Optimal. Leaf size=452 \[ -\frac{20 b^2 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{13/2} (b d-a e)^2 (-a B e-A b e+2 b B d)}{13 e^7 (a+b x)}+\frac{10 b \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{11/2} (b d-a e)^3 (-a B e-2 A b e+3 b B d)}{11 e^7 (a+b x)}-\frac{2 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{9/2} (b d-a e)^4 (-a B e-5 A b e+6 b B d)}{9 e^7 (a+b x)}+\frac{2 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{7/2} (b d-a e)^5 (B d-A e)}{7 e^7 (a+b x)}-\frac{2 b^4 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{17/2} (-5 a B e-A b e+6 b B d)}{17 e^7 (a+b x)}+\frac{2 b^3 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{15/2} (b d-a e) (-2 a B e-A b e+3 b B d)}{3 e^7 (a+b x)}+\frac{2 b^5 B \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{19/2}}{19 e^7 (a+b x)} \]
[Out]
_______________________________________________________________________________________
Rubi [A] time = 0.709342, antiderivative size = 452, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.057 \[ -\frac{20 b^2 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{13/2} (b d-a e)^2 (-a B e-A b e+2 b B d)}{13 e^7 (a+b x)}+\frac{10 b \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{11/2} (b d-a e)^3 (-a B e-2 A b e+3 b B d)}{11 e^7 (a+b x)}-\frac{2 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{9/2} (b d-a e)^4 (-a B e-5 A b e+6 b B d)}{9 e^7 (a+b x)}+\frac{2 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{7/2} (b d-a e)^5 (B d-A e)}{7 e^7 (a+b x)}-\frac{2 b^4 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{17/2} (-5 a B e-A b e+6 b B d)}{17 e^7 (a+b x)}+\frac{2 b^3 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{15/2} (b d-a e) (-2 a B e-A b e+3 b B d)}{3 e^7 (a+b x)}+\frac{2 b^5 B \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{19/2}}{19 e^7 (a+b x)} \]
Antiderivative was successfully verified.
[In] Int[(A + B*x)*(d + e*x)^(5/2)*(a^2 + 2*a*b*x + b^2*x^2)^(5/2),x]
[Out]
_______________________________________________________________________________________
Rubi in Sympy [A] time = 74.8555, size = 452, normalized size = 1. \[ \frac{B \left (2 a + 2 b x\right ) \left (d + e x\right )^{\frac{7}{2}} \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{5}{2}}}{19 b e} + \frac{2 \left (d + e x\right )^{\frac{7}{2}} \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{5}{2}} \left (19 A b e - 7 B a e - 12 B b d\right )}{323 b e^{2}} + \frac{4 \left (5 a + 5 b x\right ) \left (d + e x\right )^{\frac{7}{2}} \left (a e - b d\right ) \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{3}{2}} \left (19 A b e - 7 B a e - 12 B b d\right )}{4845 b e^{3}} + \frac{32 \left (d + e x\right )^{\frac{7}{2}} \left (a e - b d\right )^{2} \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{3}{2}} \left (19 A b e - 7 B a e - 12 B b d\right )}{12597 b e^{4}} + \frac{64 \left (3 a + 3 b x\right ) \left (d + e x\right )^{\frac{7}{2}} \left (a e - b d\right )^{3} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}} \left (19 A b e - 7 B a e - 12 B b d\right )}{138567 b e^{5}} + \frac{256 \left (d + e x\right )^{\frac{7}{2}} \left (a e - b d\right )^{4} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}} \left (19 A b e - 7 B a e - 12 B b d\right )}{415701 b e^{6}} + \frac{512 \left (d + e x\right )^{\frac{7}{2}} \left (a e - b d\right )^{5} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}} \left (19 A b e - 7 B a e - 12 B b d\right )}{2909907 b e^{7} \left (a + b x\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((B*x+A)*(e*x+d)**(5/2)*(b**2*x**2+2*a*b*x+a**2)**(5/2),x)
[Out]
_______________________________________________________________________________________
Mathematica [A] time = 1.36544, size = 492, normalized size = 1.09 \[ \frac{2 \sqrt{(a+b x)^2} (d+e x)^{7/2} \left (46189 a^5 e^5 (9 A e-2 B d+7 B e x)+20995 a^4 b e^4 \left (11 A e (7 e x-2 d)+B \left (8 d^2-28 d e x+63 e^2 x^2\right )\right )-3230 a^3 b^2 e^3 \left (3 B \left (16 d^3-56 d^2 e x+126 d e^2 x^2-231 e^3 x^3\right )-13 A e \left (8 d^2-28 d e x+63 e^2 x^2\right )\right )+646 a^2 b^3 e^2 \left (15 A e \left (-16 d^3+56 d^2 e x-126 d e^2 x^2+231 e^3 x^3\right )+B \left (128 d^4-448 d^3 e x+1008 d^2 e^2 x^2-1848 d e^3 x^3+3003 e^4 x^4\right )\right )-19 a b^4 e \left (5 B \left (256 d^5-896 d^4 e x+2016 d^3 e^2 x^2-3696 d^2 e^3 x^3+6006 d e^4 x^4-9009 e^5 x^5\right )-17 A e \left (128 d^4-448 d^3 e x+1008 d^2 e^2 x^2-1848 d e^3 x^3+3003 e^4 x^4\right )\right )+b^5 \left (19 A e \left (-256 d^5+896 d^4 e x-2016 d^3 e^2 x^2+3696 d^2 e^3 x^3-6006 d e^4 x^4+9009 e^5 x^5\right )+3 B \left (1024 d^6-3584 d^5 e x+8064 d^4 e^2 x^2-14784 d^3 e^3 x^3+24024 d^2 e^4 x^4-36036 d e^5 x^5+51051 e^6 x^6\right )\right )\right )}{2909907 e^7 (a+b x)} \]
Antiderivative was successfully verified.
