3.2206 \(\int (a+b x)^{5/2} (A+B x) (d+e x)^{5/2} \, dx\)

Optimal. Leaf size=412 \[ -\frac{5 (b d-a e)^6 (2 A b e-B (a e+b d)) \tanh ^{-1}\left (\frac{\sqrt{e} \sqrt{a+b x}}{\sqrt{b} \sqrt{d+e x}}\right )}{1024 b^{9/2} e^{9/2}}+\frac{5 \sqrt{a+b x} \sqrt{d+e x} (b d-a e)^5 (2 A b e-B (a e+b d))}{1024 b^4 e^4}-\frac{5 (a+b x)^{3/2} \sqrt{d+e x} (b d-a e)^4 (2 A b e-B (a e+b d))}{1536 b^4 e^3}+\frac{(a+b x)^{5/2} \sqrt{d+e x} (b d-a e)^3 (2 A b e-B (a e+b d))}{384 b^4 e^2}+\frac{(a+b x)^{7/2} \sqrt{d+e x} (b d-a e)^2 (2 A b e-B (a e+b d))}{64 b^4 e}+\frac{(a+b x)^{7/2} (d+e x)^{3/2} (b d-a e) (2 A b e-B (a e+b d))}{24 b^3 e}+\frac{(a+b x)^{7/2} (d+e x)^{5/2} (2 A b e-B (a e+b d))}{12 b^2 e}+\frac{B (a+b x)^{7/2} (d+e x)^{7/2}}{7 b e} \]

[Out]

(5*(b*d - a*e)^5*(2*A*b*e - B*(b*d + a*e))*Sqrt[a + b*x]*Sqrt[d + e*x])/(1024*b^
4*e^4) - (5*(b*d - a*e)^4*(2*A*b*e - B*(b*d + a*e))*(a + b*x)^(3/2)*Sqrt[d + e*x
])/(1536*b^4*e^3) + ((b*d - a*e)^3*(2*A*b*e - B*(b*d + a*e))*(a + b*x)^(5/2)*Sqr
t[d + e*x])/(384*b^4*e^2) + ((b*d - a*e)^2*(2*A*b*e - B*(b*d + a*e))*(a + b*x)^(
7/2)*Sqrt[d + e*x])/(64*b^4*e) + ((b*d - a*e)*(2*A*b*e - B*(b*d + a*e))*(a + b*x
)^(7/2)*(d + e*x)^(3/2))/(24*b^3*e) + ((2*A*b*e - B*(b*d + a*e))*(a + b*x)^(7/2)
*(d + e*x)^(5/2))/(12*b^2*e) + (B*(a + b*x)^(7/2)*(d + e*x)^(7/2))/(7*b*e) - (5*
(b*d - a*e)^6*(2*A*b*e - B*(b*d + a*e))*ArcTanh[(Sqrt[e]*Sqrt[a + b*x])/(Sqrt[b]
*Sqrt[d + e*x])])/(1024*b^(9/2)*e^(9/2))

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Rubi [A]  time = 0.964434, antiderivative size = 412, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167 \[ -\frac{5 (b d-a e)^6 (2 A b e-B (a e+b d)) \tanh ^{-1}\left (\frac{\sqrt{e} \sqrt{a+b x}}{\sqrt{b} \sqrt{d+e x}}\right )}{1024 b^{9/2} e^{9/2}}+\frac{5 \sqrt{a+b x} \sqrt{d+e x} (b d-a e)^5 (2 A b e-B (a e+b d))}{1024 b^4 e^4}-\frac{5 (a+b x)^{3/2} \sqrt{d+e x} (b d-a e)^4 (2 A b e-B (a e+b d))}{1536 b^4 e^3}+\frac{(a+b x)^{5/2} \sqrt{d+e x} (b d-a e)^3 (2 A b e-B (a e+b d))}{384 b^4 e^2}+\frac{(a+b x)^{7/2} \sqrt{d+e x} (b d-a e)^2 (2 A b e-B (a e+b d))}{64 b^4 e}+\frac{(a+b x)^{7/2} (d+e x)^{3/2} (b d-a e) (2 A b e-B (a e+b d))}{24 b^3 e}+\frac{(a+b x)^{7/2} (d+e x)^{5/2} (2 A b e-B (a e+b d))}{12 b^2 e}+\frac{B (a+b x)^{7/2} (d+e x)^{7/2}}{7 b e} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x)^(5/2)*(A + B*x)*(d + e*x)^(5/2),x]

