Optimal. Leaf size=210 \[ \frac {\sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^4 \log (d+e x)}{e^5 (a+b x)}-\frac {b x \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^3}{e^4 (a+b x)}+\frac {(a+b x) \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^2}{2 e^3}-\frac {(a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)}{3 e^2}+\frac {(a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}{4 e} \]
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Rubi [A] time = 0.10, antiderivative size = 210, 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, 21, 43} \begin {gather*} -\frac {b x \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^3}{e^4 (a+b x)}+\frac {(a+b x) \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^2}{2 e^3}-\frac {(a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)}{3 e^2}+\frac {\sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^4 \log (d+e x)}{e^5 (a+b x)}+\frac {(a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}{4 e} \end {gather*}
Antiderivative was successfully verified.
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Rule 21
Rule 43
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} \, 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 )^3}{d+e x} \, dx}{b^2 \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)^4}{d+e x} \, dx}{a b+b^2 x}\\ &=\frac {\left (b \sqrt {a^2+2 a b x+b^2 x^2}\right ) \int \left (-\frac {b (b d-a e)^3}{e^4}+\frac {b (b d-a e)^2 (a+b x)}{e^3}-\frac {b (b d-a e) (a+b x)^2}{e^2}+\frac {b (a+b x)^3}{e}+\frac {(-b d+a e)^4}{e^4 (d+e x)}\right ) \, dx}{a b+b^2 x}\\ &=-\frac {b (b d-a e)^3 x \sqrt {a^2+2 a b x+b^2 x^2}}{e^4 (a+b x)}+\frac {(b d-a e)^2 (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}{2 e^3}-\frac {(b d-a e) (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}{3 e^2}+\frac {(a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}{4 e}+\frac {(b d-a e)^4 \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 = 0.06, size = 133, normalized size = 0.63 \begin {gather*} \frac {\sqrt {(a+b x)^2} \left (b e x \left (48 a^3 e^3+36 a^2 b e^2 (e x-2 d)+8 a b^2 e \left (6 d^2-3 d e x+2 e^2 x^2\right )+b^3 \left (-12 d^3+6 d^2 e x-4 d e^2 x^2+3 e^3 x^3\right )\right )+12 (b d-a e)^4 \log (d+e x)\right )}{12 e^5 (a+b x)} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 1.88, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{3/2}}{d+e x} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 0.41, size = 179, normalized size = 0.85 \begin {gather*} \frac {3 \, b^{4} e^{4} x^{4} - 4 \, {\left (b^{4} d e^{3} - 4 \, a b^{3} e^{4}\right )} x^{3} + 6 \, {\left (b^{4} d^{2} e^{2} - 4 \, a b^{3} d e^{3} + 6 \, a^{2} b^{2} e^{4}\right )} x^{2} - 12 \, {\left (b^{4} d^{3} e - 4 \, a b^{3} d^{2} e^{2} + 6 \, a^{2} b^{2} d e^{3} - 4 \, 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 (e x + d\right )}{12 \, e^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.17, size = 266, normalized size = 1.27 \begin {gather*} {\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 )} e^{\left (-5\right )} \log \left ({\left | x e + d \right |}\right ) + \frac {1}{12} \, {\left (3 \, b^{4} x^{4} e^{3} \mathrm {sgn}\left (b x + a\right ) - 4 \, b^{4} d x^{3} e^{2} \mathrm {sgn}\left (b x + a\right ) + 6 \, b^{4} d^{2} x^{2} e \mathrm {sgn}\left (b x + a\right ) - 12 \, b^{4} d^{3} x \mathrm {sgn}\left (b x + a\right ) + 16 \, a b^{3} x^{3} e^{3} \mathrm {sgn}\left (b x + a\right ) - 24 \, a b^{3} d x^{2} e^{2} \mathrm {sgn}\left (b x + a\right ) + 48 \, a b^{3} d^{2} x e \mathrm {sgn}\left (b x + a\right ) + 36 \, a^{2} b^{2} x^{2} e^{3} \mathrm {sgn}\left (b x + a\right ) - 72 \, a^{2} b^{2} d x e^{2} \mathrm {sgn}\left (b x + a\right ) + 48 \, a^{3} b x e^{3} \mathrm {sgn}\left (b x + a\right )\right )} e^{\left (-4\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 225, normalized size = 1.07 \begin {gather*} \frac {\left (\left (b x +a \right )^{2}\right )^{\frac {3}{2}} \left (3 b^{4} e^{4} x^{4}+16 a \,b^{3} e^{4} x^{3}-4 b^{4} d \,e^{3} x^{3}+36 a^{2} b^{2} e^{4} x^{2}-24 a \,b^{3} d \,e^{3} x^{2}+6 b^{4} d^{2} e^{2} x^{2}+12 a^{4} e^{4} \ln \left (e x +d \right )-48 a^{3} b d \,e^{3} \ln \left (e x +d \right )+48 a^{3} b \,e^{4} x +72 a^{2} b^{2} d^{2} e^{2} \ln \left (e x +d \right )-72 a^{2} b^{2} d \,e^{3} x -48 a \,b^{3} d^{3} e \ln \left (e x +d \right )+48 a \,b^{3} d^{2} e^{2} x +12 b^{4} d^{4} \ln \left (e x +d \right )-12 b^{4} d^{3} e x \right )}{12 \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 [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}}{d+e\,x} \,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}}}{d + e x}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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