Optimal. Leaf size=340 \[ \frac {\sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^5 (B d-A e) \log (d+e x)}{e^7 (a+b x)}-\frac {b x \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^4 (B d-A e)}{e^6 (a+b x)}+\frac {(a+b x) \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^3 (B d-A e)}{2 e^5}-\frac {(a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^2 (B d-A e)}{3 e^4}+\frac {(a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e) (B d-A e)}{4 e^3}-\frac {(a+b x)^4 \sqrt {a^2+2 a b x+b^2 x^2} (B d-A e)}{5 e^2}+\frac {B (a+b x)^5 \sqrt {a^2+2 a b x+b^2 x^2}}{6 b e} \]
________________________________________________________________________________________
Rubi [A] time = 0.25, antiderivative size = 340, 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 {b x \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^4 (B d-A e)}{e^6 (a+b x)}+\frac {(a+b x) \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^3 (B d-A e)}{2 e^5}-\frac {(a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^2 (B d-A e)}{3 e^4}+\frac {(a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e) (B d-A e)}{4 e^3}-\frac {(a+b x)^4 \sqrt {a^2+2 a b x+b^2 x^2} (B d-A e)}{5 e^2}+\frac {\sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^5 (B d-A e) \log (d+e x)}{e^7 (a+b x)}+\frac {B (a+b x)^5 \sqrt {a^2+2 a b x+b^2 x^2}}{6 b e} \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} \, 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} \, 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 (-B d+A e)}{e^6}-\frac {b^6 (b d-a e)^3 (-B d+A e) (a+b x)}{e^5}+\frac {b^6 (b d-a e)^2 (-B d+A e) (a+b x)^2}{e^4}-\frac {b^6 (b d-a e) (-B d+A e) (a+b x)^3}{e^3}+\frac {b^6 (-B d+A e) (a+b x)^4}{e^2}+\frac {B \left (a b+b^2 x\right )^5}{e}-\frac {b^5 (b d-a e)^5 (-B d+A e)}{e^6 (d+e x)}\right ) \, dx}{b^4 \left (a b+b^2 x\right )}\\ &=-\frac {b (b d-a e)^4 (B d-A e) x \sqrt {a^2+2 a b x+b^2 x^2}}{e^6 (a+b x)}+\frac {(b d-a e)^3 (B d-A e) (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}{2 e^5}-\frac {(b d-a e)^2 (B d-A e) (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}{3 e^4}+\frac {(b d-a e) (B d-A e) (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}{4 e^3}-\frac {(B d-A e) (a+b x)^4 \sqrt {a^2+2 a b x+b^2 x^2}}{5 e^2}+\frac {B (a+b x)^5 \sqrt {a^2+2 a b x+b^2 x^2}}{6 b e}+\frac {(b d-a e)^5 (B d-A 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 = 0.24, size = 386, normalized size = 1.14 \begin {gather*} \frac {\sqrt {(a+b x)^2} \left (e x \left (60 a^5 B e^5+150 a^4 b e^4 (2 A e-2 B d+B e x)+100 a^3 b^2 e^3 \left (3 A e (e x-2 d)+B \left (6 d^2-3 d e x+2 e^2 x^2\right )\right )+50 a^2 b^3 e^2 \left (2 A e \left (6 d^2-3 d e x+2 e^2 x^2\right )+B \left (-12 d^3+6 d^2 e x-4 d e^2 x^2+3 e^3 x^3\right )\right )+5 a b^4 e \left (5 A e \left (-12 d^3+6 d^2 e x-4 d e^2 x^2+3 e^3 x^3\right )+B \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 )+b^5 \left (A e \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 )+B \left (-60 d^5+30 d^4 e x-20 d^3 e^2 x^2+15 d^2 e^3 x^3-12 d e^4 x^4+10 e^5 x^5\right )\right )\right )+60 (b d-a e)^5 (B d-A e) \log (d+e x)\right )}{60 e^7 (a+b x)} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [F] time = 5.50, 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} \, dx \end {gather*}
Verification is not applicable to the result.
