3.1852 \(\int (A+B x) (d+e x)^{5/2} \left (a^2+2 a b x+b^2 x^2\right )^{5/2} \, dx\)

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]

(2*(b*d - a*e)^5*(B*d - A*e)*(d + e*x)^(7/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(7*e
^7*(a + b*x)) - (2*(b*d - a*e)^4*(6*b*B*d - 5*A*b*e - a*B*e)*(d + e*x)^(9/2)*Sqr
t[a^2 + 2*a*b*x + b^2*x^2])/(9*e^7*(a + b*x)) + (10*b*(b*d - a*e)^3*(3*b*B*d - 2
*A*b*e - a*B*e)*(d + e*x)^(11/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(11*e^7*(a + b*x
)) - (20*b^2*(b*d - a*e)^2*(2*b*B*d - A*b*e - a*B*e)*(d + e*x)^(13/2)*Sqrt[a^2 +
2*a*b*x + b^2*x^2])/(13*e^7*(a + b*x)) + (2*b^3*(b*d - a*e)*(3*b*B*d - A*b*e -
2*a*B*e)*(d + e*x)^(15/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(3*e^7*(a + b*x)) - (2*
b^4*(6*b*B*d - A*b*e - 5*a*B*e)*(d + e*x)^(17/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/
(17*e^7*(a + b*x)) + (2*b^5*B*(d + e*x)^(19/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(1
9*e^7*(a + b*x))

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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]

(2*(b*d - a*e)^5*(B*d - A*e)*(d + e*x)^(7/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(7*e
^7*(a + b*x)) - (2*(b*d - a*e)^4*(6*b*B*d - 5*A*b*e - a*B*e)*(d + e*x)^(9/2)*Sqr
t[a^2 + 2*a*b*x + b^2*x^2])/(9*e^7*(a + b*x)) + (10*b*(b*d - a*e)^3*(3*b*B*d - 2
*A*b*e - a*B*e)*(d + e*x)^(11/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(11*e^7*(a + b*x
)) - (20*b^2*(b*d - a*e)^2*(2*b*B*d - A*b*e - a*B*e)*(d + e*x)^(13/2)*Sqrt[a^2 +
2*a*b*x + b^2*x^2])/(13*e^7*(a + b*x)) + (2*b^3*(b*d - a*e)*(3*b*B*d - A*b*e -
2*a*B*e)*(d + e*x)^(15/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(3*e^7*(a + b*x)) - (2*
b^4*(6*b*B*d - A*b*e - 5*a*B*e)*(d + e*x)^(17/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/
(17*e^7*(a + b*x)) + (2*b^5*B*(d + e*x)^(19/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(1
9*e^7*(a + b*x))

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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]

B*(2*a + 2*b*x)*(d + e*x)**(7/2)*(a**2 + 2*a*b*x + b**2*x**2)**(5/2)/(19*b*e) +
2*(d + e*x)**(7/2)*(a**2 + 2*a*b*x + b**2*x**2)**(5/2)*(19*A*b*e - 7*B*a*e - 12*
B*b*d)/(323*b*e**2) + 4*(5*a + 5*b*x)*(d + e*x)**(7/2)*(a*e - b*d)*(a**2 + 2*a*b
*x + b**2*x**2)**(3/2)*(19*A*b*e - 7*B*a*e - 12*B*b*d)/(4845*b*e**3) + 32*(d + e
*x)**(7/2)*(a*e - b*d)**2*(a**2 + 2*a*b*x + b**2*x**2)**(3/2)*(19*A*b*e - 7*B*a*
e - 12*B*b*d)/(12597*b*e**4) + 64*(3*a + 3*b*x)*(d + e*x)**(7/2)*(a*e - b*d)**3*
sqrt(a**2 + 2*a*b*x + b**2*x**2)*(19*A*b*e - 7*B*a*e - 12*B*b*d)/(138567*b*e**5)
+ 256*(d + e*x)**(7/2)*(a*e - b*d)**4*sqrt(a**2 + 2*a*b*x + b**2*x**2)*(19*A*b*
e - 7*B*a*e - 12*B*b*d)/(415701*b*e**6) + 512*(d + e*x)**(7/2)*(a*e - b*d)**5*sq
rt(a**2 + 2*a*b*x + b**2*x**2)*(19*A*b*e - 7*B*a*e - 12*B*b*d)/(2909907*b*e**7*(
a + b*x))

