Optimal. Leaf size=169 \[ \frac {e (a+b x) (d+e x)^{m+1} (b (2 B d-A e (1-m))-a B e (m+1)) \, _2F_1\left (2,m+1;m+2;\frac {b (d+e x)}{b d-a e}\right )}{2 b (m+1) \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^3}-\frac {(A b-a B) (d+e x)^{m+1}}{2 b (a+b x) \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)} \]
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Rubi [A] time = 0.16, antiderivative size = 168, normalized size of antiderivative = 0.99, number of steps used = 3, number of rules used = 3, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {770, 78, 68} \[ \frac {e (a+b x) (d+e x)^{m+1} (-a B e (m+1)-A b e (1-m)+2 b B d) \, _2F_1\left (2,m+1;m+2;\frac {b (d+e x)}{b d-a e}\right )}{2 b (m+1) \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^3}-\frac {(A b-a B) (d+e x)^{m+1}}{2 b (a+b x) \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)} \]
Antiderivative was successfully verified.
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Rule 68
Rule 78
Rule 770
Rubi steps
\begin {align*} \int \frac {(A+B x) (d+e x)^m}{\left (a^2+2 a b x+b^2 x^2\right )^{3/2}} \, dx &=\frac {\left (b^2 \left (a b+b^2 x\right )\right ) \int \frac {(A+B x) (d+e x)^m}{\left (a b+b^2 x\right )^3} \, dx}{\sqrt {a^2+2 a b x+b^2 x^2}}\\ &=-\frac {(A b-a B) (d+e x)^{1+m}}{2 b (b d-a e) (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {\left ((2 b B d-A b e (1-m)-a B e (1+m)) \left (a b+b^2 x\right )\right ) \int \frac {(d+e x)^m}{\left (a b+b^2 x\right )^2} \, dx}{2 (b d-a e) \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=-\frac {(A b-a B) (d+e x)^{1+m}}{2 b (b d-a e) (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {e (2 b B d-A b e (1-m)-a B e (1+m)) (a+b x) (d+e x)^{1+m} \, _2F_1\left (2,1+m;2+m;\frac {b (d+e x)}{b d-a e}\right )}{2 b (b d-a e)^3 (1+m) \sqrt {a^2+2 a b x+b^2 x^2}}\\ \end {align*}
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Mathematica [A] time = 0.15, size = 120, normalized size = 0.71 \[ \frac {(a+b x) (d+e x)^{m+1} \left (\frac {e (a+b x)^2 (-a B e (m+1)+A b e (m-1)+2 b B d) \, _2F_1\left (2,m+1;m+2;\frac {b (d+e x)}{b d-a e}\right )}{(m+1) (b d-a e)^2}+a B-A b\right )}{2 b \left ((a+b x)^2\right )^{3/2} (b d-a e)} \]
Antiderivative was successfully verified.
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fricas [F] time = 1.28, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} {\left (B x + A\right )} {\left (e x + d\right )}^{m}}{b^{4} x^{4} + 4 \, a b^{3} x^{3} + 6 \, a^{2} b^{2} x^{2} + 4 \, a^{3} b x + a^{4}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (B x + A\right )} {\left (e x + d\right )}^{m}}{{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 1.94, size = 0, normalized size = 0.00 \[ \int \frac {\left (B x +A \right ) \left (e x +d \right )^{m}}{\left (b^{2} x^{2}+2 a b x +a^{2}\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (B x + A\right )} {\left (e x + d\right )}^{m}}{{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}}}\,{d x} \]
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 \[ \int \frac {\left (A+B\,x\right )\,{\left (d+e\,x\right )}^m}{{\left (a^2+2\,a\,b\,x+b^2\,x^2\right )}^{3/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: HeuristicGCDFailed} \]
Verification of antiderivative is not currently implemented for this CAS.
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