[Out]

(5*(b*d - a*e)^5*(2*A*b*e - B*(b*d + a*e))*Sqrt[a + b*x]*Sqrt[d + e*x])/(1024*b^
4*e^4) - (5*(b*d - a*e)^4*(2*A*b*e - B*(b*d + a*e))*(a + b*x)^(3/2)*Sqrt[d + e*x
])/(1536*b^4*e^3) + ((b*d - a*e)^3*(2*A*b*e - B*(b*d + a*e))*(a + b*x)^(5/2)*Sqr
t[d + e*x])/(384*b^4*e^2) + ((b*d - a*e)^2*(2*A*b*e - B*(b*d + a*e))*(a + b*x)^(
7/2)*Sqrt[d + e*x])/(64*b^4*e) + ((b*d - a*e)*(2*A*b*e - B*(b*d + a*e))*(a + b*x
)^(7/2)*(d + e*x)^(3/2))/(24*b^3*e) + ((2*A*b*e - B*(b*d + a*e))*(a + b*x)^(7/2)
*(d + e*x)^(5/2))/(12*b^2*e) + (B*(a + b*x)^(7/2)*(d + e*x)^(7/2))/(7*b*e) - (5*
(b*d - a*e)^6*(2*A*b*e - B*(b*d + a*e))*ArcTanh[(Sqrt[e]*Sqrt[a + b*x])/(Sqrt[b]
*Sqrt[d + e*x])])/(1024*b^(9/2)*e^(9/2))

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Rubi in Sympy [A]  time = 96.3893, size = 377, normalized size = 0.92 \[ \frac{B \left (a + b x\right )^{\frac{7}{2}} \left (d + e x\right )^{\frac{7}{2}}}{7 b e} - \frac{\left (a + b x\right )^{\frac{5}{2}} \left (d + e x\right )^{\frac{7}{2}} \left (- A b e + \frac{B \left (a e + b d\right )}{2}\right )}{6 b e^{2}} + \frac{\left (a + b x\right )^{\frac{3}{2}} \left (d + e x\right )^{\frac{7}{2}} \left (a e - b d\right ) \left (2 A b e - B a e - B b d\right )}{24 b e^{3}} - \frac{\left (a + b x\right )^{\frac{3}{2}} \left (d + e x\right )^{\frac{5}{2}} \left (a e - b d\right )^{2} \left (- A b e + \frac{B \left (a e + b d\right )}{2}\right )}{32 b^{2} e^{3}} + \frac{5 \left (a + b x\right )^{\frac{3}{2}} \left (d + e x\right )^{\frac{3}{2}} \left (a e - b d\right )^{3} \left (- A b e + \frac{B \left (a e + b d\right )}{2}\right )}{192 b^{3} e^{3}} - \frac{5 \left (a + b x\right )^{\frac{3}{2}} \sqrt{d + e x} \left (a e - b d\right )^{4} \left (- A b e + \frac{B \left (a e + b d\right )}{2}\right )}{256 b^{4} e^{3}} + \frac{5 \sqrt{a + b x} \sqrt{d + e x} \left (a e - b d\right )^{5} \left (- A b e + \frac{B \left (a e + b d\right )}{2}\right )}{512 b^{4} e^{4}} + \frac{5 \left (a e - b d\right )^{6} \left (- A b e + \frac{B \left (a e + b d\right )}{2}\right ) \operatorname{atanh}{\left (\frac{\sqrt{b} \sqrt{d + e x}}{\sqrt{e} \sqrt{a + b x}} \right )}}{512 b^{\frac{9}{2}} e^{\frac{9}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)**(5/2)*(B*x+A)*(e*x+d)**(5/2),x)

[Out]