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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]

(2*Sqrt[(a + b*x)^2]*(d + e*x)^(7/2)*(46189*a^5*e^5*(-2*B*d + 9*A*e + 7*B*e*x) +
20995*a^4*b*e^4*(11*A*e*(-2*d + 7*e*x) + B*(8*d^2 - 28*d*e*x + 63*e^2*x^2)) - 3
230*a^3*b^2*e^3*(-13*A*e*(8*d^2 - 28*d*e*x + 63*e^2*x^2) + 3*B*(16*d^3 - 56*d^2*
e*x + 126*d*e^2*x^2 - 231*e^3*x^3)) + 646*a^2*b^3*e^2*(15*A*e*(-16*d^3 + 56*d^2*
e*x - 126*d*e^2*x^2 + 231*e^3*x^3) + B*(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)) - 19*a*b^4*e*(-17*A*e*(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) + 5*B*(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)) + b^5*(
19*A*e*(-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) + 3*B*(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))))/(2909907*e
^7*(a + b*x))

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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]

2/2909907*(e*x+d)^(7/2)*(153153*B*b^5*e^6*x^6+171171*A*b^5*e^6*x^5+855855*B*a*b^
4*e^6*x^5-108108*B*b^5*d*e^5*x^5+969969*A*a*b^4*e^6*x^4-114114*A*b^5*d*e^5*x^4+1
939938*B*a^2*b^3*e^6*x^4-570570*B*a*b^4*d*e^5*x^4+72072*B*b^5*d^2*e^4*x^4+223839
0*A*a^2*b^3*e^6*x^3-596904*A*a*b^4*d*e^5*x^3+70224*A*b^5*d^2*e^4*x^3+2238390*B*a
^3*b^2*e^6*x^3-1193808*B*a^2*b^3*d*e^5*x^3+351120*B*a*b^4*d^2*e^4*x^3-44352*B*b^
5*d^3*e^3*x^3+2645370*A*a^3*b^2*e^6*x^2-1220940*A*a^2*b^3*d*e^5*x^2+325584*A*a*b
^4*d^2*e^4*x^2-38304*A*b^5*d^3*e^3*x^2+1322685*B*a^4*b*e^6*x^2-1220940*B*a^3*b^2
*d*e^5*x^2+651168*B*a^2*b^3*d^2*e^4*x^2-191520*B*a*b^4*d^3*e^3*x^2+24192*B*b^5*d
^4*e^2*x^2+1616615*A*a^4*b*e^6*x-1175720*A*a^3*b^2*d*e^5*x+542640*A*a^2*b^3*d^2*
e^4*x-144704*A*a*b^4*d^3*e^3*x+17024*A*b^5*d^4*e^2*x+323323*B*a^5*e^6*x-587860*B
*a^4*b*d*e^5*x+542640*B*a^3*b^2*d^2*e^4*x-289408*B*a^2*b^3*d^3*e^3*x+85120*B*a*b
^4*d^4*e^2*x-10752*B*b^5*d^5*e*x+415701*A*a^5*e^6-461890*A*a^4*b*d*e^5+335920*A*
a^3*b^2*d^2*e^4-155040*A*a^2*b^3*d^3*e^3+41344*A*a*b^4*d^4*e^2-4864*A*b^5*d^5*e-
92378*B*a^5*d*e^5+167960*B*a^4*b*d^2*e^4-155040*B*a^3*b^2*d^3*e^3+82688*B*a^2*b^
3*d^4*e^2-24320*B*a*b^4*d^5*e+3072*B*b^5*d^6)*((b*x+a)^2)^(5/2)/e^7/(b*x+a)^5

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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]

