Optimal. Leaf size=422 \[ \frac {5 b^2 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^2 (-a B e-A b e+2 b B d)}{e^7 (a+b x) (d+e x)^2}-\frac {5 b \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^3 (-a B e-2 A b e+3 b B d)}{3 e^7 (a+b x) (d+e x)^3}+\frac {\sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^4 (-a B e-5 A b e+6 b B d)}{4 e^7 (a+b x) (d+e x)^4}-\frac {\sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^5 (B d-A e)}{5 e^7 (a+b x) (d+e x)^5}-\frac {b^4 \sqrt {a^2+2 a b x+b^2 x^2} \log (d+e x) (-5 a B e-A b e+6 b B d)}{e^7 (a+b x)}-\frac {5 b^3 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e) (-2 a B e-A b e+3 b B d)}{e^7 (a+b x) (d+e x)}+\frac {b^5 B x \sqrt {a^2+2 a b x+b^2 x^2}}{e^6 (a+b x)} \]
________________________________________________________________________________________
Rubi [A] time = 0.36, antiderivative size = 422, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.061, Rules used = {770, 77} \begin {gather*} -\frac {5 b^3 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e) (-2 a B e-A b e+3 b B d)}{e^7 (a+b x) (d+e x)}+\frac {5 b^2 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^2 (-a B e-A b e+2 b B d)}{e^7 (a+b x) (d+e x)^2}-\frac {5 b \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^3 (-a B e-2 A b e+3 b B d)}{3 e^7 (a+b x) (d+e x)^3}+\frac {\sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^4 (-a B e-5 A b e+6 b B d)}{4 e^7 (a+b x) (d+e x)^4}-\frac {\sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^5 (B d-A e)}{5 e^7 (a+b x) (d+e x)^5}-\frac {b^4 \sqrt {a^2+2 a b x+b^2 x^2} \log (d+e x) (-5 a B e-A b e+6 b B d)}{e^7 (a+b x)}+\frac {b^5 B x \sqrt {a^2+2 a b x+b^2 x^2}}{e^6 (a+b x)} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 77
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)^6} \, dx &=\frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \frac {\left (a b+b^2 x\right )^5 (A+B x)}{(d+e x)^6} \, 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^{10} B}{e^6}-\frac {b^5 (b d-a e)^5 (-B d+A e)}{e^6 (d+e x)^6}+\frac {b^5 (b d-a e)^4 (-6 b B d+5 A b e+a B e)}{e^6 (d+e x)^5}-\frac {5 b^6 (b d-a e)^3 (-3 b B d+2 A b e+a B e)}{e^6 (d+e x)^4}+\frac {10 b^7 (b d-a e)^2 (-2 b B d+A b e+a B e)}{e^6 (d+e x)^3}-\frac {5 b^8 (b d-a e) (-3 b B d+A b e+2 a B e)}{e^6 (d+e x)^2}+\frac {b^9 (-6 b B d+A b e+5 a B e)}{e^6 (d+e x)}\right ) \, dx}{b^4 \left (a b+b^2 x\right )}\\ &=\frac {b^5 B x \sqrt {a^2+2 a b x+b^2 x^2}}{e^6 (a+b x)}-\frac {(b d-a e)^5 (B d-A e) \sqrt {a^2+2 a b x+b^2 x^2}}{5 e^7 (a+b x) (d+e x)^5}+\frac {(b d-a e)^4 (6 b B d-5 A b e-a B e) \sqrt {a^2+2 a b x+b^2 x^2}}{4 e^7 (a+b x) (d+e x)^4}-\frac {5 b (b d-a e)^3 (3 b B d-2 A b e-a B e) \sqrt {a^2+2 a b x+b^2 x^2}}{3 e^7 (a+b x) (d+e x)^3}+\frac {5 b^2 (b d-a e)^2 (2 b B d-A b e-a B e) \sqrt {a^2+2 a b x+b^2 x^2}}{e^7 (a+b x) (d+e x)^2}-\frac {5 b^3 (b d-a e) (3 b B d-A b e-2 a B e) \sqrt {a^2+2 a b x+b^2 x^2}}{e^7 (a+b x) (d+e x)}-\frac {b^4 (6 b B d-A b e-5 a B e) \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 = 0.29, size = 490, normalized size = 1.16 \begin {gather*} -\frac {\sqrt {(a+b x)^2} \left (3 a^5 e^5 (4 A e+B (d+5 e x))+5 a^4 b e^4 \left (3 A e (d+5 e x)+2 B \left (d^2+5 d e x+10 e^2 x^2\right )\right )+10 a^3 b^2 e^3 \left (2 A e \left (d^2+5 d e x+10 e^2 x^2\right )+3 B \left (d^3+5 d^2 e x+10 d e^2 x^2+10 e^3 x^3\right )\right )+30 a^2 b^3 e^2 \left (A e \left (d^3+5 d^2 e x+10 d e^2 x^2+10 e^3 x^3\right )+4 B \left (d^4+5 d^3 e x+10 d^2 e^2 x^2+10 d e^3 x^3+5 e^4 x^4\right )\right )+5 a b^4 e \left (12 A e \left (d^4+5 d^3 e x+10 d^2 e^2 x^2+10 d e^3 x^3+5 e^4 x^4\right )-B d \left (137 d^4+625 d^3 e x+1100 d^2 e^2 x^2+900 d e^3 x^3+300 e^4 x^4\right )\right )+60 b^4 (d+e x)^5 \log (d+e x) (-5 a B e-A b e+6 b B d)+b^5 \left (6 B \left (87 d^6+375 d^5 e x+600 d^4 e^2 x^2+400 d^3 e^3 x^3+50 d^2 e^4 x^4-50 d e^5 x^5-10 e^6 x^6\right )-A d e \left (137 d^4+625 d^3 e x+1100 d^2 e^2 x^2+900 d e^3 x^3+300 e^4 x^4\right )\right )\right )}{60 e^7 (a+b x) (d+e x)^5} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [F] time = 180.07, size = 0, normalized size = 0.00 \begin {gather*} \text {\$Aborted} \end {gather*}
Verification is not applicable to the result.
[In]
[Out]
________________________________________________________________________________________
fricas [B] time = 0.44, size = 802, normalized size = 1.90 \begin {gather*} \frac {60 \, B b^{5} e^{6} x^{6} + 300 \, B b^{5} d e^{5} x^{5} - 522 \, B b^{5} d^{6} - 12 \, A a^{5} e^{6} + 137 \, {\left (5 \, B a b^{4} + A b^{5}\right )} d^{5} e - 60 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} d^{4} e^{2} - 30 \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} d^{3} e^{3} - 10 \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} d^{2} e^{4} - 3 \, {\left (B a^{5} + 5 \, A a^{4} b\right )} d e^{5} - 300 \, {\left (B b^{5} d^{2} e^{4} - {\left (5 \, B a b^{4} + A b^{5}\right )} d e^{5} + {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} e^{6}\right )} x^{4} - 300 \, {\left (8 \, B b^{5} d^{3} e^{3} - 3 \, {\left (5 \, B a b^{4} + A b^{5}\right )} d^{2} e^{4} + 2 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} d e^{5} + {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} e^{6}\right )} x^{3} - 100 \, {\left (36 \, B b^{5} d^{4} e^{2} - 11 \, {\left (5 \, B a b^{4} + A b^{5}\right )} d^{3} e^{3} + 6 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} d^{2} e^{4} + 3 \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} d e^{5} + {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} e^{6}\right )} x^{2} - 5 \, {\left (450 \, B b^{5} d^{5} e - 125 \, {\left (5 \, B a b^{4} + A b^{5}\right )} d^{4} e^{2} + 60 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} d^{3} e^{3} + 30 \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} d^{2} e^{4} + 10 \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} d e^{5} + 3 \, {\left (B a^{5} + 5 \, A a^{4} b\right )} e^{6}\right )} x - 60 \, {\left (6 \, B b^{5} d^{6} - {\left (5 \, B a b^{4} + A b^{5}\right )} d^{5} e + {\left (6 \, B b^{5} d e^{5} - {\left (5 \, B a b^{4} + A b^{5}\right )} e^{6}\right )} x^{5} + 5 \, {\left (6 \, B b^{5} d^{2} e^{4} - {\left (5 \, B a b^{4} + A b^{5}\right )} d e^{5}\right )} x^{4} + 10 \, {\left (6 \, B b^{5} d^{3} e^{3} - {\left (5 \, B a b^{4} + A b^{5}\right )} d^{2} e^{4}\right )} x^{3} + 10 \, {\left (6 \, B b^{5} d^{4} e^{2} - {\left (5 \, B a b^{4} + A b^{5}\right )} d^{3} e^{3}\right )} x^{2} + 5 \, {\left (6 \, B b^{5} d^{5} e - {\left (5 \, B a b^{4} + A b^{5}\right )} d^{4} e^{2}\right )} x\right )} \log \left (e x + d\right )}{60 \, {\left (e^{12} x^{5} + 5 \, d e^{11} x^{4} + 10 \, d^{2} e^{10} x^{3} + 10 \, d^{3} e^{9} x^{2} + 5 \, d^{4} e^{8} x + d^{5} e^{7}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [B] time = 0.27, size = 863, normalized size = 2.05
