Optimal. Leaf size=219 \[ -\frac {\sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^3 (d+e x)^{m+1}}{e^4 (m+1) (a+b x)}+\frac {3 b \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^2 (d+e x)^{m+2}}{e^4 (m+2) (a+b x)}-\frac {3 b^2 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e) (d+e x)^{m+3}}{e^4 (m+3) (a+b x)}+\frac {b^3 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{m+4}}{e^4 (m+4) (a+b x)} \]
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Rubi [A] time = 0.09, antiderivative size = 219, 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 {\sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^3 (d+e x)^{m+1}}{e^4 (m+1) (a+b x)}+\frac {3 b \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^2 (d+e x)^{m+2}}{e^4 (m+2) (a+b x)}-\frac {3 b^2 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e) (d+e x)^{m+3}}{e^4 (m+3) (a+b x)}+\frac {b^3 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{m+4}}{e^4 (m+4) (a+b x)} \end {gather*}
Antiderivative was successfully verified.
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Rule 43
Rule 646
Rubi steps
\begin {align*} \int (d+e x)^m \left (a^2+2 a b x+b^2 x^2\right )^{3/2} \, dx &=\frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \left (a b+b^2 x\right )^3 (d+e x)^m \, dx}{b^2 \left (a b+b^2 x\right )}\\ &=\frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \left (-\frac {b^3 (b d-a e)^3 (d+e x)^m}{e^3}+\frac {3 b^4 (b d-a e)^2 (d+e x)^{1+m}}{e^3}-\frac {3 b^5 (b d-a e) (d+e x)^{2+m}}{e^3}+\frac {b^6 (d+e x)^{3+m}}{e^3}\right ) \, dx}{b^2 \left (a b+b^2 x\right )}\\ &=-\frac {(b d-a e)^3 (d+e x)^{1+m} \sqrt {a^2+2 a b x+b^2 x^2}}{e^4 (1+m) (a+b x)}+\frac {3 b (b d-a e)^2 (d+e x)^{2+m} \sqrt {a^2+2 a b x+b^2 x^2}}{e^4 (2+m) (a+b x)}-\frac {3 b^2 (b d-a e) (d+e x)^{3+m} \sqrt {a^2+2 a b x+b^2 x^2}}{e^4 (3+m) (a+b x)}+\frac {b^3 (d+e x)^{4+m} \sqrt {a^2+2 a b x+b^2 x^2}}{e^4 (4+m) (a+b x)}\\ \end {align*}
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Mathematica [A] time = 0.12, size = 113, normalized size = 0.52 \begin {gather*} \frac {\left ((a+b x)^2\right )^{3/2} (d+e x)^{m+1} \left (-\frac {3 b^2 (d+e x)^2 (b d-a e)}{m+3}+\frac {3 b (d+e x) (b d-a e)^2}{m+2}-\frac {(b d-a e)^3}{m+1}+\frac {b^3 (d+e x)^3}{m+4}\right )}{e^4 (a+b x)^3} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 1.19, size = 0, normalized size = 0.00 \begin {gather*} \int (d+e x)^m \left (a^2+2 a b x+b^2 x^2\right )^{3/2} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [B] time = 0.42, size = 496, normalized size = 2.26 \begin {gather*} \frac {{\left (a^{3} d e^{3} m^{3} - 6 \, b^{3} d^{4} + 24 \, a b^{2} d^{3} e - 36 \, a^{2} b d^{2} e^{2} + 24 \, a^{3} d e^{3} + {\left (b^{3} e^{4} m^{3} + 6 \, b^{3} e^{4} m^{2} + 11 \, b^{3} e^{4} m + 6 \, b^{3} e^{4}\right )} x^{4} + {\left (24 \, a b^{2} e^{4} + {\left (b^{3} d e^{3} + 3 \, a b^{2} e^{4}\right )} m^{3} + 3 \, {\left (b^{3} d e^{3} + 7 \, a b^{2} e^{4}\right )} m^{2} + 2 \, {\left (b^{3} d e^{3} + 21 \, a b^{2} e^{4}\right )} m\right )} x^{3} - 3 \, {\left (a^{2} b d^{2} e^{2} - 3 \, a^{3} d e^{3}\right )} m^{2} + 3 \, {\left (12 \, a^{2} b e^{4} + {\left (a b^{2} d e^{3} + a^{2} b e^{4}\right )} m^{3} - {\left (b^{3} d^{2} e^{2} - 5 \, a b^{2} d e^{3} - 8 \, a^{2} b e^{4}\right )} m^{2} - {\left (b^{3} d^{2} e^{2} - 4 \, a b^{2} d e^{3} - 19 \, a^{2} b e^{4}\right )} m\right )} x^{2} + {\left (6 \, a b^{2} d^{3} e - 21 \, a^{2} b d^{2} e^{2} + 26 \, a^{3} d e^{3}\right )} m + {\left (24 \, a^{3} e^{4} + {\left (3 \, a^{2} b d e^{3} + a^{3} e^{4}\right )} m^{3} - 3 \, {\left (2 \, a b^{2} d^{2} e^{2} - 7 \, a^{2} b d e^{3} - 3 \, a^{3} e^{4}\right )} m^{2} + 2 \, {\left (3 \, b^{3} d^{3} e - 12 \, a b^{2} d^{2} e^{2} + 18 \, a^{2} b d e^{3} + 13 \, a^{3} e^{4}\right )} m\right )} x\right )} {\left (e x + d\right )}^{m}}{e^{4} m^{4} + 10 \, e^{4} m^{3} + 35 \, e^{4} m^{2} + 50 \, e^{4} m + 24 \, e^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.25, size = 1075, normalized size = 4.91
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.06, size = 402, normalized size = 1.84 \begin {gather*} \frac {\left (\left (b x +a \right )^{2}\right )^{\frac {3}{2}} \left (b^{3} e^{3} m^{3} x^{3}+3 a \,b^{2} e^{3} m^{3} x^{2}+6 b^{3} e^{3} m^{2} x^{3}+3 a^{2} b \,e^{3} m^{3} x +21 a \,b^{2} e^{3} m^{2} x^{2}-3 b^{3} d \,e^{2} m^{2} x^{2}+11 b^{3} e^{3} m \,x^{3}+a^{3} e^{3} m^{3}+24 a^{2} b \,e^{3} m^{2} x -6 a \,b^{2} d \,e^{2} m^{2} x +42 a \,b^{2} e^{3} m \,x^{2}-9 b^{3} d \,e^{2} m \,x^{2}+6 b^{3} e^{3} x^{3}+9 a^{3} e^{3} m^{2}-3 a^{2} b d \,e^{2} m^{2}+57 a^{2} b \,e^{3} m x -30 a \,b^{2} d \,e^{2} m x +24 a \,b^{2} e^{3} x^{2}+6 b^{3} d^{2} e m x -6 b^{3} d \,e^{2} x^{2}+26 a^{3} e^{3} m -21 a^{2} b d \,e^{2} m +36 a^{2} b \,e^{3} x +6 a \,b^{2} d^{2} e m -24 a \,b^{2} d \,e^{2} x +6 b^{3} d^{2} e x +24 a^{3} e^{3}-36 a^{2} b d \,e^{2}+24 a \,b^{2} d^{2} e -6 b^{3} d^{3}\right ) \left (e x +d \right )^{m +1}}{\left (b x +a \right )^{3} \left (m^{4}+10 m^{3}+35 m^{2}+50 m +24\right ) e^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.23, size = 305, normalized size = 1.39 \begin {gather*} \frac {{\left ({\left (m^{3} + 6 \, m^{2} + 11 \, m + 6\right )} b^{3} e^{4} x^{4} - 3 \, {\left (m^{2} + 7 \, m + 12\right )} a^{2} b d^{2} e^{2} + {\left (m^{3} + 9 \, m^{2} + 26 \, m + 24\right )} a^{3} d e^{3} + 6 \, a b^{2} d^{3} e {\left (m + 4\right )} - 6 \, b^{3} d^{4} + {\left ({\left (m^{3} + 3 \, m^{2} + 2 \, m\right )} b^{3} d e^{3} + 3 \, {\left (m^{3} + 7 \, m^{2} + 14 \, m + 8\right )} a b^{2} e^{4}\right )} x^{3} - 3 \, {\left ({\left (m^{2} + m\right )} b^{3} d^{2} e^{2} - {\left (m^{3} + 5 \, m^{2} + 4 \, m\right )} a b^{2} d e^{3} - {\left (m^{3} + 8 \, m^{2} + 19 \, m + 12\right )} a^{2} b e^{4}\right )} x^{2} - {\left (6 \, {\left (m^{2} + 4 \, m\right )} a b^{2} d^{2} e^{2} - 3 \, {\left (m^{3} + 7 \, m^{2} + 12 \, m\right )} a^{2} b d e^{3} - {\left (m^{3} + 9 \, m^{2} + 26 \, m + 24\right )} a^{3} e^{4} - 6 \, b^{3} d^{3} e m\right )} x\right )} {\left (e x + d\right )}^{m}}{{\left (m^{4} + 10 \, m^{3} + 35 \, m^{2} + 50 \, m + 24\right )} e^{4}} \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 {\left (d+e\,x\right )}^m\,{\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 \left (d + e x\right )^{m} \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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