2/153153*(9009*b^5*e^8*x^8 - 256*b^5*d^8 + 2176*a*b^4*d^7*e - 8160*a^2*b^3*d^6*e
^2 + 17680*a^3*b^2*d^5*e^3 - 24310*a^4*b*d^4*e^4 + 21879*a^5*d^3*e^5 + 3003*(7*b
^5*d*e^7 + 17*a*b^4*e^8)*x^7 + 231*(55*b^5*d^2*e^6 + 527*a*b^4*d*e^7 + 510*a^2*b
^3*e^8)*x^6 + 63*(b^5*d^3*e^5 + 1207*a*b^4*d^2*e^6 + 4590*a^2*b^3*d*e^7 + 2210*a
^3*b^2*e^8)*x^5 - 35*(2*b^5*d^4*e^4 - 17*a*b^4*d^3*e^5 - 5406*a^2*b^3*d^2*e^6 -
10166*a^3*b^2*d*e^7 - 2431*a^4*b*e^8)*x^4 + (80*b^5*d^5*e^3 - 680*a*b^4*d^4*e^4
+ 2550*a^2*b^3*d^3*e^5 + 249730*a^3*b^2*d^2*e^6 + 230945*a^4*b*d*e^7 + 21879*a^5
*e^8)*x^3 - 3*(32*b^5*d^6*e^2 - 272*a*b^4*d^5*e^3 + 1020*a^2*b^3*d^4*e^4 - 2210*
a^3*b^2*d^3*e^5 - 60775*a^4*b*d^2*e^6 - 21879*a^5*d*e^7)*x^2 + (128*b^5*d^7*e -
1088*a*b^4*d^6*e^2 + 4080*a^2*b^3*d^5*e^3 - 8840*a^3*b^2*d^4*e^4 + 12155*a^4*b*d
^3*e^5 + 65637*a^5*d^2*e^6)*x)*sqrt(e*x + d)*A/e^6 + 2/2909907*(153153*b^5*e^9*x
^9 + 3072*b^5*d^9 - 24320*a*b^4*d^8*e + 82688*a^2*b^3*d^7*e^2 - 155040*a^3*b^2*d
^6*e^3 + 167960*a^4*b*d^5*e^4 - 92378*a^5*d^4*e^5 + 9009*(39*b^5*d*e^8 + 95*a*b^
4*e^9)*x^8 + 3003*(69*b^5*d^2*e^7 + 665*a*b^4*d*e^8 + 646*a^2*b^3*e^9)*x^7 + 231
*(3*b^5*d^3*e^6 + 5225*a*b^4*d^2*e^7 + 20026*a^2*b^3*d*e^8 + 9690*a^3*b^2*e^9)*x
^6 - 63*(12*b^5*d^4*e^5 - 95*a*b^4*d^3*e^6 - 45866*a^2*b^3*d^2*e^7 - 87210*a^3*b
^2*d*e^8 - 20995*a^4*b*e^9)*x^5 + 7*(120*b^5*d^5*e^4 - 950*a*b^4*d^4*e^5 + 3230*
a^2*b^3*d^3*e^6 + 513570*a^3*b^2*d^2*e^7 + 482885*a^4*b*d*e^8 + 46189*a^5*e^9)*x
^4 - (960*b^5*d^6*e^3 - 7600*a*b^4*d^5*e^4 + 25840*a^2*b^3*d^4*e^5 - 48450*a^3*b
^2*d^3*e^6 - 2372435*a^4*b*d^2*e^7 - 877591*a^5*d*e^8)*x^3 + 3*(384*b^5*d^7*e^2
- 3040*a*b^4*d^6*e^3 + 10336*a^2*b^3*d^5*e^4 - 19380*a^3*b^2*d^4*e^5 + 20995*a^4
*b*d^3*e^6 + 230945*a^5*d^2*e^7)*x^2 - (1536*b^5*d^8*e - 12160*a*b^4*d^7*e^2 + 4
1344*a^2*b^3*d^6*e^3 - 77520*a^3*b^2*d^5*e^4 + 83980*a^4*b*d^4*e^5 - 46189*a^5*d
^3*e^6)*x)*sqrt(e*x + d)*B/e^7