B*(a + b*x)**(7/2)*(d + e*x)**(7/2)/(7*b*e) - (a + b*x)**(5/2)*(d + e*x)**(7/2)*
(-A*b*e + B*(a*e + b*d)/2)/(6*b*e**2) + (a + b*x)**(3/2)*(d + e*x)**(7/2)*(a*e -
b*d)*(2*A*b*e - B*a*e - B*b*d)/(24*b*e**3) - (a + b*x)**(3/2)*(d + e*x)**(5/2)*
(a*e - b*d)**2*(-A*b*e + B*(a*e + b*d)/2)/(32*b**2*e**3) + 5*(a + b*x)**(3/2)*(d
+ e*x)**(3/2)*(a*e - b*d)**3*(-A*b*e + B*(a*e + b*d)/2)/(192*b**3*e**3) - 5*(a
+ b*x)**(3/2)*sqrt(d + e*x)*(a*e - b*d)**4*(-A*b*e + B*(a*e + b*d)/2)/(256*b**4*
e**3) + 5*sqrt(a + b*x)*sqrt(d + e*x)*(a*e - b*d)**5*(-A*b*e + B*(a*e + b*d)/2)/
(512*b**4*e**4) + 5*(a*e - b*d)**6*(-A*b*e + B*(a*e + b*d)/2)*atanh(sqrt(b)*sqrt
(d + e*x)/(sqrt(e)*sqrt(a + b*x)))/(512*b**(9/2)*e**(9/2))

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Mathematica [A]  time = 1.25709, size = 575, normalized size = 1.4 \[ \frac{\sqrt{a+b x} \sqrt{d+e x} \left (-105 a^6 B e^6+70 a^5 b e^5 (3 A e+7 B d+B e x)-7 a^4 b^2 e^4 \left (10 A e (17 d+2 e x)+B \left (113 d^2+46 d e x+8 e^2 x^2\right )\right )+4 a^3 b^3 e^3 \left (7 A e \left (99 d^2+28 d e x+4 e^2 x^2\right )+B \left (75 d^3+127 d^2 e x+64 d e^2 x^2+12 e^3 x^3\right )\right )+a^2 b^4 e^2 \left (84 A e \left (33 d^3+198 d^2 e x+212 d e^2 x^2+72 e^3 x^3\right )+B \left (-791 d^4+508 d^3 e x+9840 d^2 e^2 x^2+12752 d e^3 x^3+4736 e^4 x^4\right )\right )+2 a b^5 e \left (7 A e \left (-85 d^4+56 d^3 e x+1272 d^2 e^2 x^2+1696 d e^3 x^3+640 e^4 x^4\right )+B \left (245 d^5-161 d^4 e x+128 d^3 e^2 x^2+6376 d^2 e^3 x^3+9344 d e^4 x^4+3712 e^5 x^5\right )\right )+b^6 \left (14 A e \left (15 d^5-10 d^4 e x+8 d^3 e^2 x^2+432 d^2 e^3 x^3+640 d e^4 x^4+256 e^5 x^5\right )+B \left (-105 d^6+70 d^5 e x-56 d^4 e^2 x^2+48 d^3 e^3 x^3+4736 d^2 e^4 x^4+7424 d e^5 x^5+3072 e^6 x^6\right )\right )\right )}{21504 b^4 e^4}+\frac{5 (b d-a e)^6 (a B e-2 A b e+b B d) \log \left (2 \sqrt{b} \sqrt{e} \sqrt{a+b x} \sqrt{d+e x}+a e+b d+2 b e x\right )}{2048 b^{9/2} e^{9/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x)^(5/2)*(A + B*x)*(d + e*x)^(5/2),x]

[Out]

