Optimal. Leaf size=162 \[ -\frac {(d+e x)^3}{b \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {3 e (a+b x) (d+e x)^2}{2 b^2 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {3 e (a+b x) (b d-a e)^2 \log (a+b x)}{b^4 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {3 e^2 x (a+b x) (b d-a e)}{b^3 \sqrt {a^2+2 a b x+b^2 x^2}} \]
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Rubi [A] time = 0.09, antiderivative size = 162, 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 = {768, 646, 43} \begin {gather*} \frac {3 e^2 x (a+b x) (b d-a e)}{b^3 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {(d+e x)^3}{b \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {3 e (a+b x) (d+e x)^2}{2 b^2 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {3 e (a+b x) (b d-a e)^2 \log (a+b x)}{b^4 \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
Rule 768
Rubi steps
\begin {align*} \int \frac {(a+b x) (d+e x)^3}{\left (a^2+2 a b x+b^2 x^2\right )^{3/2}} \, dx &=-\frac {(d+e x)^3}{b \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {(3 e) \int \frac {(d+e x)^2}{\sqrt {a^2+2 a b x+b^2 x^2}} \, dx}{b}\\ &=-\frac {(d+e x)^3}{b \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {\left (3 e \left (a b+b^2 x\right )\right ) \int \frac {(d+e x)^2}{a b+b^2 x} \, dx}{b \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=-\frac {(d+e x)^3}{b \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {\left (3 e \left (a b+b^2 x\right )\right ) \int \left (\frac {e (b d-a e)}{b^3}+\frac {(b d-a e)^2}{b^2 \left (a b+b^2 x\right )}+\frac {e (d+e x)}{b^2}\right ) \, dx}{b \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=\frac {3 e^2 (b d-a e) x (a+b x)}{b^3 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {3 e (a+b x) (d+e x)^2}{2 b^2 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {(d+e x)^3}{b \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {3 e (b d-a e)^2 (a+b x) \log (a+b x)}{b^4 \sqrt {a^2+2 a b x+b^2 x^2}}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 120, normalized size = 0.74 \begin {gather*} \frac {2 a^3 e^3-2 a^2 b e^2 (3 d+2 e x)+3 a b^2 e \left (2 d^2+2 d e x-e^2 x^2\right )+6 e (a+b x) (b d-a e)^2 \log (a+b x)+b^3 \left (-2 d^3+6 d e^2 x^2+e^3 x^3\right )}{2 b^4 \sqrt {(a+b x)^2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [B] time = 1.91, size = 2469, normalized size = 15.24 \begin {gather*} \text {Result too large to show} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.45, size = 173, normalized size = 1.07 \begin {gather*} \frac {b^{3} e^{3} x^{3} - 2 \, b^{3} d^{3} + 6 \, a b^{2} d^{2} e - 6 \, a^{2} b d e^{2} + 2 \, a^{3} e^{3} + 3 \, {\left (2 \, b^{3} d e^{2} - a b^{2} e^{3}\right )} x^{2} + 2 \, {\left (3 \, a b^{2} d e^{2} - 2 \, a^{2} b e^{3}\right )} x + 6 \, {\left (a b^{2} d^{2} e - 2 \, a^{2} b d e^{2} + a^{3} e^{3} + {\left (b^{3} d^{2} e - 2 \, a b^{2} d e^{2} + a^{2} b e^{3}\right )} x\right )} \log \left (b x + a\right )}{2 \, {\left (b^{5} x + a b^{4}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.34, size = 185, normalized size = 1.14 \begin {gather*} \frac {1}{2} \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} {\left (\frac {x e^{3}}{b^{3}} + \frac {6 \, b^{7} d e^{2} - 5 \, a b^{6} e^{3}}{b^{10}}\right )} - \frac {{\left (b^{2} d^{2} e - 2 \, a b d e^{2} + a^{2} e^{3}\right )} \log \left ({\left | -3 \, {\left (x {\left | b \right |} - \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}}\right )}^{2} a b - a^{3} b - {\left (x {\left | b \right |} - \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}}\right )}^{3} {\left | b \right |} - 3 \, {\left (x {\left | b \right |} - \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}}\right )} a^{2} {\left | b \right |} \right |}\right )}{b^{3} {\left | b \right |}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.07, size = 209, normalized size = 1.29 \begin {gather*} \frac {\left (b^{3} e^{3} x^{3}+6 a^{2} b \,e^{3} x \ln \left (b x +a \right )-12 a \,b^{2} d \,e^{2} x \ln \left (b x +a \right )-3 a \,b^{2} e^{3} x^{2}+6 b^{3} d^{2} e x \ln \left (b x +a \right )+6 b^{3} d \,e^{2} x^{2}+6 a^{3} e^{3} \ln \left (b x +a \right )-12 a^{2} b d \,e^{2} \ln \left (b x +a \right )-4 a^{2} b \,e^{3} x +6 a \,b^{2} d^{2} e \ln \left (b x +a \right )+6 a \,b^{2} d \,e^{2} x +2 a^{3} e^{3}-6 a^{2} b d \,e^{2}+6 a \,b^{2} d^{2} e -2 b^{3} d^{3}\right ) \left (b x +a \right )^{2}}{2 \left (\left (b x +a \right )^{2}\right )^{\frac {3}{2}} b^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.63, size = 481, normalized size = 2.97 \begin {gather*} \frac {e^{3} x^{3}}{2 \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} b} - \frac {5 \, a e^{3} x^{2}}{2 \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} b^{2}} + \frac {6 \, a^{2} e^{3} \log \left (x + \frac {a}{b}\right )}{b^{4}} - \frac {5 \, a^{3} e^{3}}{\sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} b^{4}} + \frac {{\left (3 \, b d e^{2} + a e^{3}\right )} x^{2}}{\sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} b^{2}} + \frac {12 \, a^{3} e^{3} x}{b^{5} {\left (x + \frac {a}{b}\right )}^{2}} - \frac {a d^{3}}{2 \, b^{3} {\left (x + \frac {a}{b}\right )}^{2}} + \frac {23 \, a^{4} e^{3}}{2 \, b^{6} {\left (x + \frac {a}{b}\right )}^{2}} - \frac {3 \, {\left (3 \, b d e^{2} + a e^{3}\right )} a \log \left (x + \frac {a}{b}\right )}{b^{4}} + \frac {3 \, {\left (b d^{2} e + a d e^{2}\right )} \log \left (x + \frac {a}{b}\right )}{b^{3}} + \frac {2 \, {\left (3 \, b d e^{2} + a e^{3}\right )} a^{2}}{\sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} b^{4}} - \frac {b d^{3} + 3 \, a d^{2} e}{\sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} b^{2}} - \frac {6 \, {\left (3 \, b d e^{2} + a e^{3}\right )} a^{2} x}{b^{5} {\left (x + \frac {a}{b}\right )}^{2}} + \frac {6 \, {\left (b d^{2} e + a d e^{2}\right )} a x}{b^{4} {\left (x + \frac {a}{b}\right )}^{2}} - \frac {11 \, {\left (3 \, b d e^{2} + a e^{3}\right )} a^{3}}{2 \, b^{6} {\left (x + \frac {a}{b}\right )}^{2}} + \frac {9 \, {\left (b d^{2} e + a d e^{2}\right )} a^{2}}{2 \, b^{5} {\left (x + \frac {a}{b}\right )}^{2}} + \frac {{\left (b d^{3} + 3 \, a d^{2} e\right )} a}{2 \, b^{4} {\left (x + \frac {a}{b}\right )}^{2}} \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 = -0.01 \begin {gather*} \int \frac {\left (a+b\,x\right )\,{\left (d+e\,x\right )}^3}{{\left (a^2+2\,a\,b\,x+b^2\,x^2\right )}^{3/2}} \,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 (d + e x\right )^{3}}{\left (\left (a + b x\right )^{2}\right )^{\frac {3}{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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