Optimal. Leaf size=254 \[ -\frac {\sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^5 \log (d+e x)}{e^6 (a+b x)}+\frac {b x \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^4}{e^5 (a+b x)}-\frac {(a+b x) \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^3}{2 e^4}+\frac {(a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^2}{3 e^3}-\frac {(a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)}{4 e^2}+\frac {(a+b x)^4 \sqrt {a^2+2 a b x+b^2 x^2}}{5 e} \]
________________________________________________________________________________________
Rubi [A] time = 0.13, antiderivative size = 254, 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 {b x \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^4}{e^5 (a+b x)}-\frac {(a+b x) \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^3}{2 e^4}+\frac {(a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^2}{3 e^3}-\frac {(a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)}{4 e^2}-\frac {\sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^5 \log (d+e x)}{e^6 (a+b x)}+\frac {(a+b x)^4 \sqrt {a^2+2 a b x+b^2 x^2}}{5 e} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 43
Rule 646
Rubi steps
\begin {align*} \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{d+e x} \, dx &=\frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \frac {\left (a b+b^2 x\right )^5}{d+e x} \, dx}{b^4 \left (a b+b^2 x\right )}\\ &=\frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \left (\frac {b^6 (b d-a e)^4}{e^5}-\frac {b^5 (b d-a e)^3 \left (a b+b^2 x\right )}{e^4}+\frac {b^4 (b d-a e)^2 \left (a b+b^2 x\right )^2}{e^3}-\frac {b^3 (b d-a e) \left (a b+b^2 x\right )^3}{e^2}+\frac {b^2 \left (a b+b^2 x\right )^4}{e}-\frac {b^5 (b d-a e)^5}{e^5 (d+e x)}\right ) \, dx}{b^4 \left (a b+b^2 x\right )}\\ &=\frac {b (b d-a e)^4 x \sqrt {a^2+2 a b x+b^2 x^2}}{e^5 (a+b x)}-\frac {(b d-a e)^3 (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}{2 e^4}+\frac {(b d-a e)^2 (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}{3 e^3}-\frac {(b d-a e) (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}{4 e^2}+\frac {(a+b x)^4 \sqrt {a^2+2 a b x+b^2 x^2}}{5 e}-\frac {(b d-a e)^5 \sqrt {a^2+2 a b x+b^2 x^2} \log (d+e x)}{e^6 (a+b x)}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.09, size = 185, normalized size = 0.73 \begin {gather*} \frac {\sqrt {(a+b x)^2} \left (b e x \left (300 a^4 e^4+300 a^3 b e^3 (e x-2 d)+100 a^2 b^2 e^2 \left (6 d^2-3 d e x+2 e^2 x^2\right )+25 a b^3 e \left (-12 d^3+6 d^2 e x-4 d e^2 x^2+3 e^3 x^3\right )+b^4 \left (60 d^4-30 d^3 e x+20 d^2 e^2 x^2-15 d e^3 x^3+12 e^4 x^4\right )\right )-60 (b d-a e)^5 \log (d+e x)\right )}{60 e^6 (a+b x)} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [F] time = 2.47, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{d+e x} \, dx \end {gather*}
