Optimal. Leaf size=257 \[ -\frac {(a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2} (B d-A e)}{4 e (d+e x)^4 (b d-a e)}+\frac {3 b^2 B \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)}{e^5 (a+b x) (d+e x)}-\frac {3 b B \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^2}{2 e^5 (a+b x) (d+e x)^2}+\frac {B \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^3}{3 e^5 (a+b x) (d+e x)^3}+\frac {b^3 B \sqrt {a^2+2 a b x+b^2 x^2} \log (d+e x)}{e^5 (a+b x)} \]
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Rubi [A] time = 0.18, antiderivative size = 257, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {770, 78, 43} \begin {gather*} -\frac {(a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2} (B d-A e)}{4 e (d+e x)^4 (b d-a e)}+\frac {3 b^2 B \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)}{e^5 (a+b x) (d+e x)}-\frac {3 b B \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^2}{2 e^5 (a+b x) (d+e x)^2}+\frac {B \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^3}{3 e^5 (a+b x) (d+e x)^3}+\frac {b^3 B \sqrt {a^2+2 a b x+b^2 x^2} \log (d+e x)}{e^5 (a+b x)} \end {gather*}
Antiderivative was successfully verified.
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Rule 43
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)^5} \, 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)^5} \, 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}}{4 e (b d-a e) (d+e x)^4}+\frac {\left (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)^4} \, dx}{b^2 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}}{4 e (b d-a e) (d+e x)^4}+\frac {\left (B \sqrt {a^2+2 a b x+b^2 x^2}\right ) \int \left (-\frac {b^3 (b d-a e)^3}{e^3 (d+e x)^4}+\frac {3 b^4 (b d-a e)^2}{e^3 (d+e x)^3}-\frac {3 b^5 (b d-a e)}{e^3 (d+e x)^2}+\frac {b^6}{e^3 (d+e x)}\right ) \, dx}{b^2 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}}{4 e (b d-a e) (d+e x)^4}+\frac {B (b d-a e)^3 \sqrt {a^2+2 a b x+b^2 x^2}}{3 e^5 (a+b x) (d+e x)^3}-\frac {3 b B (b d-a e)^2 \sqrt {a^2+2 a b x+b^2 x^2}}{2 e^5 (a+b x) (d+e x)^2}+\frac {3 b^2 B (b d-a e) \sqrt {a^2+2 a b x+b^2 x^2}}{e^5 (a+b x) (d+e x)}+\frac {b^3 B \sqrt {a^2+2 a b x+b^2 x^2} \log (d+e x)}{e^5 (a+b x)}\\ \end {align*}
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Mathematica [A] time = 0.14, size = 240, normalized size = 0.93 \begin {gather*} -\frac {\sqrt {(a+b x)^2} \left (a^3 e^3 (3 A e+B (d+4 e x))+3 a^2 b e^2 \left (A e (d+4 e x)+B \left (d^2+4 d e x+6 e^2 x^2\right )\right )+3 a b^2 e \left (A e \left (d^2+4 d e x+6 e^2 x^2\right )+3 B \left (d^3+4 d^2 e x+6 d e^2 x^2+4 e^3 x^3\right )\right )+b^3 \left (3 A e \left (d^3+4 d^2 e x+6 d e^2 x^2+4 e^3 x^3\right )-B d \left (25 d^3+88 d^2 e x+108 d e^2 x^2+48 e^3 x^3\right )\right )-12 b^3 B (d+e x)^4 \log (d+e x)\right )}{12 e^5 (a+b x) (d+e x)^4} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 180.31, size = 0, normalized size = 0.00 \begin {gather*} \text {\$Aborted} \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 0.43, size = 354, normalized size = 1.38 \begin {gather*} \frac {25 \, B b^{3} d^{4} - 3 \, A a^{3} e^{4} - 3 \, {\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} - {\left (B a^{3} + 3 \, A a^{2} b\right )} d e^{3} + 12 \, {\left (4 \, B b^{3} d e^{3} - {\left (3 \, B a b^{2} + A b^{3}\right )} e^{4}\right )} x^{3} + 18 \, {\left (6 \, B b^{3} d^{2} e^{2} - {\left (3 \, B a b^{2} + A b^{3}\right )} d e^{3} - {\left (B a^{2} b + A a b^{2}\right )} e^{4}\right )} x^{2} + 4 \, {\left (22 \, B