[In]
[Out]
________________________________________________________________________________________
fricas [B] time = 0.42, size = 555, normalized size = 1.63 \begin {gather*} \frac {10 \, B b^{5} e^{6} x^{6} - 12 \, {\left (B b^{5} d e^{5} - {\left (5 \, B a b^{4} + A b^{5}\right )} e^{6}\right )} x^{5} + 15 \, {\left (B b^{5} d^{2} e^{4} - {\left (5 \, B a b^{4} + A b^{5}\right )} d e^{5} + 5 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} e^{6}\right )} x^{4} - 20 \, {\left (B b^{5} d^{3} e^{3} - {\left (5 \, B a b^{4} + A b^{5}\right )} d^{2} e^{4} + 5 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} d e^{5} - 10 \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} e^{6}\right )} x^{3} + 30 \, {\left (B b^{5} d^{4} e^{2} - {\left (5 \, B a b^{4} + A b^{5}\right )} d^{3} e^{3} + 5 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} d^{2} e^{4} - 10 \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} d e^{5} + 5 \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} e^{6}\right )} x^{2} - 60 \, {\left (B b^{5} d^{5} e - {\left (5 \, B a b^{4} + A b^{5}\right )} d^{4} e^{2} + 5 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} d^{3} e^{3} - 10 \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} d^{2} e^{4} + 5 \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} d e^{5} - {\left (B a^{5} + 5 \, A a^{4} b\right )} e^{6}\right )} x + 60 \, {\left (B b^{5} d^{6} + A a^{5} e^{6} - {\left (5 \, B a b^{4} + A b^{5}\right )} d^{5} e + 5 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} d^{4} e^{2} - 10 \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} d^{3} e^{3} + 5 \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} d^{2} e^{4} - {\left (B a^{5} + 5 \, A a^{4} b\right )} d e^{5}\right )} \log \left (e x + d\right )}{60 \, e^{7}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [B] time = 0.24, size = 920, normalized size = 2.71
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [B] time = 0.07, size = 754, normalized size = 2.22 \begin {gather*} \frac {\left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}} \left (10 B \,b^{5} e^{6} x^{6}+12 A \,b^{5} e^{6} x^{5}+60 B a \,b^{4} e^{6} x^{5}-12 B \,b^{5} d \,e^{5} x^{5}+75 A a \,b^{4} e^{6} x^{4}-15 A \,b^{5} d \,e^{5} x^{4}+150 B \,a^{2} b^{3} e^{6} x^{4}-75 B a \,b^{4} d \,e^{5} x^{4}+15 B \,b^{5} d^{2} e^{4} x^{4}+200 A \,a^{2} b^{3} e^{6} x^{3}-100 A a \,b^{4} d \,e^{5} x^{3}+20 A \,b^{5} d^{2} e^{4} x^{3}+200 B \,a^{3} b^{2} e^{6} x^{3}-200 B \,a^{2} b^{3} d \,e^{5} x^{3}+100 B a \,b^{4} d^{2} e^{4} x^{3}-20 B \,b^{5} d^{3} e^{3} x^{3}+300 A \,a^{3} b^{2} e^{6} x^{2}-300 A \,a^{2} b^{3} d \,e^{5} x^{2}+150 A a \,b^{4} d^{2} e^{4} x^{2}-30 A \,b^{5} d^{3} e^{3} x^{2}+150 B \,a^{4} b \,e^{6} x^{2}-300 B \,a^{3} b^{2} d \,e^{5} x^{2}+300 B \,a^{2} b^{3} d^{2} e^{4} x^{2}-150 B a \,b^{4} d^{3} e^{3} x^{2}+30 B \,b^{5} d^{4} e^{2} x^{2}+60 A \,a^{5} e^{6} \ln \left (e x +d \right )-300 A \,a^{4} b d \,e^{5} \ln \left (e x +d \right )+300 A \,a^{4} b \,e^{6} x +600 A \,a^{3} b^{2} d^{2} e^{4} \ln \left (e x +d \right )-600 A \,a^{3} b^{2} d \,e^{5} x -600 A \,a^{2} b^{3} d^{3} e^{3} \ln \left (e x +d \right )+600 A \,a^{2} b^{3} d^{2} e^{4} x +300 A a \,b^{4} d^{4} e^{2} \ln \left (e x +d \right )-300 A a \,b^{4} d^{3} e^{3} x -60 A \,b^{5} d^{5} e \ln \left (e x +d \right )+60 A \,b^{5} d^{4} e^{2} x -60 B \,a^{5} d \,e^{5} \ln \left (e x +d \right )+60 B \,a^{5} e^{6} x +300 B \,a^{4} b \,d^{2} e^{4} \ln \left (e x +d \right )-300 B \,a^{4} b d \,e^{5} x -600 B \,a^{3} b^{2} d^{3} e^{3} \ln \left (e x +d \right )+600 B \,a^{3} b^{2} d^{2} e^{4} x +600 B \,a^{2} b^{3} d^{4} e^{2} \ln \left (e x +d \right )-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 )+300 B a \,b^{4} d^{4} e^{2} x +60 B \,b^{5} d^{6} \ln \left (e x +d \right )-60 B \,b^{5} d^{5} e x \right )}{60 \left (b x +a \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}}{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 (A + B x\right ) \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]
________________________________________________________________________________________