Verification is not applicable to the result.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.39, size = 259, normalized size = 1.02 \begin {gather*} \frac {12 \, b^{5} e^{5} x^{5} - 15 \, {\left (b^{5} d e^{4} - 5 \, a b^{4} e^{5}\right )} x^{4} + 20 \, {\left (b^{5} d^{2} e^{3} - 5 \, a b^{4} d e^{4} + 10 \, a^{2} b^{3} e^{5}\right )} x^{3} - 30 \, {\left (b^{5} d^{3} e^{2} - 5 \, a b^{4} d^{2} e^{3} + 10 \, a^{2} b^{3} d e^{4} - 10 \, a^{3} b^{2} e^{5}\right )} x^{2} + 60 \, {\left (b^{5} d^{4} e - 5 \, a b^{4} d^{3} e^{2} + 10 \, a^{2} b^{3} d^{2} e^{3} - 10 \, a^{3} b^{2} d e^{4} + 5 \, a^{4} b e^{5}\right )} x - 60 \, {\left (b^{5} d^{5} - 5 \, a b^{4} d^{4} e + 10 \, a^{2} b^{3} d^{3} e^{2} - 10 \, a^{3} b^{2} d^{2} e^{3} + 5 \, a^{4} b d e^{4} - a^{5} e^{5}\right )} \log \left (e x + d\right )}{60 \, e^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [B] time = 0.19, size = 385, normalized size = 1.52 \begin {gather*} -{\left (b^{5} d^{5} \mathrm {sgn}\left (b x + a\right ) - 5 \, a b^{4} d^{4} e \mathrm {sgn}\left (b x + a\right ) + 10 \, a^{2} b^{3} d^{3} e^{2} \mathrm {sgn}\left (b x + a\right ) - 10 \, a^{3} b^{2} d^{2} e^{3} \mathrm {sgn}\left (b x + a\right ) + 5 \, a^{4} b d e^{4} \mathrm {sgn}\left (b x + a\right ) - a^{5} e^{5} \mathrm {sgn}\left (b x + a\right )\right )} e^{\left (-6\right )} \log \left ({\left | x e + d \right |}\right ) + \frac {1}{60} \, {\left (12 \, b^{5} x^{5} e^{4} \mathrm {sgn}\left (b x + a\right ) - 15 \, b^{5} d x^{4} e^{3} \mathrm {sgn}\left (b x + a\right ) + 20 \, b^{5} d^{2} x^{3} e^{2} \mathrm {sgn}\left (b x + a\right ) - 30 \, b^{5} d^{3} x^{2} e \mathrm {sgn}\left (b x + a\right ) + 60 \, b^{5} d^{4} x \mathrm {sgn}\left (b x + a\right ) + 75 \, a b^{4} x^{4} e^{4} \mathrm {sgn}\left (b x + a\right ) - 100 \, a b^{4} d x^{3} e^{3} \mathrm {sgn}\left (b x + a\right ) + 150 \, a b^{4} d^{2} x^{2} e^{2} \mathrm {sgn}\left (b x + a\right ) - 300 \, a b^{4} d^{3} x e \mathrm {sgn}\left (b x + a\right ) + 200 \, a^{2} b^{3} x^{3} e^{4} \mathrm {sgn}\left (b x + a\right ) - 300 \, a^{2} b^{3} d x^{2} e^{3} \mathrm {sgn}\left (b x + a\right ) + 600 \, a^{2} b^{3} d^{2} x e^{2} \mathrm {sgn}\left (b x + a\right ) + 300 \, a^{3} b^{2} x^{2} e^{4} \mathrm {sgn}\left (b x + a\right ) - 600 \, a^{3} b^{2} d x e^{3} \mathrm {sgn}\left (b x + a\right ) + 300 \, a^{4} b x e^{4} \mathrm {sgn}\left (b x + a\right )\right )} e^{\left (-5\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.06, size = 318, normalized size = 1.25 \begin {gather*} \frac {\left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}} \left (12 b^{5} e^{5} x^{5}+75 a \,b^{4} e^{5} x^{4}-15 b^{5} d \,e^{4} x^{4}+200 a^{2} b^{3} e^{5} x^{3}-100 a \,b^{4} d \,e^{4} x^{3}+20 b^{5} d^{2} e^{3} x^{3}+300 a^{3} b^{2} e^{5} x^{2}-300 a^{2} b^{3} d \,e^{4} x^{2}+150 a \,b^{4} d^{2} e^{3} x^{2}-30 b^{5} d^{3} e^{2} x^{2}+60 a^{5} e^{5} \ln \left (e x +d \right )-300 a^{4} b d \,e^{4} \ln \left (e x +d \right )+300 a^{4} b \,e^{5} x +600 a^{3} b^{2} d^{2} e^{3} \ln \left (e x +d \right )-600 a^{3} b^{2} d \,e^{4} x -600 a^{2} b^{3} d^{3} e^{2} \ln \left (e x +d \right )+600 a^{2} b^{3} d^{2} e^{3} x +300 a \,b^{4} d^{4} e \ln \left (e x +d \right )-300 a \,b^{4} d^{3} e^{2} x -60 b^{5} d^{5} \ln \left (e x +d \right )+60 b^{5} d^{4} e x \right )}{60 \left (b x +a \right )^{5} e^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\left (a^2+2\,a\,b\,x+b^2\,x^2\right )}^{5/2}}{d+e\,x} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (\left (a + b x\right )^{2}\right )^{\frac {5}{2}}}{d + e x}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________