(Sqrt[a + b*x]*Sqrt[d + e*x]*(-105*a^6*B*e^6 + 70*a^5*b*e^5*(7*B*d + 3*A*e + B*e
*x) - 7*a^4*b^2*e^4*(10*A*e*(17*d + 2*e*x) + B*(113*d^2 + 46*d*e*x + 8*e^2*x^2))
+ 4*a^3*b^3*e^3*(7*A*e*(99*d^2 + 28*d*e*x + 4*e^2*x^2) + B*(75*d^3 + 127*d^2*e*
x + 64*d*e^2*x^2 + 12*e^3*x^3)) + a^2*b^4*e^2*(84*A*e*(33*d^3 + 198*d^2*e*x + 21
2*d*e^2*x^2 + 72*e^3*x^3) + B*(-791*d^4 + 508*d^3*e*x + 9840*d^2*e^2*x^2 + 12752
*d*e^3*x^3 + 4736*e^4*x^4)) + 2*a*b^5*e*(7*A*e*(-85*d^4 + 56*d^3*e*x + 1272*d^2*
e^2*x^2 + 1696*d*e^3*x^3 + 640*e^4*x^4) + B*(245*d^5 - 161*d^4*e*x + 128*d^3*e^2
*x^2 + 6376*d^2*e^3*x^3 + 9344*d*e^4*x^4 + 3712*e^5*x^5)) + b^6*(14*A*e*(15*d^5
- 10*d^4*e*x + 8*d^3*e^2*x^2 + 432*d^2*e^3*x^3 + 640*d*e^4*x^4 + 256*e^5*x^5) +
B*(-105*d^6 + 70*d^5*e*x - 56*d^4*e^2*x^2 + 48*d^3*e^3*x^3 + 4736*d^2*e^4*x^4 +
7424*d*e^5*x^5 + 3072*e^6*x^6))))/(21504*b^4*e^4) + (5*(b*d - a*e)^6*(b*B*d - 2*
A*b*e + a*B*e)*Log[b*d + a*e + 2*b*e*x + 2*Sqrt[b]*Sqrt[e]*Sqrt[a + b*x]*Sqrt[d
+ e*x]])/(2048*b^(9/2)*e^(9/2))

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Maple [B]  time = 0.041, size = 2851, normalized size = 6.9 \[ \text{output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^(5/2)*(B*x+A)*(e*x+d)^(5/2),x)

[Out]

