Optimal. Leaf size=302 \[ -\frac {e^3 (a+b x) \log (a+b x) (B d-A e)}{\sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^5}+\frac {e^3 (a+b x) (B d-A e) \log (d+e x)}{\sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^5}-\frac {e^2 (B d-A e)}{\sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^4}+\frac {e (B d-A e)}{2 (a+b x) \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^3}-\frac {B d-A e}{3 (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^2}-\frac {A b-a B}{4 b (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)} \]
________________________________________________________________________________________
Rubi [A] time = 0.26, antiderivative size = 302, 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 {e^2 (B d-A e)}{\sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^4}-\frac {e^3 (a+b x) \log (a+b x) (B d-A e)}{\sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^5}+\frac {e^3 (a+b x) (B d-A e) \log (d+e x)}{\sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^5}+\frac {e (B d-A e)}{2 (a+b x) \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^3}-\frac {B d-A e}{3 (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^2}-\frac {A b-a B}{4 b (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 77
Rule 770
Rubi steps
\begin {align*} \int \frac {A+B x}{(d+e x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx &=\frac {\left (b^4 \left (a b+b^2 x\right )\right ) \int \frac {A+B x}{\left (a b+b^2 x\right )^5 (d+e x)} \, dx}{\sqrt {a^2+2 a b x+b^2 x^2}}\\ &=\frac {\left (b^4 \left (a b+b^2 x\right )\right ) \int \left (\frac {A b-a B}{b^5 (b d-a e) (a+b x)^5}+\frac {B d-A e}{b^4 (b d-a e)^2 (a+b x)^4}+\frac {e (-B d+A e)}{b^4 (b d-a e)^3 (a+b x)^3}-\frac {e^2 (-B d+A e)}{b^4 (b d-a e)^4 (a+b x)^2}+\frac {e^3 (-B d+A e)}{b^4 (b d-a e)^5 (a+b x)}-\frac {e^4 (-B d+A e)}{b^5 (b d-a e)^5 (d+e x)}\right ) \, dx}{\sqrt {a^2+2 a b x+b^2 x^2}}\\ &=-\frac {e^2 (B d-A e)}{(b d-a e)^4 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {A b-a B}{4 b (b d-a e) (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {B d-A e}{3 (b d-a e)^2 (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {e (B d-A e)}{2 (b d-a e)^3 (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {e^3 (B d-A e) (a+b x) \log (a+b x)}{(b d-a e)^5 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {e^3 (B d-A e) (a+b x) \log (d+e x)}{(b d-a e)^5 \sqrt {a^2+2 a b x+b^2 x^2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.15, size = 182, normalized size = 0.60 \begin {gather*} \frac {12 e^3 (a+b x)^3 \log (a+b x) (A e-B d)+12 e^3 (a+b x)^3 (B d-A e) \log (d+e x)+12 e^2 (a+b x)^2 (b d-a e) (A e-B d)+\frac {3 (a B-A b) (b d-a e)^4}{b (a+b x)}-6 e (a+b x) (b d-a e)^2 (A e-B d)+4 (b d-a e)^3 (A e-B d)}{12 \left ((a+b x)^2\right )^{3/2} (b d-a e)^5} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [F] time = 180.00, size = 0, normalized size = 0.00 \begin {gather*} \text {\$Aborted} \end {gather*}
Verification is not applicable to the result.
