3.2194 \(\int (a+b x)^{3/2} (A+B x) (d+e x)^{5/2} \, dx\)

Optimal. Leaf size=358 \[ -\frac{(b d-a e)^5 (7 a B e-12 A b e+5 b B d) \tanh ^{-1}\left (\frac{\sqrt{e} \sqrt{a+b x}}{\sqrt{b} \sqrt{d+e x}}\right )}{512 b^{9/2} e^{7/2}}+\frac{\sqrt{a+b x} \sqrt{d+e x} (b d-a e)^4 (7 a B e-12 A b e+5 b B d)}{512 b^4 e^3}-\frac{(a+b x)^{3/2} \sqrt{d+e x} (b d-a e)^3 (7 a B e-12 A b e+5 b B d)}{768 b^4 e^2}-\frac{(a+b x)^{5/2} \sqrt{d+e x} (b d-a e)^2 (7 a B e-12 A b e+5 b B d)}{192 b^4 e}-\frac{(a+b x)^{5/2} (d+e x)^{3/2} (b d-a e) (7 a B e-12 A b e+5 b B d)}{96 b^3 e}-\frac{(a+b x)^{5/2} (d+e x)^{5/2} (7 a B e-12 A b e+5 b B d)}{60 b^2 e}+\frac{B (a+b x)^{5/2} (d+e x)^{7/2}}{6 b e} \]

[Out]

((b*d - a*e)^4*(5*b*B*d - 12*A*b*e + 7*a*B*e)*Sqrt[a + b*x]*Sqrt[d + e*x])/(512*
b^4*e^3) - ((b*d - a*e)^3*(5*b*B*d - 12*A*b*e + 7*a*B*e)*(a + b*x)^(3/2)*Sqrt[d
+ e*x])/(768*b^4*e^2) - ((b*d - a*e)^2*(5*b*B*d - 12*A*b*e + 7*a*B*e)*(a + b*x)^
(5/2)*Sqrt[d + e*x])/(192*b^4*e) - ((b*d - a*e)*(5*b*B*d - 12*A*b*e + 7*a*B*e)*(
a + b*x)^(5/2)*(d + e*x)^(3/2))/(96*b^3*e) - ((5*b*B*d - 12*A*b*e + 7*a*B*e)*(a
+ b*x)^(5/2)*(d + e*x)^(5/2))/(60*b^2*e) + (B*(a + b*x)^(5/2)*(d + e*x)^(7/2))/(
6*b*e) - ((b*d - a*e)^5*(5*b*B*d - 12*A*b*e + 7*a*B*e)*ArcTanh[(Sqrt[e]*Sqrt[a +
b*x])/(Sqrt[b]*Sqrt[d + e*x])])/(512*b^(9/2)*e^(7/2))

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Rubi [A]  time = 0.814931, antiderivative size = 358, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167 \[ -\frac{(b d-a e)^5 (7 a B e-12 A b e+5 b B d) \tanh ^{-1}\left (\frac{\sqrt{e} \sqrt{a+b x}}{\sqrt{b} \sqrt{d+e x}}\right )}{512 b^{9/2} e^{7/2}}+\frac{\sqrt{a+b x} \sqrt{d+e x} (b d-a e)^4 (7 a B e-12 A b e+5 b B d)}{512 b^4 e^3}-\frac{(a+b x)^{3/2} \sqrt{d+e x} (b d-a e)^3 (7 a B e-12 A b e+5 b B d)}{768 b^4 e^2}-\frac{(a+b x)^{5/2} \sqrt{d+e x} (b d-a e)^2 (7 a B e-12 A b e+5 b B d)}{192 b^4 e}-\frac{(a+b x)^{5/2} (d+e x)^{3/2} (b d-a e) (7 a B e-12 A b e+5 b B d)}{96 b^3 e}-\frac{(a+b x)^{5/2} (d+e x)^{5/2} (7 a B e-12 A b e+5 b B d)}{60 b^2 e}+\frac{B (a+b x)^{5/2} (d+e x)^{7/2}}{6 b e} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x)^(3/2)*(A + B*x)*(d + e*x)^(5/2),x]