-1/43008*(b*x+a)^(1/2)*(e*x+d)^(1/2)*(-1260*ln(1/2*(2*b*x*e+2*(b*e*x^2+a*e*x+b*d
*x+a*d)^(1/2)*(b*e)^(1/2)+a*e+b*d)/(b*e)^(1/2))*a^5*d*e^6*A*b^2+644*b^5*(b*e*x^2
+a*e*x+b*d*x+a*d)^(1/2)*x*d^4*a*B*e^2*(b*e)^(1/2)-1016*(b*e*x^2+a*e*x+b*d*x+a*d)
^(1/2)*x*d^3*a^2*B*e^3*b^4*(b*e)^(1/2)-1016*(b*e*x^2+a*e*x+b*d*x+a*d)^(1/2)*x*a^
3*d^2*B*e^4*b^3*(b*e)^(1/2)+644*e^5*(b*e*x^2+a*e*x+b*d*x+a*d)^(1/2)*x*a^4*d*B*b^
2*(b*e)^(1/2)-1568*b^5*(b*e*x^2+a*e*x+b*d*x+a*d)^(1/2)*x*d^3*a*A*e^3*(b*e)^(1/2)
-1568*e^5*(b*e*x^2+a*e*x+b*d*x+a*d)^(1/2)*x*a^3*d*A*b^3*(b*e)^(1/2)-33264*(b*e*x
^2+a*e*x+b*d*x+a*d)^(1/2)*x*a^2*d^2*A*e^4*b^4*(b*e)^(1/2)+210*e^7*ln(1/2*(2*b*x*
e+2*(b*e*x^2+a*e*x+b*d*x+a*d)^(1/2)*(b*e)^(1/2)+a*e+b*d)/(b*e)^(1/2))*a^6*A*b+21
0*b^7*ln(1/2*(2*b*x*e+2*(b*e*x^2+a*e*x+b*d*x+a*d)^(1/2)*(b*e)^(1/2)+a*e+b*d)/(b*
e)^(1/2))*d^6*A*e+210*(b*e*x^2+a*e*x+b*d*x+a*d)^(1/2)*a^6*B*e^6*(b*e)^(1/2)+210*
(b*e*x^2+a*e*x+b*d*x+a*d)^(1/2)*d^6*B*b^6*(b*e)^(1/2)-6144*B*x^6*b^6*e^6*(b*e)^(
1/2)*(b*e*x^2+a*e*x+b*d*x+a*d)^(1/2)-7168*A*x^5*b^6*e^6*(b*e)^(1/2)*(b*e*x^2+a*e
*x+b*d*x+a*d)^(1/2)+3150*e^5*ln(1/2*(2*b*x*e+2*(b*e*x^2+a*e*x+b*d*x+a*d)^(1/2)*(
b*e)^(1/2)+a*e+b*d)/(b*e)^(1/2))*a^4*d^2*A*b^3-4200*ln(1/2*(2*b*x*e+2*(b*e*x^2+a
*e*x+b*d*x+a*d)^(1/2)*(b*e)^(1/2)+a*e+b*d)/(b*e)^(1/2))*a^3*d^3*A*e^4*b^4+3150*b
^5*ln(1/2*(2*b*x*e+2*(b*e*x^2+a*e*x+b*d*x+a*d)^(1/2)*(b*e)^(1/2)+a*e+b*d)/(b*e)^
(1/2))*d^4*a^2*A*e^3-1260*ln(1/2*(2*b*x*e+2*(b*e*x^2+a*e*x+b*d*x+a*d)^(1/2)*(b*e
)^(1/2)+a*e+b*d)/(b*e)^(1/2))*d^5*a*b^6*A*e^2+525*ln(1/2*(2*b*x*e+2*(b*e*x^2+a*e
*x+b*d*x+a*d)^(1/2)*(b*e)^(1/2)+a*e+b*d)/(b*e)^(1/2))*a^6*d*B*e^6*b-945*ln(1/2*(
2*b*x*e+2*(b*e*x^2+a*e*x+b*d*x+a*d)^(1/2)*(b*e)^(1/2)+a*e+b*d)/(b*e)^(1/2))*a^5*