[In] Integrate[(A + B*x)*(d + e*x)^(5/2)*(a^2 + 2*a*b*x + b^2*x^2)^(5/2),x]
[Out]
_______________________________________________________________________________________
Maple [A] time = 0.014, size = 689, normalized size = 1.5 \[{\frac{306306\,B{x}^{6}{b}^{5}{e}^{6}+342342\,A{x}^{5}{b}^{5}{e}^{6}+1711710\,B{x}^{5}a{b}^{4}{e}^{6}-216216\,B{x}^{5}{b}^{5}d{e}^{5}+1939938\,A{x}^{4}a{b}^{4}{e}^{6}-228228\,A{x}^{4}{b}^{5}d{e}^{5}+3879876\,B{x}^{4}{a}^{2}{b}^{3}{e}^{6}-1141140\,B{x}^{4}a{b}^{4}d{e}^{5}+144144\,B{x}^{4}{b}^{5}{d}^{2}{e}^{4}+4476780\,A{x}^{3}{a}^{2}{b}^{3}{e}^{6}-1193808\,A{x}^{3}a{b}^{4}d{e}^{5}+140448\,A{x}^{3}{b}^{5}{d}^{2}{e}^{4}+4476780\,B{x}^{3}{a}^{3}{b}^{2}{e}^{6}-2387616\,B{x}^{3}{a}^{2}{b}^{3}d{e}^{5}+702240\,B{x}^{3}a{b}^{4}{d}^{2}{e}^{4}-88704\,B{x}^{3}{b}^{5}{d}^{3}{e}^{3}+5290740\,A{x}^{2}{a}^{3}{b}^{2}{e}^{6}-2441880\,A{x}^{2}{a}^{2}{b}^{3}d{e}^{5}+651168\,A{x}^{2}a{b}^{4}{d}^{2}{e}^{4}-76608\,A{x}^{2}{b}^{5}{d}^{3}{e}^{3}+2645370\,B{x}^{2}{a}^{4}b{e}^{6}-2441880\,B{x}^{2}{a}^{3}{b}^{2}d{e}^{5}+1302336\,B{x}^{2}{a}^{2}{b}^{3}{d}^{2}{e}^{4}-383040\,B{x}^{2}a{b}^{4}{d}^{3}{e}^{3}+48384\,B{x}^{2}{b}^{5}{d}^{4}{e}^{2}+3233230\,Ax{a}^{4}b{e}^{6}-2351440\,Ax{a}^{3}{b}^{2}d{e}^{5}+1085280\,Ax{a}^{2}{b}^{3}{d}^{2}{e}^{4}-289408\,Axa{b}^{4}{d}^{3}{e}^{3}+34048\,Ax{b}^{5}{d}^{4}{e}^{2}+646646\,Bx{a}^{5}{e}^{6}-1175720\,Bx{a}^{4}bd{e}^{5}+1085280\,Bx{a}^{3}{b}^{2}{d}^{2}{e}^{4}-578816\,Bx{a}^{2}{b}^{3}{d}^{3}{e}^{3}+170240\,Bxa{b}^{4}{d}^{4}{e}^{2}-21504\,Bx{b}^{5}{d}^{5}e+831402\,A{a}^{5}{e}^{6}-923780\,Ad{e}^{5}{a}^{4}b+671840\,A{a}^{3}{b}^{2}{d}^{2}{e}^{4}-310080\,A{a}^{2}{b}^{3}{d}^{3}{e}^{3}+82688\,Aa{b}^{4}{d}^{4}{e}^{2}-9728\,A{b}^{5}{d}^{5}e-184756\,Bd{e}^{5}{a}^{5}+335920\,B{a}^{4}b{d}^{2}{e}^{4}-310080\,B{a}^{3}{b}^{2}{d}^{3}{e}^{3}+165376\,B{a}^{2}{b}^{3}{d}^{4}{e}^{2}-48640\,Ba{b}^{4}{d}^{5}e+6144\,B{b}^{5}{d}^{6}}{2909907\,{e}^{7} \left ( bx+a \right ) ^{5}} \left ( ex+d \right ) ^{{\frac{7}{2}}} \left ( \left ( bx+a \right ) ^{2} \right ) ^{{\frac{5}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((B*x+A)*(e*x+d)^(5/2)*(b^2*x^2+2*a*b*x+a^2)^(5/2),x)
[Out]
_______________________________________________________________________________________
Maxima [A] time = 0.737061, size = 1458, normalized size = 3.23 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b^2*x^2 + 2*a*b*x + a^2)^(5/2)*(B*x + A)*(e*x + d)^(5/2),x, algorithm="maxima")
[Out]
_______________________________________________________________________________________
Fricas [A] time = 0.302228, size = 1341, normalized size = 2.97 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b^2*x^2 + 2*a*b*x + a^2)^(5/2)*(B*x + A)*(e*x + d)^(5/2),x, algorithm="fricas")
[Out]
_______________________________________________________________________________________
Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x+A)*(e*x+d)**(5/2)*(b**2*x**2+2*a*b*x+a**2)**(5/2),x)
[Out]
_______________________________________________________________________________________
GIAC/XCAS [A] time = 0.394059, size = 1, normalized size = 0. \[ \mathit{Done} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b^2*x^2 + 2*a*b*x + a^2)^(5/2)*(B*x + A)*(e*x + d)^(5/2),x, algorithm="giac")
[Out]