[Out]

((b*d - a*e)^4*(5*b*B*d - 12*A*b*e + 7*a*B*e)*Sqrt[a + b*x]*Sqrt[d + e*x])/(512*
b^4*e^3) - ((b*d - a*e)^3*(5*b*B*d - 12*A*b*e + 7*a*B*e)*(a + b*x)^(3/2)*Sqrt[d
+ e*x])/(768*b^4*e^2) - ((b*d - a*e)^2*(5*b*B*d - 12*A*b*e + 7*a*B*e)*(a + b*x)^
(5/2)*Sqrt[d + e*x])/(192*b^4*e) - ((b*d - a*e)*(5*b*B*d - 12*A*b*e + 7*a*B*e)*(
a + b*x)^(5/2)*(d + e*x)^(3/2))/(96*b^3*e) - ((5*b*B*d - 12*A*b*e + 7*a*B*e)*(a
+ b*x)^(5/2)*(d + e*x)^(5/2))/(60*b^2*e) + (B*(a + b*x)^(5/2)*(d + e*x)^(7/2))/(
6*b*e) - ((b*d - a*e)^5*(5*b*B*d - 12*A*b*e + 7*a*B*e)*ArcTanh[(Sqrt[e]*Sqrt[a +
b*x])/(Sqrt[b]*Sqrt[d + e*x])])/(512*b^(9/2)*e^(7/2))

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Rubi in Sympy [A]  time = 75.2077, size = 350, normalized size = 0.98 \[ \frac{B \left (a + b x\right )^{\frac{5}{2}} \left (d + e x\right )^{\frac{7}{2}}}{6 b e} + \frac{\left (a + b x\right )^{\frac{3}{2}} \left (d + e x\right )^{\frac{7}{2}} \left (12 A b e - 7 B a e - 5 B b d\right )}{60 b e^{2}} + \frac{\sqrt{a + b x} \left (d + e x\right )^{\frac{7}{2}} \left (a e - b d\right ) \left (12 A b e - 7 B a e - 5 B b d\right )}{160 b e^{3}} + \frac{\sqrt{a + b x} \left (d + e x\right )^{\frac{5}{2}} \left (a e - b d\right )^{2} \left (12 A b e - 7 B a e - 5 B b d\right )}{960 b^{2} e^{3}} - \frac{\sqrt{a + b x} \left (d + e x\right )^{\frac{3}{2}} \left (a e - b d\right )^{3} \left (12 A b e - 7 B a e - 5 B b d\right )}{768 b^{3} e^{3}} + \frac{\sqrt{a + b x} \sqrt{d + e x} \left (a e - b d\right )^{4} \left (12 A b e - 7 B a e - 5 B b d\right )}{512 b^{4} e^{3}} - \frac{\left (a e - b d\right )^{5} \left (12 A b e - 7 B a e - 5 B b d\right ) \operatorname{atanh}{\left (\frac{\sqrt{e} \sqrt{a + b x}}{\sqrt{b} \sqrt{d + e x}} \right )}}{512 b^{\frac{9}{2}} e^{\frac{7}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)**(3/2)*(B*x+A)*(e*x+d)**(5/2),x)

[Out]