d^2*B*e^5*b^2+525*ln(1/2*(2*b*x*e+2*(b*e*x^2+a*e*x+b*d*x+a*d)^(1/2)*(b*e)^(1/2)+
a*e+b*d)/(b*e)^(1/2))*a^4*d^3*B*e^4*b^3+525*ln(1/2*(2*b*x*e+2*(b*e*x^2+a*e*x+b*d
*x+a*d)^(1/2)*(b*e)^(1/2)+a*e+b*d)/(b*e)^(1/2))*a^3*d^4*B*e^3*b^4-945*ln(1/2*(2*
b*x*e+2*(b*e*x^2+a*e*x+b*d*x+a*d)^(1/2)*(b*e)^(1/2)+a*e+b*d)/(b*e)^(1/2))*d^5*a^
2*B*b^5*e^2+525*ln(1/2*(2*b*x*e+2*(b*e*x^2+a*e*x+b*d*x+a*d)^(1/2)*(b*e)^(1/2)+a*
e+b*d)/(b*e)^(1/2))*d^6*a*B*b^6*e-420*(b*e*x^2+a*e*x+b*d*x+a*d)^(1/2)*a^5*e^6*A*
b*(b*e)^(1/2)-420*(b*e*x^2+a*e*x+b*d*x+a*d)^(1/2)*d^5*b^6*A*e*(b*e)^(1/2)-105*e^
7*ln(1/2*(2*b*x*e+2*(b*e*x^2+a*e*x+b*d*x+a*d)^(1/2)*(b*e)^(1/2)+a*e+b*d)/(b*e)^(
1/2))*a^7*B-105*b^7*ln(1/2*(2*b*x*e+2*(b*e*x^2+a*e*x+b*d*x+a*d)^(1/2)*(b*e)^(1/2
)+a*e+b*d)/(b*e)^(1/2))*d^7*B-37376*B*x^4*a*b^5*d*e^5*(b*e)^(1/2)*(b*e*x^2+a*e*x
+b*d*x+a*d)^(1/2)-47488*A*x^3*a*b^5*d*e^5*(b*e)^(1/2)*(b*e*x^2+a*e*x+b*d*x+a*d)^
(1/2)-25504*B*x^3*a^2*b^4*d*e^5*(b*e)^(1/2)*(b*e*x^2+a*e*x+b*d*x+a*d)^(1/2)-2550
4*B*x^3*a*b^5*d^2*e^4*(b*e)^(1/2)*(b*e*x^2+a*e*x+b*d*x+a*d)^(1/2)-35616*A*x^2*a^
2*b^4*d*e^5*(b*e)^(1/2)*(b*e*x^2+a*e*x+b*d*x+a*d)^(1/2)-35616*A*x^2*a*b^5*d^2*e^
4*(b*e)^(1/2)*(b*e*x^2+a*e*x+b*d*x+a*d)^(1/2)-512*B*x^2*a^3*b^3*d*e^5*(b*e)^(1/2
)*(b*e*x^2+a*e*x+b*d*x+a*d)^(1/2)-19680*B*x^2*a^2*b^4*d^2*e^4*(b*e)^(1/2)*(b*e*x
^2+a*e*x+b*d*x+a*d)^(1/2)-512*B*x^2*a*b^5*d^3*e^3*(b*e)^(1/2)*(b*e*x^2+a*e*x+b*d
*x+a*d)^(1/2)+112*B*x^2*b^6*d^4*e^2*(b*e)^(1/2)*(b*e*x^2+a*e*x+b*d*x+a*d)^(1/2)-
14848*B*x^5*a*b^5*e^6*(b*e)^(1/2)*(b*e*x^2+a*e*x+b*d*x+a*d)^(1/2)-14848*B*x^5*b^
6*d*e^5*(b*e)^(1/2)*(b*e*x^2+a*e*x+b*d*x+a*d)^(1/2)+280*(b*e*x^2+a*e*x+b*d*x+a*d
)^(1/2)*x*a^4*e^6*A*b^2*(b*e)^(1/2)+280*(b*e*x^2+a*e*x+b*d*x+a*d)^(1/2)*x*d^4*b^
6*A*e^2*(b*e)^(1/2)-140*(b*e*x^2+a*e*x+b*d*x+a*d)^(1/2)*x*a^5*B*e^6*b*(b*e)^(1/2