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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]

2/2909907*(153153*B*b^5*e^9*x^9 + 3072*B*b^5*d^9 + 415701*A*a^5*d^3*e^6 - 4864*(
5*B*a*b^4 + A*b^5)*d^8*e + 41344*(2*B*a^2*b^3 + A*a*b^4)*d^7*e^2 - 155040*(B*a^3
*b^2 + A*a^2*b^3)*d^6*e^3 + 167960*(B*a^4*b + 2*A*a^3*b^2)*d^5*e^4 - 92378*(B*a^
5 + 5*A*a^4*b)*d^4*e^5 + 9009*(39*B*b^5*d*e^8 + 19*(5*B*a*b^4 + A*b^5)*e^9)*x^8
+ 3003*(69*B*b^5*d^2*e^7 + 133*(5*B*a*b^4 + A*b^5)*d*e^8 + 323*(2*B*a^2*b^3 + A*
a*b^4)*e^9)*x^7 + 231*(3*B*b^5*d^3*e^6 + 1045*(5*B*a*b^4 + A*b^5)*d^2*e^7 + 1001
3*(2*B*a^2*b^3 + A*a*b^4)*d*e^8 + 9690*(B*a^3*b^2 + A*a^2*b^3)*e^9)*x^6 - 63*(12
*B*b^5*d^4*e^5 - 19*(5*B*a*b^4 + A*b^5)*d^3*e^6 - 22933*(2*B*a^2*b^3 + A*a*b^4)*
d^2*e^7 - 87210*(B*a^3*b^2 + A*a^2*b^3)*d*e^8 - 20995*(B*a^4*b + 2*A*a^3*b^2)*e^
9)*x^5 + 7*(120*B*b^5*d^5*e^4 - 190*(5*B*a*b^4 + A*b^5)*d^4*e^5 + 1615*(2*B*a^2*
b^3 + A*a*b^4)*d^3*e^6 + 513570*(B*a^3*b^2 + A*a^2*b^3)*d^2*e^7 + 482885*(B*a^4*
b + 2*A*a^3*b^2)*d*e^8 + 46189*(B*a^5 + 5*A*a^4*b)*e^9)*x^4 - (960*B*b^5*d^6*e^3
- 415701*A*a^5*e^9 - 1520*(5*B*a*b^4 + A*b^5)*d^5*e^4 + 12920*(2*B*a^2*b^3 + A*
a*b^4)*d^4*e^5 - 48450*(B*a^3*b^2 + A*a^2*b^3)*d^3*e^6 - 2372435*(B*a^4*b + 2*A*
a^3*b^2)*d^2*e^7 - 877591*(B*a^5 + 5*A*a^4*b)*d*e^8)*x^3 + 3*(384*B*b^5*d^7*e^2
+ 415701*A*a^5*d*e^8 - 608*(5*B*a*b^4 + A*b^5)*d^6*e^3 + 5168*(2*B*a^2*b^3 + A*a
*b^4)*d^5*e^4 - 19380*(B*a^3*b^2 + A*a^2*b^3)*d^4*e^5 + 20995*(B*a^4*b + 2*A*a^3
*b^2)*d^3*e^6 + 230945*(B*a^5 + 5*A*a^4*b)*d^2*e^7)*x^2 - (1536*B*b^5*d^8*e - 12
47103*A*a^5*d^2*e^7 - 2432*(5*B*a*b^4 + A*b^5)*d^7*e^2 + 20672*(2*B*a^2*b^3 + A*
a*b^4)*d^6*e^3 - 77520*(B*a^3*b^2 + A*a^2*b^3)*d^5*e^4 + 83980*(B*a^4*b + 2*A*a^
3*b^2)*d^4*e^5 - 46189*(B*a^5 + 5*A*a^4*b)*d^3*e^6)*x)*sqrt(e*x + d)/e^7

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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]

Timed out

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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]

Done