B*(a + b*x)**(5/2)*(d + e*x)**(7/2)/(6*b*e) + (a + b*x)**(3/2)*(d + e*x)**(7/2)*
(12*A*b*e - 7*B*a*e - 5*B*b*d)/(60*b*e**2) + sqrt(a + b*x)*(d + e*x)**(7/2)*(a*e
- b*d)*(12*A*b*e - 7*B*a*e - 5*B*b*d)/(160*b*e**3) + sqrt(a + b*x)*(d + e*x)**(
5/2)*(a*e - b*d)**2*(12*A*b*e - 7*B*a*e - 5*B*b*d)/(960*b**2*e**3) - sqrt(a + b*
x)*(d + e*x)**(3/2)*(a*e - b*d)**3*(12*A*b*e - 7*B*a*e - 5*B*b*d)/(768*b**3*e**3
) + sqrt(a + b*x)*sqrt(d + e*x)*(a*e - b*d)**4*(12*A*b*e - 7*B*a*e - 5*B*b*d)/(5
12*b**4*e**3) - (a*e - b*d)**5*(12*A*b*e - 7*B*a*e - 5*B*b*d)*atanh(sqrt(e)*sqrt
(a + b*x)/(sqrt(b)*sqrt(d + e*x)))/(512*b**(9/2)*e**(7/2))

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Mathematica [A]  time = 0.739197, size = 445, normalized size = 1.24 \[ \frac{\sqrt{a+b x} \sqrt{d+e x} \left (-105 a^5 B e^5+5 a^4 b e^4 (36 A e+83 B d+14 B e x)-2 a^3 b^2 e^3 \left (60 A e (7 d+e x)+B \left (273 d^2+136 d e x+28 e^2 x^2\right )\right )+6 a^2 b^3 e^2 \left (4 A e \left (64 d^2+23 d e x+4 e^2 x^2\right )+B \left (25 d^3+58 d^2 e x+36 d e^2 x^2+8 e^3 x^3\right )\right )+a b^4 e \left (24 A e \left (35 d^3+233 d^2 e x+256 d e^2 x^2+88 e^3 x^3\right )+B \left (-245 d^4+160 d^3 e x+3384 d^2 e^2 x^2+4448 d e^3 x^3+1664 e^4 x^4\right )\right )+b^5 \left (12 A e \left (-15 d^4+10 d^3 e x+248 d^2 e^2 x^2+336 d e^3 x^3+128 e^4 x^4\right )+5 B \left (15 d^5-10 d^4 e x+8 d^3 e^2 x^2+432 d^2 e^3 x^3+640 d e^4 x^4+256 e^5 x^5\right )\right )\right )}{7680 b^4 e^3}+\frac{(b d-a e)^5 (-7 a B e+12 A b e-5 b B d) \log \left (2 \sqrt{b} \sqrt{e} \sqrt{a+b x} \sqrt{d+e x}+a e+b d+2 b e x\right )}{1024 b^{9/2} e^{7/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x)^(3/2)*(A + B*x)*(d + e*x)^(5/2),x]

[Out]

(Sqrt[a + b*x]*Sqrt[d + e*x]*(-105*a^5*B*e^5 + 5*a^4*b*e^4*(83*B*d + 36*A*e + 14
*B*e*x) - 2*a^3*b^2*e^3*(60*A*e*(7*d + e*x) + B*(273*d^2 + 136*d*e*x + 28*e^2*x^
2)) + 6*a^2*b^3*e^2*(4*A*e*(64*d^2 + 23*d*e*x + 4*e^2*x^2) + B*(25*d^3 + 58*d^2*
e*x + 36*d*e^2*x^2 + 8*e^3*x^3)) + a*b^4*e*(24*A*e*(35*d^3 + 233*d^2*e*x + 256*d
*e^2*x^2 + 88*e^3*x^3) + B*(-245*d^4 + 160*d^3*e*x + 3384*d^2*e^2*x^2 + 4448*d*e
^3*x^3 + 1664*e^4*x^4)) + b^5*(12*A*e*(-15*d^4 + 10*d^3*e*x + 248*d^2*e^2*x^2 +
336*d*e^3*x^3 + 128*e^4*x^4) + 5*B*(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))))/(7680*b^4*e^3) + ((b*d - a*e)^5*(-5*
b*B*d + 12*A*b*e - 7*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]])/(1024*b^(9/2)*e^(7/2))

