3.2207 \(\int (a+b x)^{5/2} (A+B x) (d+e x)^{3/2} \, dx\)

Optimal. Leaf size=358 \[ \frac{(b d-a e)^5 (5 a B e-12 A b e+7 b B d) \tanh ^{-1}\left (\frac{\sqrt{e} \sqrt{a+b x}}{\sqrt{b} \sqrt{d+e x}}\right )}{512 b^{7/2} e^{9/2}}-\frac{\sqrt{a+b x} \sqrt{d+e x} (b d-a e)^4 (5 a B e-12 A b e+7 b B d)}{512 b^3 e^4}+\frac{(a+b x)^{3/2} \sqrt{d+e x} (b d-a e)^3 (5 a B e-12 A b e+7 b B d)}{768 b^3 e^3}-\frac{(a+b x)^{5/2} \sqrt{d+e x} (b d-a e)^2 (5 a B e-12 A b e+7 b B d)}{960 b^3 e^2}-\frac{(a+b x)^{7/2} \sqrt{d+e x} (b d-a e) (5 a B e-12 A b e+7 b B d)}{160 b^3 e}-\frac{(a+b x)^{7/2} (d+e x)^{3/2} (5 a B e-12 A b e+7 b B d)}{60 b^2 e}+\frac{B (a+b x)^{7/2} (d+e x)^{5/2}}{6 b e} \]

[Out]

-((b*d - a*e)^4*(7*b*B*d - 12*A*b*e + 5*a*B*e)*Sqrt[a + b*x]*Sqrt[d + e*x])/(512
*b^3*e^4) + ((b*d - a*e)^3*(7*b*B*d - 12*A*b*e + 5*a*B*e)*(a + b*x)^(3/2)*Sqrt[d
+ e*x])/(768*b^3*e^3) - ((b*d - a*e)^2*(7*b*B*d - 12*A*b*e + 5*a*B*e)*(a + b*x)
^(5/2)*Sqrt[d + e*x])/(960*b^3*e^2) - ((b*d - a*e)*(7*b*B*d - 12*A*b*e + 5*a*B*e
)*(a + b*x)^(7/2)*Sqrt[d + e*x])/(160*b^3*e) - ((7*b*B*d - 12*A*b*e + 5*a*B*e)*(
a + b*x)^(7/2)*(d + e*x)^(3/2))/(60*b^2*e) + (B*(a + b*x)^(7/2)*(d + e*x)^(5/2))
/(6*b*e) + ((b*d - a*e)^5*(7*b*B*d - 12*A*b*e + 5*a*B*e)*ArcTanh[(Sqrt[e]*Sqrt[a
+ b*x])/(Sqrt[b]*Sqrt[d + e*x])])/(512*b^(7/2)*e^(9/2))

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Rubi [A]  time = 0.776741, 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 \[ \frac{(b d-a e)^5 (5 a B e-12 A b e+7 b B d) \tanh ^{-1}\left (\frac{\sqrt{e} \sqrt{a+b x}}{\sqrt{b} \sqrt{d+e x}}\right )}{512 b^{7/2} e^{9/2}}-\frac{\sqrt{a+b x} \sqrt{d+e x} (b d-a e)^4 (5 a B e-12 A b e+7 b B d)}{512 b^3 e^4}+\frac{(a+b x)^{3/2} \sqrt{d+e x} (b d-a e)^3 (5 a B e-12 A b e+7 b B d)}{768 b^3 e^3}-\frac{(a+b x)^{5/2} \sqrt{d+e x} (b d-a e)^2 (5 a B e-12 A b e+7 b B d)}{960 b^3 e^2}-\frac{(a+b x)^{7/2} \sqrt{d+e x} (b d-a e) (5 a B e-12 A b e+7 b B d)}{160 b^3 e}-\frac{(a+b x)^{7/2} (d+e x)^{3/2} (5 a B e-12 A b e+7 b B d)}{60 b^2 e}+\frac{B (a+b x)^{7/2} (d+e x)^{5/2}}{6 b e} \]

Antiderivative was successfully verified.

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

[Out]

-((b*d - a*e)^4*(7*b*B*d - 12*A*b*e + 5*a*B*e)*Sqrt[a + b*x]*Sqrt[d + e*x])/(512
*b^3*e^4) + ((b*d - a*e)^3*(7*b*B*d - 12*A*b*e + 5*a*B*e)*(a + b*x)^(3/2)*Sqrt[d
+ e*x])/(768*b^3*e^3) - ((b*d - a*e)^2*(7*b*B*d - 12*A*b*e + 5*a*B*e)*(a + b*x)
^(5/2)*Sqrt[d + e*x])/(960*b^3*e^2) - ((b*d - a*e)*(7*b*B*d - 12*A*b*e + 5*a*B*e
)*(a + b*x)^(7/2)*Sqrt[d + e*x])/(160*b^3*e) - ((7*b*B*d - 12*A*b*e + 5*a*B*e)*(
a + b*x)^(7/2)*(d + e*x)^(3/2))/(60*b^2*e) + (B*(a + b*x)^(7/2)*(d + e*x)^(5/2))
/(6*b*e) + ((b*d - a*e)^5*(7*b*B*d - 12*A*b*e + 5*a*B*e)*ArcTanh[(Sqrt[e]*Sqrt[a
+ b*x])/(Sqrt[b]*Sqrt[d + e*x])])/(512*b^(7/2)*e^(9/2))