)-140*(b*e*x^2+a*e*x+b*d*x+a*d)^(1/2)*x*d^5*B*b^6*e*(b*e)^(1/2)+2380*e^5*(b*e*x^
2+a*e*x+b*d*x+a*d)^(1/2)*a^4*d*A*b^2*(b*e)^(1/2)-5544*(b*e*x^2+a*e*x+b*d*x+a*d)^
(1/2)*a^3*d^2*A*e^4*b^3*(b*e)^(1/2)-5544*(b*e*x^2+a*e*x+b*d*x+a*d)^(1/2)*a^2*d^3
*A*e^3*b^4*(b*e)^(1/2)+2380*b^5*(b*e*x^2+a*e*x+b*d*x+a*d)^(1/2)*d^4*a*A*e^2*(b*e
)^(1/2)-980*(b*e*x^2+a*e*x+b*d*x+a*d)^(1/2)*a^5*B*d*e^5*b*(b*e)^(1/2)+1582*(b*e*
x^2+a*e*x+b*d*x+a*d)^(1/2)*a^4*d^2*B*e^4*b^2*(b*e)^(1/2)-600*(b*e*x^2+a*e*x+b*d*
x+a*d)^(1/2)*a^3*d^3*B*e^3*b^3*(b*e)^(1/2)+1582*(b*e*x^2+a*e*x+b*d*x+a*d)^(1/2)*
d^4*a^2*B*e^2*b^4*(b*e)^(1/2)-980*(b*e*x^2+a*e*x+b*d*x+a*d)^(1/2)*d^5*B*a*b^5*e*
(b*e)^(1/2)-17920*A*x^4*a*b^5*e^6*(b*e)^(1/2)*(b*e*x^2+a*e*x+b*d*x+a*d)^(1/2)-17
920*A*x^4*b^6*d*e^5*(b*e)^(1/2)*(b*e*x^2+a*e*x+b*d*x+a*d)^(1/2)-9472*B*x^4*a^2*b
^4*e^6*(b*e)^(1/2)*(b*e*x^2+a*e*x+b*d*x+a*d)^(1/2)-9472*B*x^4*b^6*d^2*e^4*(b*e)^
(1/2)*(b*e*x^2+a*e*x+b*d*x+a*d)^(1/2)-12096*A*x^3*a^2*b^4*e^6*(b*e)^(1/2)*(b*e*x
^2+a*e*x+b*d*x+a*d)^(1/2)-12096*A*x^3*b^6*d^2*e^4*(b*e)^(1/2)*(b*e*x^2+a*e*x+b*d
*x+a*d)^(1/2)-96*B*x^3*a^3*b^3*e^6*(b*e)^(1/2)*(b*e*x^2+a*e*x+b*d*x+a*d)^(1/2)-9
6*B*x^3*b^6*d^3*e^3*(b*e)^(1/2)*(b*e*x^2+a*e*x+b*d*x+a*d)^(1/2)-224*A*x^2*a^3*b^
3*e^6*(b*e)^(1/2)*(b*e*x^2+a*e*x+b*d*x+a*d)^(1/2)-224*A*x^2*b^6*d^3*e^3*(b*e)^(1
/2)*(b*e*x^2+a*e*x+b*d*x+a*d)^(1/2)+112*B*x^2*a^4*b^2*e^6*(b*e)^(1/2)*(b*e*x^2+a
*e*x+b*d*x+a*d)^(1/2))/(b*e*x^2+a*e*x+b*d*x+a*d)^(1/2)/e^4/b^4/(b*e)^(1/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*(b*x + a)^(5/2)*(e*x + d)^(5/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.381798, size = 1, normalized size = 0. \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*(b*x + a)^(5/2)*(e*x + d)^(5/2),x, algorithm="fricas")