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Maple [B]  time = 0.033, size = 2198, normalized size = 6.1 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^(3/2)*(B*x+A)*(e*x+d)^(5/2),x)

[Out]

-1/15360*(b*x+a)^(1/2)*(e*x+d)^(1/2)*(900*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*b^5*A*e^2-105*e^6*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*B+
75*b^6*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*B+240*(b*e*x^2+a*e*x+b*d*x+a*d)^(1/2)*x*a^3*e^5*A*b^2*(b*e)^(1/2)
-240*(b*e*x^2+a*e*x+b*d*x+a*d)^(1/2)*x*d^3*b^5*A*(b*e)^(1/2)*e^2-140*(b*e*x^2+a*
e*x+b*d*x+a*d)^(1/2)*x*a^4*B*e^5*b*(b*e)^(1/2)-696*(b*e*x^2+a*e*x+b*d*x+a*d)^(1/
2)*x*a^2*d^2*B*b^3*(b*e)^(1/2)*e^3+180*e^6*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*A*b-675*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*B*e^4*b^2
+300*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*B*b^3*e^3+225*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^2*d^4*B*b^4*e^2-270*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^5*B*a*e-360*(
b*e*x^2+a*e*x+b*d*x+a*d)^(1/2)*a^4*e^5*A*b*(b*e)^(1/2)+360*(b*e*x^2+a*e*x+b*d*x+
a*d)^(1/2)*d^4*b^5*A*(b*e)^(1/2)*e+544*(b*e*x^2+a*e*x+b*d*x+a*d)^(1/2)*x*a^3*B*d
*e^4*b^2*(b*e)^(1/2)-11184*(b*e*x^2+a*e*x+b*d*x+a*d)^(1/2)*x*a*d^2*A*b^4*(b*e)^(
1/2)*e^3-1104*e^4*(b*e*x^2+a*e*x+b*d*x+a*d)^(1/2)*x*a^2*d*A*b^3*(b*e)^(1/2)-320*
(b*e*x^2+a*e*x+b*d*x+a*d)^(1/2)*x*d^3*B*a*b^4*(b*e)^(1/2)*e^2-8896*B*x^3*a*b^4*d
*e^4*(b*e*x^2+a*e*x+b*d*x+a*d)^(1/2)*(b*e)^(1/2)-12288*A*x^2*a*b^4*d*e^4*(b*e*x^
2+a*e*x+b*d*x+a*d)^(1/2)*(b*e)^(1/2)-432*B*x^2*a^2*b^3*d*e^4*(b*e*x^2+a*e*x+b*d*
x+a*d)^(1/2)*(b*e)^(1/2)-6768*B*x^2*a*b^4*d^2*e^3*(b*e*x^2+a*e*x+b*d*x+a*d)^(1/2
)*(b*e)^(1/2)+450*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*B*e^5*b-2560*B*x^5*b^5*e^5*(b*e*x^2+a*e*x+b*d*x+a*d)
