Optimal. Leaf size=222 \[ \frac {(a+b x) (d+e x)^3 (b d-a e)}{3 b^2 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {(a+b x) (d+e x)^4}{4 b \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {(a+b x) (b d-a e)^4 \log (a+b x)}{b^5 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {e x (a+b x) (b d-a e)^3}{b^4 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {(a+b x) (d+e x)^2 (b d-a e)^2}{2 b^3 \sqrt {a^2+2 a b x+b^2 x^2}} \]
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Rubi [A] time = 0.08, antiderivative size = 222, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {646, 43} \begin {gather*} \frac {e x (a+b x) (b d-a e)^3}{b^4 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {(a+b x) (d+e x)^2 (b d-a e)^2}{2 b^3 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {(a+b x) (d+e x)^3 (b d-a e)}{3 b^2 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {(a+b x) (d+e x)^4}{4 b \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {(a+b x) (b d-a e)^4 \log (a+b x)}{b^5 \sqrt {a^2+2 a b x+b^2 x^2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 43
Rule 646
Rubi steps
\begin {align*} \int \frac {(d+e x)^4}{\sqrt {a^2+2 a b x+b^2 x^2}} \, dx &=\frac {\left (a b+b^2 x\right ) \int \frac {(d+e x)^4}{a b+b^2 x} \, dx}{\sqrt {a^2+2 a b x+b^2 x^2}}\\ &=\frac {\left (a b+b^2 x\right ) \int \left (\frac {e (b d-a e)^3}{b^5}+\frac {(b d-a e)^4}{b^4 \left (a b+b^2 x\right )}+\frac {e (b d-a e)^2 (d+e x)}{b^4}+\frac {e (b d-a e) (d+e x)^2}{b^3}+\frac {e (d+e x)^3}{b^2}\right ) \, dx}{\sqrt {a^2+2 a b x+b^2 x^2}}\\ &=\frac {e (b d-a e)^3 x (a+b x)}{b^4 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {(b d-a e)^2 (a+b x) (d+e x)^2}{2 b^3 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {(b d-a e) (a+b x) (d+e x)^3}{3 b^2 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {(a+b x) (d+e x)^4}{4 b \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {(b d-a e)^4 (a+b x) \log (a+b x)}{b^5 \sqrt {a^2+2 a b x+b^2 x^2}}\\ \end {align*}
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Mathematica [A] time = 0.08, size = 130, normalized size = 0.59 \begin {gather*} \frac {(a+b x) \left (b e x \left (-12 a^3 e^3+6 a^2 b e^2 (8 d+e x)-4 a b^2 e \left (18 d^2+6 d e x+e^2 x^2\right )+b^3 \left (48 d^3+36 d^2 e x+16 d e^2 x^2+3 e^3 x^3\right )\right )+12 (b d-a e)^4 \log (a+b x)\right )}{12 b^5 \sqrt {(a+b x)^2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [B] time = 1.06, size = 634, normalized size = 2.86 \begin {gather*} \frac {12 a^3 e^4 x-48 a^2 b d e^3 x-6 a^2 b e^4 x^2+72 a b^2 d^2 e^2 x+24 a b^2 d e^3 x^2+4 a b^2 e^4 x^3-48 b^3 d^3 e x-36 b^3 d^2 e^2 x^2-16 b^3 d e^3 x^3-3 b^3 e^4 x^4}{24 b^3 \sqrt {b^2}}+\frac {\sqrt {a^2+2 a b x+b^2 x^2} \left (-25 a^3 e^4+88 a^2 b d e^3+13 a^2 b e^4 x-108 a b^2 d^2 e^2-40 a b^2 d e^3 x-7 a b^2 e^4 x^2+48 b^3 d^3 e+36 b^3 d^2 e^2 x+16 b^3 d e^3 x^2+3 b^3 e^4 x^3\right )}{24 b^5}+\frac {\left (-a^4 \sqrt {b^2} e^4-a^4 b e^4+4 a^3 b^2 d e^3+4 a^3 \sqrt {b^2} b d e^3-6 a^2 b^3 d^2 e^2-6 a^2 \left (b^2\right )^{3/2} d^2 e^2+4 a b^4 d^3 e+4 a \sqrt {b^2} b^3 d^3 e+b^5 \left (-d^4\right )-\sqrt {b^2} b^4 d^4\right ) \log \left (\sqrt {a^2+2 a b x+b^2 x^2}-a-\sqrt {b^2} x\right )}{2 b^5 \sqrt {b^2}}+\frac {\left (a^4 \sqrt {b^2} e^4-a^4 b e^4+4 a^3 b^2 d e^3-4 a^3 \sqrt {b^2} b d e^3-6 a^2 b^3 d^2 e^2+6 a^2 \left (b^2\right )^{3/2} d^2 e^2+4 a b^4 d^3 e-4 a \sqrt {b^2} b^3 d^3 e+b^5 \left (-d^4\right )+\sqrt {b^2} b^4 d^4\right ) \log \left (\sqrt {a^2+2 a b x+b^2 x^2}+a-\sqrt {b^2} x\right )}{2 b^5 \sqrt {b^2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.40, size = 181, normalized size = 0.82 \begin {gather*} \frac {3 \, b^{4} e^{4} x^{4} + 4 \, {\left (4 \, b^{4} d e^{3} - a b^{3} e^{4}\right )} x^{3} + 6 \, {\left (6 \, b^{4} d^{2} e^{2} - 4 \, a b^{3} d e^{3} + a^{2} b^{2} e^{4}\right )} x^{2} + 12 \, {\left (4 \, b^{4} d^{3} e - 6 \, a b^{3} d^{2} e^{2} + 4 \, a^{2} b^{2} d e^{3} - a^{3} b e^{4}\right )} x + 12 \, {\left (b^{4} d^{4} - 4 \, a b^{3} d^{3} e + 6 \, a^{2} b^{2} d^{2} e^{2} - 4 \, a^{3} b d e^{3} + a^{4} e^{4}\right )} \log \left (b x + a\right )}{12 \, b^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.18, size = 264, normalized size = 1.19 \begin {gather*} \frac {3 \, b^{3} x^{4} e^{4} \mathrm {sgn}\left (b x + a\right ) + 16 \, b^{3} d x^{3} e^{3} \mathrm {sgn}\left (b x + a\right ) + 36 \, b^{3} d^{2} x^{2} e^{2} \mathrm {sgn}\left (b x + a\right ) + 48 \, b^{3} d^{3} x e \mathrm {sgn}\left (b x + a\right ) - 4 \, a b^{2} x^{3} e^{4} \mathrm {sgn}\left (b x + a\right ) - 24 \, a b^{2} d x^{2} e^{3} \mathrm {sgn}\left (b x + a\right ) - 72 \, a b^{2} d^{2} x e^{2} \mathrm {sgn}\left (b x + a\right ) + 6 \, a^{2} b x^{2} e^{4} \mathrm {sgn}\left (b x + a\right ) + 48 \, a^{2} b d x e^{3} \mathrm {sgn}\left (b x + a\right ) - 12 \, a^{3} x e^{4} \mathrm {sgn}\left (b x + a\right )}{12 \, b^{4}} + \frac {{\left (b^{4} d^{4} \mathrm {sgn}\left (b x + a\right ) - 4 \, a b^{3} d^{3} e \mathrm {sgn}\left (b x + a\right ) + 6 \, a^{2} b^{2} d^{2} e^{2} \mathrm {sgn}\left (b x + a\right ) - 4 \, a^{3} b d e^{3} \mathrm {sgn}\left (b x + a\right ) + a^{4} e^{4} \mathrm {sgn}\left (b x + a\right )\right )} \log \left ({\left | b x + a \right |}\right )}{b^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 223, normalized size = 1.00 \begin {gather*} \frac {\left (b x +a \right ) \left (3 b^{4} e^{4} x^{4}-4 a \,b^{3} e^{4} x^{3}+16 b^{4} d \,e^{3} x^{3}+6 a^{2} b^{2} e^{4} x^{2}-24 a \,b^{3} d \,e^{3} x^{2}+36 b^{4} d^{2} e^{2} x^{2}+12 a^{4} e^{4} \ln \left (b x +a \right )-48 a^{3} b d \,e^{3} \ln \left (b x +a \right )-12 a^{3} b \,e^{4} x +72 a^{2} b^{2} d^{2} e^{2} \ln \left (b x +a \right )+48 a^{2} b^{2} d \,e^{3} x -48 a \,b^{3} d^{3} e \ln \left (b x +a \right )-72 a \,b^{3} d^{2} e^{2} x +12 b^{4} d^{4} \ln \left (b x +a \right )+48 b^{4} d^{3} e x \right )}{12 \sqrt {\left (b x +a \right )^{2}}\, b^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.07, size = 348, normalized size = 1.57 \begin {gather*} \frac {\sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} e^{4} x^{3}}{4 \, b^{2}} + \frac {3 \, d^{2} e^{2} x^{2}}{b} - \frac {10 \, a d e^{3} x^{2}}{3 \, b^{2}} + \frac {13 \, a^{2} e^{4} x^{2}}{12 \, b^{3}} + \frac {4 \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} d e^{3} x^{2}}{3 \, b^{2}} - \frac {7 \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} a e^{4} x^{2}}{12 \, b^{3}} - \frac {6 \, a d^{2} e^{2} x}{b^{2}} + \frac {20 \, a^{2} d e^{3} x}{3 \, b^{3}} - \frac {13 \, a^{3} e^{4} x}{6 \, b^{4}} + \frac {d^{4} \log \left (x + \frac {a}{b}\right )}{b} - \frac {4 \, a d^{3} e \log \left (x + \frac {a}{b}\right )}{b^{2}} + \frac {6 \, a^{2} d^{2} e^{2} \log \left (x + \frac {a}{b}\right )}{b^{3}} - \frac {4 \, a^{3} d e^{3} \log \left (x + \frac {a}{b}\right )}{b^{4}} + \frac {a^{4} e^{4} \log \left (x + \frac {a}{b}\right )}{b^{5}} + \frac {4 \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} d^{3} e}{b^{2}} - \frac {8 \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} a^{2} d e^{3}}{3 \, b^{4}} + \frac {7 \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} a^{3} e^{4}}{6 \, b^{5}} \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 (d+e\,x\right )}^4}{\sqrt {{\left (a+b\,x\right )}^2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.43, size = 136, normalized size = 0.61 \begin {gather*} x^{3} \left (- \frac {a e^{4}}{3 b^{2}} + \frac {4 d e^{3}}{3 b}\right ) + x^{2} \left (\frac {a^{2} e^{4}}{2 b^{3}} - \frac {2 a d e^{3}}{b^{2}} + \frac {3 d^{2} e^{2}}{b}\right ) + x \left (- \frac {a^{3} e^{4}}{b^{4}} + \frac {4 a^{2} d e^{3}}{b^{3}} - \frac {6 a d^{2} e^{2}}{b^{2}} + \frac {4 d^{3} e}{b}\right ) + \frac {e^{4} x^{4}}{4 b} + \frac {\left (a e - b d\right )^{4} \log {\left (a + b x \right )}}{b^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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