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Rubi in Sympy [A]  time = 76.3101, size = 350, normalized size = 0.98 \[ \frac{B \left (a + b x\right )^{\frac{7}{2}} \left (d + e x\right )^{\frac{5}{2}}}{6 b e} + \frac{\left (a + b x\right )^{\frac{5}{2}} \left (d + e x\right )^{\frac{5}{2}} \left (12 A b e - 5 B a e - 7 B b d\right )}{60 b e^{2}} + \frac{\left (a + b x\right )^{\frac{3}{2}} \left (d + e x\right )^{\frac{5}{2}} \left (a e - b d\right ) \left (12 A b e - 5 B a e - 7 B b d\right )}{96 b e^{3}} + \frac{\left (a + b x\right )^{\frac{3}{2}} \left (d + e x\right )^{\frac{3}{2}} \left (a e - b d\right )^{2} \left (12 A b e - 5 B a e - 7 B b d\right )}{192 b^{2} e^{3}} - \frac{\left (a + b x\right )^{\frac{3}{2}} \sqrt{d + e x} \left (a e - b d\right )^{3} \left (12 A b e - 5 B a e - 7 B b d\right )}{256 b^{3} e^{3}} + \frac{\sqrt{a + b x} \sqrt{d + e x} \left (a e - b d\right )^{4} \left (12 A b e - 5 B a e - 7 B b d\right )}{512 b^{3} e^{4}} + \frac{\left (a e - b d\right )^{5} \left (12 A b e - 5 B a e - 7 B b d\right ) \operatorname{atanh}{\left (\frac{\sqrt{e} \sqrt{a + b x}}{\sqrt{b} \sqrt{d + e x}} \right )}}{512 b^{\frac{7}{2}} e^{\frac{9}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

B*(a + b*x)**(7/2)*(d + e*x)**(5/2)/(6*b*e) + (a + b*x)**(5/2)*(d + e*x)**(5/2)*
(12*A*b*e - 5*B*a*e - 7*B*b*d)/(60*b*e**2) + (a + b*x)**(3/2)*(d + e*x)**(5/2)*(
a*e - b*d)*(12*A*b*e - 5*B*a*e - 7*B*b*d)/(96*b*e**3) + (a + b*x)**(3/2)*(d + e*
x)**(3/2)*(a*e - b*d)**2*(12*A*b*e - 5*B*a*e - 7*B*b*d)/(192*b**2*e**3) - (a + b
*x)**(3/2)*sqrt(d + e*x)*(a*e - b*d)**3*(12*A*b*e - 5*B*a*e - 7*B*b*d)/(256*b**3
*e**3) + sqrt(a + b*x)*sqrt(d + e*x)*(a*e - b*d)**4*(12*A*b*e - 5*B*a*e - 7*B*b*
d)/(512*b**3*e**4) + (a*e - b*d)**5*(12*A*b*e - 5*B*a*e - 7*B*b*d)*atanh(sqrt(e)
*sqrt(a + b*x)/(sqrt(b)*sqrt(d + e*x)))/(512*b**(7/2)*e**(9/2))