^(1/2)*(b*e)^(1/2)-3072*A*x^4*b^5*e^5*(b*e*x^2+a*e*x+b*d*x+a*d)^(1/2)*(b*e)^(1/2
)-900*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*e^5*A*b^2+1800*e^4*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^2*A*b^3-1800*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^2*d^3*A*b^4*e^3-
6400*B*x^4*b^5*d*e^4*(b*e*x^2+a*e*x+b*d*x+a*d)^(1/2)*(b*e)^(1/2)-4224*A*x^3*a*b^
4*e^5*(b*e*x^2+a*e*x+b*d*x+a*d)^(1/2)*(b*e)^(1/2)-8064*A*x^3*b^5*d*e^4*(b*e*x^2+
a*e*x+b*d*x+a*d)^(1/2)*(b*e)^(1/2)-96*B*x^3*a^2*b^3*e^5*(b*e*x^2+a*e*x+b*d*x+a*d
)^(1/2)*(b*e)^(1/2)-4320*B*x^3*b^5*d^2*e^3*(b*e*x^2+a*e*x+b*d*x+a*d)^(1/2)*(b*e)
^(1/2)-192*A*x^2*a^2*b^3*e^5*(b*e*x^2+a*e*x+b*d*x+a*d)^(1/2)*(b*e)^(1/2)-5952*A*
x^2*b^5*d^2*e^3*(b*e*x^2+a*e*x+b*d*x+a*d)^(1/2)*(b*e)^(1/2)+112*B*x^2*a^3*b^2*e^
5*(b*e*x^2+a*e*x+b*d*x+a*d)^(1/2)*(b*e)^(1/2)-80*B*x^2*b^5*d^3*e^2*(b*e*x^2+a*e*
x+b*d*x+a*d)^(1/2)*(b*e)^(1/2)-3072*A*a^2*b^3*d^2*e^3*(b*e*x^2+a*e*x+b*d*x+a*d)^
(1/2)*(b*e)^(1/2)-3328*B*x^4*a*b^4*e^5*(b*e*x^2+a*e*x+b*d*x+a*d)^(1/2)*(b*e)^(1/
2)+100*(b*e*x^2+a*e*x+b*d*x+a*d)^(1/2)*x*d^4*B*b^5*(b*e)^(1/2)*e+1680*e^4*(b*e*x
^2+a*e*x+b*d*x+a*d)^(1/2)*a^3*d*A*b^2*(b*e)^(1/2)-1680*(b*e*x^2+a*e*x+b*d*x+a*d)
^(1/2)*a*d^3*A*b^4*(b*e)^(1/2)*e^2-830*e^4*(b*e*x^2+a*e*x+b*d*x+a*d)^(1/2)*a^4*d
*B*b*(b*e)^(1/2)+1092*(b*e*x^2+a*e*x+b*d*x+a*d)^(1/2)*a^3*d^2*B*b^2*(b*e)^(1/2)*
e^3-300*(b*e*x^2+a*e*x+b*d*x+a*d)^(1/2)*a^2*d^3*B*b^3*(b*e)^(1/2)*e^2+490*(b*e*x
^2+a*e*x+b*d*x+a*d)^(1/2)*a*d^4*B*b^4*(b*e)^(1/2)*e-180*b^6*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*e+210*(b*e
*x^2+a*e*x+b*d*x+a*d)^(1/2)*a^5*B*e^5*(b*e)^(1/2)-150*(b*e*x^2+a*e*x+b*d*x+a*d)^
(1/2)*d^5*B*b^5*(b*e)^(1/2))/(b*e*x^2+a*e*x+b*d*x+a*d)^(1/2)/b^4/(b*e)^(1/2)/e^3