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [B] time = 0.07, size = 1012, normalized size = 2.40 \begin {gather*} \frac {\left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}} \left (60 A \,b^{5} e^{6} x^{5} \ln \left (e x +d \right )+300 B a \,b^{4} e^{6} x^{5} \ln \left (e x +d \right )-360 B \,b^{5} d \,e^{5} x^{5} \ln \left (e x +d \right )+60 B \,b^{5} e^{6} x^{6}+300 A \,b^{5} d \,e^{5} x^{4} \ln \left (e x +d \right )+1500 B a \,b^{4} d \,e^{5} x^{4} \ln \left (e x +d \right )-1800 B \,b^{5} d^{2} e^{4} x^{4} \ln \left (e x +d \right )+300 B \,b^{5} d \,e^{5} x^{5}-300 A a \,b^{4} e^{6} x^{4}+600 A \,b^{5} d^{2} e^{4} x^{3} \ln \left (e x +d \right )+300 A \,b^{5} d \,e^{5} x^{4}-600 B \,a^{2} b^{3} e^{6} x^{4}+3000 B a \,b^{4} d^{2} e^{4} x^{3} \ln \left (e x +d \right )+1500 B a \,b^{4} d \,e^{5} x^{4}-3600 B \,b^{5} d^{3} e^{3} x^{3} \ln \left (e x +d \right )-300 B \,b^{5} d^{2} e^{4} x^{4}-300 A \,a^{2} b^{3} e^{6} x^{3}-600 A a \,b^{4} d \,e^{5} x^{3}+600 A \,b^{5} d^{3} e^{3} x^{2} \ln \left (e x +d \right )+900 A \,b^{5} d^{2} e^{4} x^{3}-300 B \,a^{3} b^{2} e^{6} x^{3}-1200 B \,a^{2} b^{3} d \,e^{5} x^{3}+3000 B a \,b^{4} d^{3} e^{3} x^{2} \ln \left (e x +d \right )+4500 B a \,b^{4} d^{2} e^{4} x^{3}-3600 B \,b^{5} d^{4} e^{2} x^{2} \ln \left (e x +d \right )-2400 B \,b^{5} d^{3} e^{3} x^{3}-200 A \,a^{3} b^{2} e^{6} x^{2}-300 A \,a^{2} b^{3} d \,e^{5} x^{2}-600 A a \,b^{4} d^{2} e^{4} x^{2}+300 A \,b^{5} d^{4} e^{2} x \ln \left (e x +d \right )+1100 A \,b^{5} d^{3} e^{3} x^{2}-100 B \,a^{4} b \,e^{6} x^{2}-300 B \,a^{3} b^{2} d \,e^{5} x^{2}-1200 B \,a^{2} b^{3} d^{2} e^{4} x^{2}+1500 B a \,b^{4} d^{4} e^{2} x \ln \left (e x +d \right )+5500 B a \,b^{4} d^{3} e^{3} x^{2}-1800 B \,b^{5} d^{5} e x \ln \left (e x +d \right )-3600 B \,b^{5} d^{4} e^{2} x^{2}-75 A \,a^{4} b \,e^{6} x -100 A \,a^{3} b^{2} d \,e^{5} x -150 A \,a^{2} b^{3} d^{2} e^{4} x -300 A a \,b^{4} d^{3} e^{3} x +60 A \,b^{5} d^{5} e \ln \left (e x +d \right )+625 A \,b^{5} d^{4} e^{2} x -15 B \,a^{5} e^{6} x -50 B \,a^{4} b d \,e^{5} x -150 B \,a^{3} b^{2} d^{2} e^{4} x -600 B \,a^{2} b^{3} d^{3} e^{3} x +300 B a \,b^{4} d^{5} e \ln \left (e x +d \right )+3125 B a \,b^{4} d^{4} e^{2} x -360 B \,b^{5} d^{6} \ln \left (e x +d \right )-2250 B \,b^{5} d^{5} e x -12 A \,a^{5} e^{6}-15 A \,a^{4} b d \,e^{5}-20 A \,a^{3} b^{2} d^{2} e^{4}-30 A \,a^{2} b^{3} d^{3} e^{3}-60 A a \,b^{4} d^{4} e^{2}+137 A \,b^{5} d^{5} e -3 B \,a^{5} d \,e^{5}-10 B \,a^{4} b \,d^{2} e^{4}-30 B \,a^{3} b^{2} d^{3} e^{3}-120 B \,a^{2} b^{3} d^{4} e^{2}+685 B a \,b^{4} d^{5} e -522 B \,b^{5} d^{6}\right )}{60 \left (b x +a \right )^{5} \left (e x +d \right )^{5} 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 )}^6} \,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 )^{6}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________