b^{3} d^{3} e - 3 \, {\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} - {\left (B a^{3} + 3 \, A a^{2} b\right )} e^{4}\right )} x + 12 \, {\left (B b^{3} e^{4} x^{4} + 4 \, B b^{3} d e^{3} x^{3} + 6 \, B b^{3} d^{2} e^{2} x^{2} + 4 \, B b^{3} d^{3} e x + B b^{3} d^{4}\right )} \log \left (e x + d\right )}{12 \, {\left (e^{9} x^{4} + 4 \, d e^{8} x^{3} + 6 \, d^{2} e^{7} x^{2} + 4 \, d^{3} e^{6} x + d^{4} e^{5}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.23, size = 419, normalized size = 1.63 \begin {gather*} B b^{3} e^{\left (-5\right )} \log \left ({\left | x e + d \right |}\right ) \mathrm {sgn}\left (b x + a\right ) + \frac {{\left (12 \, {\left (4 \, B b^{3} d e^{2} \mathrm {sgn}\left (b x + a\right ) - 3 \, B a b^{2} e^{3} \mathrm {sgn}\left (b x + a\right ) - A b^{3} e^{3} \mathrm {sgn}\left (b x + a\right )\right )} x^{3} + 18 \, {\left (6 \, B b^{3} d^{2} e \mathrm {sgn}\left (b x + a\right ) - 3 \, B a b^{2} d e^{2} \mathrm {sgn}\left (b x + a\right ) - A b^{3} d e^{2} \mathrm {sgn}\left (b x + a\right ) - B a^{2} b e^{3} \mathrm {sgn}\left (b x + a\right ) - A a b^{2} e^{3} \mathrm {sgn}\left (b x + a\right )\right )} x^{2} + 4 \, {\left (22 \, B b^{3} d^{3} \mathrm {sgn}\left (b x + a\right ) - 9 \, B a b^{2} d^{2} e \mathrm {sgn}\left (b x + a\right ) - 3 \, A b^{3} d^{2} e \mathrm {sgn}\left (b x + a\right ) - 3 \, B a^{2} b d e^{2} \mathrm {sgn}\left (b x + a\right ) - 3 \, A a b^{2} d e^{2} \mathrm {sgn}\left (b x + a\right ) - B a^{3} e^{3} \mathrm {sgn}\left (b x + a\right ) - 3 \, A a^{2} b e^{3} \mathrm {sgn}\left (b x + a\right )\right )} x + {\left (25 \, B b^{3} d^{4} \mathrm {sgn}\left (b x + a\right ) - 9 \, B a b^{2} d^{3} e \mathrm {sgn}\left (b x + a\right ) - 3 \, A b^{3} d^{3} e \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 ) - 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 ) - 3 \, A a^{3} e^{4} \mathrm {sgn}\left (b x + a\right )\right )} e^{\left (-1\right )}\right )} e^{\left (-4\right )}}{12 \, {\left (x e + d\right )}^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.06, size = 394, normalized size = 1.53 \begin {gather*} -\frac {\left (\left (b x +a \right )^{2}\right )^{\frac {3}{2}} \left (-12 B \,b^{3} e^{4} x^{4} \ln \left (e x +d \right )-48 B \,b^{3} d \,e^{3} x^{3} \ln \left (e x +d \right )+12 A \,b^{3} e^{4} x^{3}+36 B a \,b^{2} e^{4} x^{3}-72 B \,b^{3} d^{2} e^{2} x^{2} \ln \left (e x +d \right )-48 B \,b^{3} d \,e^{3} x^{3}+18 A a \,b^{2} e^{4} x^{2}+18 A \,b^{3} d \,e^{3} x^{2}+18 B \,a^{2} b \,e^{4} x^{2}+54 B a \,b^{2} d \,e^{3} x^{2}-48 B \,b^{3} d^{3} e x \ln \left (e x +d \right )-108 B \,b^{3} d^{2} e^{2} x^{2}+12 A \,a^{2} b \,e^{4} x +12 A a \,b^{2} d \,e^{3} x +12 A \,b^{3} d^{2} e^{2} x +4 B \,a^{3} e^{4} x +12 B \,a^{2} b d \,e^{3} x +36 B a \,b^{2} d^{2} e^{2} x -12 B \,b^{3} d^{4} \ln \left (e x +d \right )-88 B \,b^{3} d^{3} e x +3 A \,a^{3} e^{4}+3 A \,a^{2} b d \,e^{3}+3 A a \,b^{2} d^{2} e^{2}+3 A \,b^{3} d^{3} e +B \,a^{3} d \,e^{3}+3 B \,a^{2} b \,d^{2} e^{2}+9 B a \,b^{2} d^{3} e -25 B \,b^{3} d^{4}\right )}{12 \left (b x +a \right )^{3} \left (e x +d \right )^{4} 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 [F] time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {\left (A+B\,x\right )\,{\left (a^2+2\,a\,b\,x+b^2\,x^2\right )}^{3/2}}{{\left (d+e\,x\right )}^5} \,d x \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 )^{5}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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