2+6 d e x+15 e^2 x^2\right )+B \left (d^3+6 d^2 e x+15 d e^2 x^2+20 e^3 x^3\right )\right )+b^3 \left (A e \left (d^3+6 d^2 e x+15 d e^2 x^2+20 e^3 x^3\right )+2 B \left (d^4+6 d^3 e x+15 d^2 e^2 x^2+20 d e^3 x^3+15 e^4 x^4\right )\right )\right )}{60 e^5 (a+b x) (d+e x)^6} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 180.03, 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 = 0.42, size = 317, normalized size = 1.64 \begin {gather*} -\frac {30 \, B b^{3} e^{4} x^{4} + 2 \, B b^{3} d^{4} + 10 \, A a^{3} e^{4} + {\left (3 \, B a b^{2} + A b^{3}\right )} d^{3} e + 3 \, {\left (B a^{2} b + A a b^{2}\right )} d^{2} e^{2} + 2 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} d e^{3} + 20 \, {\left (2 \, B b^{3} d e^{3} + {\left (3 \, B a b^{2} + A b^{3}\right )} e^{4}\right )} x^{3} + 15 \, {\left (2 \, B b^{3} d^{2} e^{2} + {\left (3 \, B a b^{2} + A b^{3}\right )} d e^{3} + 3 \, {\left (B a^{2} b + A a b^{2}\right )} e^{4}\right )} x^{2} + 6 \, {\left (2 \, B b^{3} d^{3} e + {\left (3 \, B a b^{2} + A b^{3}\right )} d^{2} e^{2} + 3 \, {\left (B a^{2} b + A a b^{2}\right )} d e^{3} + 2 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} e^{4}\right )} x}{60 \, {\left (e^{11} x^{6} + 6 \, d e^{10} x^{5} + 15 \, d^{2} e^{9} x^{4} + 20 \, d^{3} e^{8} x^{3} + 15 \, d^{4} e^{7} x^{2} + 6 \, d^{5} e^{6} x + d^{6} e^{5}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.18, size = 426, normalized size = 2.21 \begin {gather*} -\frac {{\left (30 \, 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 ) + 30 \, B b^{3} d^{2} x^{2} e^{2} \mathrm {sgn}\left (b x + a\right ) + 12 \, B b^{3} d^{3} x e \mathrm {sgn}\left (b x + a\right ) + 2 \, B b^{3} d^{4} \mathrm {sgn}\left (b x + a\right ) + 60 \, B a b^{2} x^{3} e^{4} \mathrm {sgn}\left (b x + a\right ) + 20 \, A b^{3} x^{3} e^{4} \mathrm {sgn}\left (b x + a\right ) + 45 \, B a b^{2} d x^{2} e^{3} \mathrm {sgn}\left (b x + a\right ) + 15 \, A b^{3} d x^{2} e^{3} \mathrm {sgn}\left (b x + a\right ) + 18 \, B a b^{2} d^{2} x e^{2} \mathrm {sgn}\left (b x + a\right ) + 6 \, 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 ) + 45 \, B a^{2} b x^{2} e^{4} \mathrm {sgn}\left (b x + a\right ) + 45 \, A a b^{2} x^{2} e^{4} \mathrm {sgn}\left (b x + a\right ) + 18 \, B a^{2} b d x e^{3} \mathrm {sgn}\left (b x + a\right ) + 18 \, A a b^{2} d x e^{3} \mathrm {sgn}\left (b x + a\right ) + 3 \, B a^{2} b d^{2} e^{2} \mathrm {sgn}\left (b x + a\right ) + 3 \, A a b^{2} d^{2} e^{2} \mathrm {sgn}\left (b x + a\right ) + 12 \, B a^{3} x e^{4} \mathrm {sgn}\left (b x + a\right ) + 36 \, A a^{2} b x e^{4} \mathrm {sgn}\left (b x + a\right ) + 2 \, B a^{3} d e^{3} \mathrm {sgn}\left (b x + a\right ) + 6 \, A a^{2} b d e^{3} \mathrm {sgn}\left (b x + a\right ) + 10 \, A a^{3} e^{4} \mathrm {sgn}\left (b x + a\right )\right )} e^{\left (-5\right )}}{60 \, {\left (x e + d\right )}^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.05, size = 316, normalized size = 1.64 \begin {gather*} -\frac {\left (30 B \,b^{3} e^{4} x^{4}+20 A \,b^{3} e^{4} x^{3}+60 B a \,b^{2} e^{4} x^{3}+40 B \,b^{3} d \,e^{3} x^{3}+45 A a \,b^{2} e^{4} x^{2}+15 A \,b^{3} d \,e^{3} x^{2}+45 B \,a^{2} b \,e^{4} x^{2}+45 B a \,b^{2} d \,e^{3} x^{2}+30 B \,b^{3} d^{2} e^{2} x^{2}+36 A \,a^{2} b \,e^{4} x +18 A a \,b^{2} d \,e^{3} x +6 A \,b^{3} d^{2} e^{2} x +12 B \,a^{3} e^{4} x +18 B \,a^{2} b d \,e^{3} x +18 B a \,b^{2} d^{2} e^{2} x +12 B \,b^{3} d^{3} e x +10 A \,a^{3} e^{4}+6 A \,a^{2} b d \,e^{3}+3 A a \,b^{2} d^{2} e^{2}+A \,b^{3} d^{3} e +2 B \,a^{3} d \,e^{3}+3 B \,a^{2} b \,d^{2} e^{2}+3 B a \,b^{2} d^{3} e +2 B \,b^{3} d^{4}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {3}{2}}}{60 \left (e x +d \right )^{6} \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 = 2.17, size = 577, normalized size = 2.99 \begin {gather*} -\frac {\left (\frac {A\,b^3\,e-3\,B\,b^3\,d+3\,B\,a\,b^2\,e}{3\,e^5}-\frac {B\,b^3\,d}{3\,e^5}\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 {A\,a^3}{6\,e}-\frac {d\,\left (\frac {B\,a^3+3\,A\,b\,a^2}{6\,e}+\frac {d\,\left (\frac {d\,\left (\frac {A\,b^3+3\,B\,a\,b^2}{6\,e}-\frac {B\,b^3\,d}{6\,e^2}\right )}{e}-\frac {a\,b\,\left (A\,b+B\,a\right )}{2\,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 )}^6}-\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}{5\,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}{5\,e^5}-\frac {d\,\left (\frac {b^2\,\left (A\,b\,e+3\,B\,a\,e-B\,b\,d\right )}{5\,e^3}-\frac {B\,b^3\,d}{5\,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 )}^5}-\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}{4\,e^5}-\frac {d\,\left (\frac {b^2\,\left (A\,b\,e+3\,B\,a\,e-2\,B\,b\,d\right )}{4\,e^4}-\frac {B\,b^3\,d}{4\,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 )}^4}-\frac {B\,b^3\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{2\,e^5\,\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] 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 )^{7}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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