Optimal. Leaf size=345 \[ -\frac {15 b^2 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^4}{e^7 (a+b x) (d+e x)}+\frac {3 b \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^5}{e^7 (a+b x) (d+e x)^2}-\frac {\sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^6}{3 e^7 (a+b x) (d+e x)^3}+\frac {b^6 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^3}{3 e^7 (a+b x)}-\frac {3 b^5 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^2 (b d-a e)}{e^7 (a+b x)}+\frac {15 b^4 x \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^2}{e^6 (a+b x)}-\frac {20 b^3 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^3 \log (d+e x)}{e^7 (a+b x)} \]
________________________________________________________________________________________
Rubi [A] time = 0.25, antiderivative size = 345, 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^6 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^3}{3 e^7 (a+b x)}-\frac {3 b^5 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^2 (b d-a e)}{e^7 (a+b x)}+\frac {15 b^4 x \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^2}{e^6 (a+b x)}-\frac {15 b^2 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^4}{e^7 (a+b x) (d+e x)}+\frac {3 b \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^5}{e^7 (a+b x) (d+e x)^2}-\frac {\sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^6}{3 e^7 (a+b x) (d+e x)^3}-\frac {20 b^3 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^3 \log (d+e x)}{e^7 (a+b x)} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
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 )^{5/2}}{(d+e x)^4} \, 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 )^5}{(d+e x)^4} \, dx}{b^4 \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)^6}{(d+e x)^4} \, dx}{a b+b^2 x}\\ &=\frac {\left (b \sqrt {a^2+2 a b x+b^2 x^2}\right ) \int \left (\frac {15 b^4 (b d-a e)^2}{e^6}+\frac {(-b d+a e)^6}{e^6 (d+e x)^4}-\frac {6 b (b d-a e)^5}{e^6 (d+e x)^3}+\frac {15 b^2 (b d-a e)^4}{e^6 (d+e x)^2}-\frac {20 b^3 (b d-a e)^3}{e^6 (d+e x)}-\frac {6 b^5 (b d-a e) (d+e x)}{e^6}+\frac {b^6 (d+e x)^2}{e^6}\right ) \, dx}{a b+b^2 x}\\ &=\frac {15 b^4 (b d-a e)^2 x \sqrt {a^2+2 a b x+b^2 x^2}}{e^6 (a+b x)}-\frac {(b d-a e)^6 \sqrt {a^2+2 a b x+b^2 x^2}}{3 e^7 (a+b x) (d+e x)^3}+\frac {3 b (b d-a e)^5 \sqrt {a^2+2 a b x+b^2 x^2}}{e^7 (a+b x) (d+e x)^2}-\frac {15 b^2 (b d-a e)^4 \sqrt {a^2+2 a b x+b^2 x^2}}{e^7 (a+b x) (d+e x)}-\frac {3 b^5 (b d-a e) (d+e x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}{e^7 (a+b x)}+\frac {b^6 (d+e x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}{3 e^7 (a+b x)}-\frac {20 b^3 (b d-a e)^3 \sqrt {a^2+2 a b x+b^2 x^2} \log (d+e x)}{e^7 (a+b x)}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.15, size = 320, normalized size = 0.93 \begin {gather*} -\frac {\sqrt {(a+b x)^2} \left (a^6 e^6+3 a^5 b e^5 (d+3 e x)+15 a^4 b^2 e^4 \left (d^2+3 d e x+3 e^2 x^2\right )-10 a^3 b^3 d e^3 \left (11 d^2+27 d e x+18 e^2 x^2\right )+15 a^2 b^4 e^2 \left (13 d^4+27 d^3 e x+9 d^2 e^2 x^2-9 d e^3 x^3-3 e^4 x^4\right )-3 a b^5 e \left (47 d^5+81 d^4 e x-9 d^3 e^2 x^2-63 d^2 e^3 x^3-15 d e^4 x^4+3 e^5 x^5\right )+60 b^3 (d+e x)^3 (b d-a e)^3 \log (d+e x)+b^6 \left (37 d^6+51 d^5 e x-39 d^4 e^2 x^2-73 d^3 e^3 x^3-15 d^2 e^4 x^4+3 d e^5 x^5-e^6 x^6\right )\right )}{3 e^7 (a+b x) (d+e x)^3} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [F] time = 6.41, 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 )^{5/2}}{(d+e x)^4} \, dx \end {gather*}
Verification is not applicable to the result.