[In]
[Out]
________________________________________________________________________________________
fricas [B] time = 0.45, size = 969, normalized size = 3.21 \begin {gather*} -\frac {{\left (B a b^{4} + 3 \, A b^{5}\right )} d^{4} - 2 \, {\left (3 \, B a^{2} b^{3} + 8 \, A a b^{4}\right )} d^{3} e + 18 \, {\left (B a^{3} b^{2} + 2 \, A a^{2} b^{3}\right )} d^{2} e^{2} - 2 \, {\left (5 \, B a^{4} b + 24 \, A a^{3} b^{2}\right )} d e^{3} - {\left (3 \, B a^{5} - 25 \, A a^{4} b\right )} e^{4} + 12 \, {\left (B b^{5} d^{2} e^{2} + A a b^{4} e^{4} - {\left (B a b^{4} + A b^{5}\right )} d e^{3}\right )} x^{3} - 6 \, {\left (B b^{5} d^{3} e - 7 \, A a^{2} b^{3} e^{4} - {\left (8 \, B a b^{4} + A b^{5}\right )} d^{2} e^{2} + {\left (7 \, B a^{2} b^{3} + 8 \, A a b^{4}\right )} d e^{3}\right )} x^{2} + 4 \, {\left (B b^{5} d^{4} + 13 \, A a^{3} b^{2} e^{4} - {\left (6 \, B a b^{4} + A b^{5}\right )} d^{3} e + 6 \, {\left (3 \, B a^{2} b^{3} + A a b^{4}\right )} d^{2} e^{2} - {\left (13 \, B a^{3} b^{2} + 18 \, A a^{2} b^{3}\right )} d e^{3}\right )} x + 12 \, {\left (B a^{4} b d e^{3} - A a^{4} b e^{4} + {\left (B b^{5} d e^{3} - A b^{5} e^{4}\right )} x^{4} + 4 \, {\left (B a b^{4} d e^{3} - A a b^{4} e^{4}\right )} x^{3} + 6 \, {\left (B a^{2} b^{3} d e^{3} - A a^{2} b^{3} e^{4}\right )} x^{2} + 4 \, {\left (B a^{3} b^{2} d e^{3} - A a^{3} b^{2} e^{4}\right )} x\right )} \log \left (b x + a\right ) - 12 \, {\left (B a^{4} b d e^{3} - A a^{4} b e^{4} + {\left (B b^{5} d e^{3} - A b^{5} e^{4}\right )} x^{4} + 4 \, {\left (B a b^{4} d e^{3} - A a b^{4} e^{4}\right )} x^{3} + 6 \, {\left (B a^{2} b^{3} d e^{3} - A a^{2} b^{3} e^{4}\right )} x^{2} + 4 \, {\left (B a^{3} b^{2} d e^{3} - A a^{3} b^{2} e^{4}\right )} x\right )} \log \left (e x + d\right )}{12 \, {\left (a^{4} b^{6} d^{5} - 5 \, a^{5} b^{5} d^{4} e + 10 \, a^{6} b^{4} d^{3} e^{2} - 10 \, a^{7} b^{3} d^{2} e^{3} + 5 \, a^{8} b^{2} d e^{4} - a^{9} b e^{5} + {\left (b^{10} d^{5} - 5 \, a b^{9} d^{4} e + 10 \, a^{2} b^{8} d^{3} e^{2} - 10 \, a^{3} b^{7} d^{2} e^{3} + 5 \, a^{4} b^{6} d e^{4} - a^{5} b^{5} e^{5}\right )} x^{4} + 4 \, {\left (a b^{9} d^{5} - 5 \, a^{2} b^{8} d^{4} e + 10 \, a^{3} b^{7} d^{3} e^{2} - 10 \, a^{4} b^{6} d^{2} e^{3} + 5 \, a^{5} b^{5} d e^{4} - a^{6} b^{4} e^{5}\right )} x^{3} + 6 \, {\left (a^{2} b^{8} d^{5} - 5 \, a^{3} b^{7} d^{4} e + 10 \, a^{4} b^{6} d^{3} e^{2} - 10 \, a^{5} b^{5} d^{2} e^{3} + 5 \, a^{6} b^{4} d e^{4} - a^{7} b^{3} e^{5}\right )} x^{2} + 4 \, {\left (a^{3} b^{7} d^{5} - 5 \, a^{4} b^{6} d^{4} e + 10 \, a^{5} b^{5} d^{3} e^{2} - 10 \, a^{6} b^{4} d^{2} e^{3} + 5 \, a^{7} b^{3} d e^{4} - a^{8} b^{2} e^{5}\right )} x\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \mathit {sage}_{0} x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [B] time = 0.08, size = 777, normalized size = 2.57 \begin {gather*} -\frac {\left (12 A \,b^{5} e^{4} x^{4} \ln \left (b x +a \right )-12 A \,b^{5} e^{4} x^{4} \ln \left (e x +d \right )-12 B \,b^{5} d \,e^{3} x^{4} \ln \left (b x +a \right )+12 B \,b^{5} d \,e^{3} x^{4} \ln \left (e x +d \right )+48 A a \,b^{4} e^{4} x^{3} \ln \left (b x +a \right )-48 A a \,b^{4} e^{4} x^{3} \ln \left (e x +d \right )-48 B a \,b^{4} d \,e^{3} x^{3} \ln \left (b x +a \right )+48 B a \,b^{4} d \,e^{3} x^{3} \ln \left (e x +d \right )+72 A \,a^{2} b^{3} e^{4} x^{2} \ln \left (b x +a \right )-72 A \,a^{2} b^{3} e^{4} x^{2} \ln \left (e x +d \right )-12 A a \,b^{4} e^{4} x^{3}+12 A \,b^{5} d \,e^{3} x^{3}-72 B \,a^{2} b^{3} d \,e^{3} x^{2} \ln \left (b x +a \right )+72 B \,a^{2} b^{3} d \,e^{3} x^{2} \ln \left (e x +d \right )+12 B a \,b^{4} d \,e^{3} x^{3}-12 B \,b^{5} d^{2} e^{2} x^{3}+48 A \,a^{3} b^{2} e^{4} x \ln \left (b x +a \right )-48 A \,a^{3} b^{2} e^{4} x \ln \left (e x +d \right )-42 A \,a^{2} b^{3} e^{4} x^{2}+48 A a \,b^{4} d \,e^{3} x^{2}-6 A \,b^{5} d^{2} e^{2} x^{2}-48 B \,a^{3} b^{2} d \,e^{3} x \ln \left (b x +a \right )+48 B \,a^{3} b^{2} d \,e^{3} x \ln \left (e x +d \right )+42 B \,a^{2} b^{3} d \,e^{3} x^{2}-48 B a \,b^{4} d^{2} e^{2} x^{2}+6 B \,b^{5} d^{3} e \,x^{2}+12 A \,a^{4} b \,e^{4} \ln \left (b x +a \right )-12 A \,a^{4} b \,e^{4} \ln \left (e x +d \right )-52 A \,a^{3} b^{2} e^{4} x +72 A \,a^{2} b^{3} d \,e^{3} x -24 A a \,b^{4} d^{2} e^{2} x +4 A \,b^{5} d^{3} e x -12 B \,a^{4} b d \,e^{3} \ln \left (b x +a \right )+12 B \,a^{4} b d \,e^{3} \ln \left (e x +d \right )+52 B \,a^{3} b^{2} d \,e^{3} x -72 B \,a^{2} b^{3} d^{2} e^{2} x +24 B a \,b^{4} d^{3} e x -4 B \,b^{5} d^{4} x -25 A \,a^{4} b \,e^{4}+48 A \,a^{3} b^{2} d \,e^{3}-36 A \,a^{2} b^{3} d^{2} e^{2}+16 A a \,b^{4} d^{3} e -3 A \,b^{5} d^{4}+3 B \,a^{5} e^{4}+10 B \,a^{4} b d \,e^{3}-18 B \,a^{3} b^{2} d^{2} e^{2}+6 B \,a^{2} b^{3} d^{3} e -B a \,b^{4} d^{4}\right ) \left (b x +a \right )}{12 \left (a e -b d \right )^{5} \left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}} b} \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 {A+B\,x}{\left (d+e\,x\right )\,{\left (a^2+2\,a\,b\,x+b^2\,x^2\right )}^{5/2}} \,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 {A + B x}{\left (d + e x\right ) \left (\left (a + b x\right )^{2}\right )^{\frac {5}{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________