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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)^(3/2)*(e*x + d)^(5/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.347556, 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)^(3/2)*(e*x + d)^(5/2),x, algorithm="fricas")

[Out]

[1/30720*(4*(1280*B*b^5*e^5*x^5 + 75*B*b^5*d^5 - 5*(49*B*a*b^4 + 36*A*b^5)*d^4*e
+ 30*(5*B*a^2*b^3 + 28*A*a*b^4)*d^3*e^2 - 6*(91*B*a^3*b^2 - 256*A*a^2*b^3)*d^2*
e^3 + 5*(83*B*a^4*b - 168*A*a^3*b^2)*d*e^4 - 15*(7*B*a^5 - 12*A*a^4*b)*e^5 + 128
*(25*B*b^5*d*e^4 + (13*B*a*b^4 + 12*A*b^5)*e^5)*x^4 + 16*(135*B*b^5*d^2*e^3 + 2*
(139*B*a*b^4 + 126*A*b^5)*d*e^4 + 3*(B*a^2*b^3 + 44*A*a*b^4)*e^5)*x^3 + 8*(5*B*b
^5*d^3*e^2 + 3*(141*B*a*b^4 + 124*A*b^5)*d^2*e^3 + 3*(9*B*a^2*b^3 + 256*A*a*b^4)
*d*e^4 - (7*B*a^3*b^2 - 12*A*a^2*b^3)*e^5)*x^2 - 2*(25*B*b^5*d^4*e - 20*(4*B*a*b
^4 + 3*A*b^5)*d^3*e^2 - 6*(29*B*a^2*b^3 + 466*A*a*b^4)*d^2*e^3 + 4*(34*B*a^3*b^2
- 69*A*a^2*b^3)*d*e^4 - 5*(7*B*a^4*b - 12*A*a^3*b^2)*e^5)*x)*sqrt(b*e)*sqrt(b*x
+ a)*sqrt(e*x + d) + 15*(5*B*b^6*d^6 - 6*(3*B*a*b^5 + 2*A*b^6)*d^5*e + 15*(B*a^
2*b^4 + 4*A*a*b^5)*d^4*e^2 + 20*(B*a^3*b^3 - 6*A*a^2*b^4)*d^3*e^3 - 15*(3*B*a^4*
b^2 - 8*A*a^3*b^3)*d^2*e^4 + 30*(B*a^5*b - 2*A*a^4*b^2)*d*e^5 - (7*B*a^6 - 12*A*
a^5*b)*e^6)*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)*sqr
t(b*e)))/(sqrt(b*e)*b^4*e^3), 1/15360*(2*(1280*B*b^5*e^5*x^5 + 75*B*b^5*d^5 - 5*
(49*B*a*b^4 + 36*A*b^5)*d^4*e + 30*(5*B*a^2*b^3 + 28*A*a*b^4)*d^3*e^2 - 6*(91*B*
a^3*b^2 - 256*A*a^2*b^3)*d^2*e^3 + 5*(83*B*a^4*b - 168*A*a^3*b^2)*d*e^4 - 15*(7*
B*a^5 - 12*A*a^4*b)*e^5 + 128*(25*B*b^5*d*e^4 + (13*B*a*b^4 + 12*A*b^5)*e^5)*x^4
+ 16*(135*B*b^5*d^2*e^3 + 2*(139*B*a*b^4 + 126*A*b^5)*d*e^4 + 3*(B*a^2*b^3 + 44
*A*a*b^4)*e^5)*x^3 + 8*(5*B*b^5*d^3*e^2 + 3*(141*B*a*b^4 + 124*A*b^5)*d^2*e^3 +
3*(9*B*a^2*b^3 + 256*A*a*b^4)*d*e^4 - (7*B*a^3*b^2 - 12*A*a^2*b^3)*e^5)*x^2 - 2*
(25*B*b^5*d^4*e - 20*(4*B*a*b^4 + 3*A*b^5)*d^3*e^2 - 6*(29*B*a^2*b^3 + 466*A*a*b
^4)*d^2*e^3 + 4*(34*B*a^3*b^2 - 69*A*a^2*b^3)*d*e^4 - 5*(7*B*a^4*b - 12*A*a^3*b^
2)*e^5)*x)*sqrt(-b*e)*sqrt(b*x + a)*sqrt(e*x + d) - 15*(5*B*b^6*d^6 - 6*(3*B*a*b
^5 + 2*A*b^6)*d^5*e + 15*(B*a^2*b^4 + 4*A*a*b^5)*d^4*e^2 + 20*(B*a^3*b^3 - 6*A*a
^2*b^4)*d^3*e^3 - 15*(3*B*a^4*b^2 - 8*A*a^3*b^3)*d^2*e^4 + 30*(B*a^5*b - 2*A*a^4
*b^2)*d*e^5 - (7*B*a^6 - 12*A*a^5*b)*e^6)*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^3)]

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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)**(3/2)*(B*x+A)*(e*x+d)**(5/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.529233, 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)^(3/2)*(e*x + d)^(5/2),x, algorithm="giac")

[Out]

Done