[In]
[Out]
________________________________________________________________________________________
fricas [B] time = 0.41, size = 576, normalized size = 1.67 \begin {gather*} \frac {b^{6} e^{6} x^{6} - 37 \, b^{6} d^{6} + 141 \, a b^{5} d^{5} e - 195 \, a^{2} b^{4} d^{4} e^{2} + 110 \, a^{3} b^{3} d^{3} e^{3} - 15 \, a^{4} b^{2} d^{2} e^{4} - 3 \, a^{5} b d e^{5} - a^{6} e^{6} - 3 \, {\left (b^{6} d e^{5} - 3 \, a b^{5} e^{6}\right )} x^{5} + 15 \, {\left (b^{6} d^{2} e^{4} - 3 \, a b^{5} d e^{5} + 3 \, a^{2} b^{4} e^{6}\right )} x^{4} + {\left (73 \, b^{6} d^{3} e^{3} - 189 \, a b^{5} d^{2} e^{4} + 135 \, a^{2} b^{4} d e^{5}\right )} x^{3} + 3 \, {\left (13 \, b^{6} d^{4} e^{2} - 9 \, a b^{5} d^{3} e^{3} - 45 \, a^{2} b^{4} d^{2} e^{4} + 60 \, a^{3} b^{3} d e^{5} - 15 \, a^{4} b^{2} e^{6}\right )} x^{2} - 3 \, {\left (17 \, b^{6} d^{5} e - 81 \, a b^{5} d^{4} e^{2} + 135 \, a^{2} b^{4} d^{3} e^{3} - 90 \, a^{3} b^{3} d^{2} e^{4} + 15 \, a^{4} b^{2} d e^{5} + 3 \, a^{5} b e^{6}\right )} x - 60 \, {\left (b^{6} d^{6} - 3 \, a b^{5} d^{5} e + 3 \, a^{2} b^{4} d^{4} e^{2} - a^{3} b^{3} d^{3} e^{3} + {\left (b^{6} d^{3} e^{3} - 3 \, a b^{5} d^{2} e^{4} + 3 \, a^{2} b^{4} d e^{5} - a^{3} b^{3} e^{6}\right )} x^{3} + 3 \, {\left (b^{6} d^{4} e^{2} - 3 \, a b^{5} d^{3} e^{3} + 3 \, a^{2} b^{4} d^{2} e^{4} - a^{3} b^{3} d e^{5}\right )} x^{2} + 3 \, {\left (b^{6} d^{5} e - 3 \, a b^{5} d^{4} e^{2} + 3 \, a^{2} b^{4} d^{3} e^{3} - a^{3} b^{3} d^{2} e^{4}\right )} x\right )} \log \left (e x + d\right )}{3 \, {\left (e^{10} x^{3} + 3 \, d e^{9} x^{2} + 3 \, d^{2} e^{8} x + d^{3} e^{7}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.19, size = 503, normalized size = 1.46 \begin {gather*} -20 \, {\left (b^{6} d^{3} \mathrm {sgn}\left (b x + a\right ) - 3 \, a b^{5} d^{2} e \mathrm {sgn}\left (b x + a\right ) + 3 \, a^{2} b^{4} d e^{2} \mathrm {sgn}\left (b x + a\right ) - a^{3} b^{3} e^{3} \mathrm {sgn}\left (b x + a\right )\right )} e^{\left (-7\right )} \log \left ({\left | x e + d \right |}\right ) + \frac {1}{3} \, {\left (b^{6} x^{3} e^{8} \mathrm {sgn}\left (b x + a\right ) - 6 \, b^{6} d x^{2} e^{7} \mathrm {sgn}\left (b x + a\right ) + 30 \, b^{6} d^{2} x e^{6} \mathrm {sgn}\left (b x + a\right ) + 9 \, a b^{5} x^{2} e^{8} \mathrm {sgn}\left (b x + a\right ) - 72 \, a b^{5} d x e^{7} \mathrm {sgn}\left (b x + a\right ) + 45 \, a^{2} b^{4} x e^{8} \mathrm {sgn}\left (b x + a\right )\right )} e^{\left (-12\right )} - \frac {{\left (37 \, b^{6} d^{6} \mathrm {sgn}\left (b x + a\right ) - 141 \, a b^{5} d^{5} e \mathrm {sgn}\left (b x + a\right ) + 195 \, a^{2} b^{4} d^{4} e^{2} \mathrm {sgn}\left (b x + a\right ) - 110 \, a^{3} b^{3} d^{3} e^{3} \mathrm {sgn}\left (b x + a\right ) + 15 \, a^{4} b^{2} d^{2} e^{4} \mathrm {sgn}\left (b x + a\right ) + 3 \, a^{5} b d e^{5} \mathrm {sgn}\left (b x + a\right ) + a^{6} e^{6} \mathrm {sgn}\left (b x + a\right ) + 45 \, {\left (b^{6} d^{4} e^{2} \mathrm {sgn}\left (b x + a\right ) - 4 \, a b^{5} d^{3} e^{3} \mathrm {sgn}\left (b x + a\right ) + 6 \, a^{2} b^{4} d^{2} e^{4} \mathrm {sgn}\left (b x + a\right ) - 4 \, a^{3} b^{3} d e^{5} \mathrm {sgn}\left (b x + a\right ) + a^{4} b^{2} e^{6} \mathrm {sgn}\left (b x + a\right )\right )} x^{2} + 9 \, {\left (9 \, b^{6} d^{5} e \mathrm {sgn}\left (b x + a\right ) - 35 \, a b^{5} d^{4} e^{2} \mathrm {sgn}\left (b x + a\right ) + 50 \, a^{2} b^{4} d^{3} e^{3} \mathrm {sgn}\left (b x + a\right ) - 30 \, a^{3} b^{3} d^{2} e^{4} \mathrm {sgn}\left (b x + a\right ) + 5 \, a^{4} b^{2} d e^{5} \mathrm {sgn}\left (b x + a\right ) + a^{5} b e^{6} \mathrm {sgn}\left (b x + a\right )\right )} x\right )} e^{\left (-7\right )}}{3 \, {\left (x e + d\right )}^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [B] time = 0.11, size = 692, normalized size = 2.01 \begin {gather*} \frac {\left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}} \left (b^{6} e^{6} x^{6}+9 a \,b^{5} e^{6} x^{5}-3 b^{6} d \,e^{5} x^{5}+60 a^{3} b^{3} e^{6} x^{3} \ln \left (e x +d \right )-180 a^{2} b^{4} d \,e^{5} x^{3} \ln \left (e x +d \right )+45 a^{2} b^{4} e^{6} x^{4}+180 a \,b^{5} d^{2} e^{4} x^{3} \ln \left (e x +d \right )-45 a \,b^{5} d \,e^{5} x^{4}-60 b^{6} d^{3} e^{3} x^{3} \ln \left (e x +d \right )+15 b^{6} d^{2} e^{4} x^{4}+180 a^{3} b^{3} d \,e^{5} x^{2} \ln \left (e x +d \right )-540 a^{2} b^{4} d^{2} e^{4} x^{2} \ln \left (e x +d \right )+135 a^{2} b^{4} d \,e^{5} x^{3}+540 a \,b^{5} d^{3} e^{3} x^{2} \ln \left (e x +d \right )-189 a \,b^{5} d^{2} e^{4} x^{3}-180 b^{6} d^{4} e^{2} x^{2} \ln \left (e x +d \right )+73 b^{6} d^{3} e^{3} x^{3}-45 a^{4} b^{2} e^{6} x^{2}+180 a^{3} b^{3} d^{2} e^{4} x \ln \left (e x +d \right )+180 a^{3} b^{3} d \,e^{5} x^{2}-540 a^{2} b^{4} d^{3} e^{3} x \ln \left (e x +d \right )-135 a^{2} b^{4} d^{2} e^{4} x^{2}+540 a \,b^{5} d^{4} e^{2} x \ln \left (e x +d \right )-27 a \,b^{5} d^{3} e^{3} x^{2}-180 b^{6} d^{5} e x \ln \left (e x +d \right )+39 b^{6} d^{4} e^{2} x^{2}-9 a^{5} b \,e^{6} x -45 a^{4} b^{2} d \,e^{5} x +60 a^{3} b^{3} d^{3} e^{3} \ln \left (e x +d \right )+270 a^{3} b^{3} d^{2} e^{4} x -180 a^{2} b^{4} d^{4} e^{2} \ln \left (e x +d \right )-405 a^{2} b^{4} d^{3} e^{3} x +180 a \,b^{5} d^{5} e \ln \left (e x +d \right )+243 a \,b^{5} d^{4} e^{2} x -60 b^{6} d^{6} \ln \left (e x +d \right )-51 b^{6} d^{5} e x -a^{6} e^{6}-3 a^{5} b d \,e^{5}-15 a^{4} b^{2} d^{2} e^{4}+110 a^{3} b^{3} d^{3} e^{3}-195 a^{2} b^{4} d^{4} e^{2}+141 a \,b^{5} d^{5} e -37 b^{6} d^{6}\right )}{3 \left (b x +a \right )^{5} \left (e x +d \right )^{3} e^{7}} \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+b\,x\right )\,{\left (a^2+2\,a\,b\,x+b^2\,x^2\right )}^{5/2}}{{\left (d+e\,x\right )}^4} \,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 (a + b x\right ) \left (\left (a + b x\right )^{2}\right )^{\frac {5}{2}}}{\left (d + e x\right )^{4}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________