[Out]

[1/86016*(4*(3072*B*b^6*e^6*x^6 - 105*B*b^6*d^6 + 70*(7*B*a*b^5 + 3*A*b^6)*d^5*e
- 7*(113*B*a^2*b^4 + 170*A*a*b^5)*d^4*e^2 + 12*(25*B*a^3*b^3 + 231*A*a^2*b^4)*d
^3*e^3 - 7*(113*B*a^4*b^2 - 396*A*a^3*b^3)*d^2*e^4 + 70*(7*B*a^5*b - 17*A*a^4*b^
2)*d*e^5 - 105*(B*a^6 - 2*A*a^5*b)*e^6 + 256*(29*B*b^6*d*e^5 + (29*B*a*b^5 + 14*
A*b^6)*e^6)*x^5 + 128*(37*B*b^6*d^2*e^4 + 2*(73*B*a*b^5 + 35*A*b^6)*d*e^5 + (37*
B*a^2*b^4 + 70*A*a*b^5)*e^6)*x^4 + 16*(3*B*b^6*d^3*e^3 + (797*B*a*b^5 + 378*A*b^
6)*d^2*e^4 + (797*B*a^2*b^4 + 1484*A*a*b^5)*d*e^5 + 3*(B*a^3*b^3 + 126*A*a^2*b^4
)*e^6)*x^3 - 8*(7*B*b^6*d^4*e^2 - 2*(16*B*a*b^5 + 7*A*b^6)*d^3*e^3 - 6*(205*B*a^
2*b^4 + 371*A*a*b^5)*d^2*e^4 - 2*(16*B*a^3*b^3 + 1113*A*a^2*b^4)*d*e^5 + 7*(B*a^
4*b^2 - 2*A*a^3*b^3)*e^6)*x^2 + 2*(35*B*b^6*d^5*e - 7*(23*B*a*b^5 + 10*A*b^6)*d^
4*e^2 + 2*(127*B*a^2*b^4 + 196*A*a*b^5)*d^3*e^3 + 2*(127*B*a^3*b^3 + 4158*A*a^2*
b^4)*d^2*e^4 - 7*(23*B*a^4*b^2 - 56*A*a^3*b^3)*d*e^5 + 35*(B*a^5*b - 2*A*a^4*b^2
)*e^6)*x)*sqrt(b*e)*sqrt(b*x + a)*sqrt(e*x + d) + 105*(B*b^7*d^7 - (5*B*a*b^6 +
2*A*b^7)*d^6*e + 3*(3*B*a^2*b^5 + 4*A*a*b^6)*d^5*e^2 - 5*(B*a^3*b^4 + 6*A*a^2*b^
5)*d^4*e^3 - 5*(B*a^4*b^3 - 8*A*a^3*b^4)*d^3*e^4 + 3*(3*B*a^5*b^2 - 10*A*a^4*b^3
)*d^2*e^5 - (5*B*a^6*b - 12*A*a^5*b^2)*d*e^6 + (B*a^7 - 2*A*a^6*b)*e^7)*log(4*(2
*b^2*e^2*x + b^2*d*e + a*b*e^2)*sqrt(b*x + a)*sqrt(e*x + d) + (8*b^2*e^2*x^2 + b
^2*d^2 + 6*a*b*d*e + a^2*e^2 + 8*(b^2*d*e + a*b*e^2)*x)*sqrt(b*e)))/(sqrt(b*e)*b
^4*e^4), 1/43008*(2*(3072*B*b^6*e^6*x^6 - 105*B*b^6*d^6 + 70*(7*B*a*b^5 + 3*A*b^
6)*d^5*e - 7*(113*B*a^2*b^4 + 170*A*a*b^5)*d^4*e^2 + 12*(25*B*a^3*b^3 + 231*A*a^
2*b^4)*d^3*e^3 - 7*(113*B*a^4*b^2 - 396*A*a^3*b^3)*d^2*e^4 + 70*(7*B*a^5*b - 17*
A*a^4*b^2)*d*e^5 - 105*(B*a^6 - 2*A*a^5*b)*e^6 + 256*(29*B*b^6*d*e^5 + (29*B*a*b
^5 + 14*A*b^6)*e^6)*x^5 + 128*(37*B*b^6*d^2*e^4 + 2*(73*B*a*b^5 + 35*A*b^6)*d*e^
5 + (37*B*a^2*b^4 + 70*A*a*b^5)*e^6)*x^4 + 16*(3*B*b^6*d^3*e^3 + (797*B*a*b^5 +
378*A*b^6)*d^2*e^4 + (797*B*a^2*b^4 + 1484*A*a*b^5)*d*e^5 + 3*(B*a^3*b^3 + 126*A
*a^2*b^4)*e^6)*x^3 - 8*(7*B*b^6*d^4*e^2 - 2*(16*B*a*b^5 + 7*A*b^6)*d^3*e^3 - 6*(
205*B*a^2*b^4 + 371*A*a*b^5)*d^2*e^4 - 2*(16*B*a^3*b^3 + 1113*A*a^2*b^4)*d*e^5 +
7*(B*a^4*b^2 - 2*A*a^3*b^3)*e^6)*x^2 + 2*(35*B*b^6*d^5*e - 7*(23*B*a*b^5 + 10*A
*b^6)*d^4*e^2 + 2*(127*B*a^2*b^4 + 196*A*a*b^5)*d^3*e^3 + 2*(127*B*a^3*b^3 + 415
8*A*a^2*b^4)*d^2*e^4 - 7*(23*B*a^4*b^2 - 56*A*a^3*b^3)*d*e^5 + 35*(B*a^5*b - 2*A
*a^4*b^2)*e^6)*x)*sqrt(-b*e)*sqrt(b*x + a)*sqrt(e*x + d) + 105*(B*b^7*d^7 - (5*B
*a*b^6 + 2*A*b^7)*d^6*e + 3*(3*B*a^2*b^5 + 4*A*a*b^6)*d^5*e^2 - 5*(B*a^3*b^4 + 6
*A*a^2*b^5)*d^4*e^3 - 5*(B*a^4*b^3 - 8*A*a^3*b^4)*d^3*e^4 + 3*(3*B*a^5*b^2 - 10*
A*a^4*b^3)*d^2*e^5 - (5*B*a^6*b - 12*A*a^5*b^2)*d*e^6 + (B*a^7 - 2*A*a^6*b)*e^7)
*arctan(1/2*(2*b*e*x + b*d + a*e)*sqrt(-b*e)/(sqrt(b*x + a)*sqrt(e*x + d)*b*e)))
/(sqrt(-b*e)*b^4*e^4)]

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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)**(5/2)*(B*x+A)*(e*x+d)**(5/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.734489, size = 1, normalized size = 0. \[ \mathit{Done} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*(b*x + a)^(5/2)*(e*x + d)^(5/2),x, algorithm="giac")

[Out]

Done