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Mathematica [A]  time = 0.632926, size = 444, normalized size = 1.24 \[ \frac{\sqrt{a+b x} \sqrt{d+e x} \left (75 a^5 B e^5-5 a^4 b e^4 (36 A e+49 B d+10 B e x)+10 a^3 b^2 e^3 \left (12 A e (7 d+e x)+B \left (15 d^2+16 d e x+4 e^2 x^2\right )\right )+6 a^2 b^3 e^2 \left (4 A e \left (64 d^2+233 d e x+124 e^2 x^2\right )+B \left (-91 d^3+58 d^2 e x+564 d e^2 x^2+360 e^3 x^3\right )\right )+a b^4 e \left (24 A e \left (-35 d^3+23 d^2 e x+256 d e^2 x^2+168 e^3 x^3\right )+B \left (415 d^4-272 d^3 e x+216 d^2 e^2 x^2+4448 d e^3 x^3+3200 e^4 x^4\right )\right )+b^5 \left (12 A e \left (15 d^4-10 d^3 e x+8 d^2 e^2 x^2+176 d e^3 x^3+128 e^4 x^4\right )+B \left (-105 d^5+70 d^4 e x-56 d^3 e^2 x^2+48 d^2 e^3 x^3+1664 d e^4 x^4+1280 e^5 x^5\right )\right )\right )}{7680 b^3 e^4}+\frac{(b d-a e)^5 (5 a B e-12 A b e+7 b B d) \log \left (2 \sqrt{b} \sqrt{e} \sqrt{a+b x} \sqrt{d+e x}+a e+b d+2 b e x\right )}{1024 b^{7/2} e^{9/2}} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[a + b*x]*Sqrt[d + e*x]*(75*a^5*B*e^5 - 5*a^4*b*e^4*(49*B*d + 36*A*e + 10*B
*e*x) + 10*a^3*b^2*e^3*(12*A*e*(7*d + e*x) + B*(15*d^2 + 16*d*e*x + 4*e^2*x^2))
+ 6*a^2*b^3*e^2*(4*A*e*(64*d^2 + 233*d*e*x + 124*e^2*x^2) + B*(-91*d^3 + 58*d^2*
e*x + 564*d*e^2*x^2 + 360*e^3*x^3)) + a*b^4*e*(24*A*e*(-35*d^3 + 23*d^2*e*x + 25
6*d*e^2*x^2 + 168*e^3*x^3) + B*(415*d^4 - 272*d^3*e*x + 216*d^2*e^2*x^2 + 4448*d
*e^3*x^3 + 3200*e^4*x^4)) + b^5*(12*A*e*(15*d^4 - 10*d^3*e*x + 8*d^2*e^2*x^2 + 1
76*d*e^3*x^3 + 128*e^4*x^4) + B*(-105*d^5 + 70*d^4*e*x - 56*d^3*e^2*x^2 + 48*d^2
*e^3*x^3 + 1664*d*e^4*x^4 + 1280*e^5*x^5))))/(7680*b^3*e^4) + ((b*d - a*e)^5*(7*
b*B*d - 12*A*b*e + 5*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^(7/2)*e^(9/2))

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Maple [B]  time = 0.032, 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)^(5/2)*(B*x+A)*(e*x+d)^(3/2),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^2-75*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+10
5*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^2-100*(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-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^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+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^2*d^4*B*b^4*e^2-450*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+320*(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)+1104*(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+11184*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)-544*(
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)+6768*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)+432*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)+270*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+3
328*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)+8064*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)+4224*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)+4320*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)+96*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)+5952*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)+192*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)+80*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)-112*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)+6400*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
)+140*(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^2-490*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)+300*(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^
3-1092*(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+830*(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+150*(b*e*
x^2+a*e*x+b*d*x+a*d)^(1/2)*a^5*B*e^5*(b*e)^(1/2)-210*(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)/e^4/b^3/(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)^(3/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.338787, 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)^(3/2),x, algorithm="fricas")

[Out]

[1/30720*(4*(1280*B*b^5*e^5*x^5 - 105*B*b^5*d^5 + 5*(83*B*a*b^4 + 36*A*b^5)*d^4*
e - 42*(13*B*a^2*b^3 + 20*A*a*b^4)*d^3*e^2 + 6*(25*B*a^3*b^2 + 256*A*a^2*b^3)*d^
2*e^3 - 35*(7*B*a^4*b - 24*A*a^3*b^2)*d*e^4 + 15*(5*B*a^5 - 12*A*a^4*b)*e^5 + 12
8*(13*B*b^5*d*e^4 + (25*B*a*b^4 + 12*A*b^5)*e^5)*x^4 + 16*(3*B*b^5*d^2*e^3 + 2*(
139*B*a*b^4 + 66*A*b^5)*d*e^4 + 9*(15*B*a^2*b^3 + 28*A*a*b^4)*e^5)*x^3 - 8*(7*B*
b^5*d^3*e^2 - 3*(9*B*a*b^4 + 4*A*b^5)*d^2*e^3 - 3*(141*B*a^2*b^3 + 256*A*a*b^4)*
d*e^4 - (5*B*a^3*b^2 + 372*A*a^2*b^3)*e^5)*x^2 + 2*(35*B*b^5*d^4*e - 4*(34*B*a*b
^4 + 15*A*b^5)*d^3*e^2 + 6*(29*B*a^2*b^3 + 46*A*a*b^4)*d^2*e^3 + 4*(20*B*a^3*b^2
+ 699*A*a^2*b^3)*d*e^4 - 5*(5*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*(7*B*b^6*d^6 - 6*(5*B*a*b^5 + 2*A*b^6)*d^5*e + 15*(3*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*(B*a^4
*b^2 - 8*A*a^3*b^3)*d^2*e^4 + 6*(3*B*a^5*b - 10*A*a^4*b^2)*d*e^5 - (5*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)*s
qrt(b*e)))/(sqrt(b*e)*b^3*e^4), 1/15360*(2*(1280*B*b^5*e^5*x^5 - 105*B*b^5*d^5 +
5*(83*B*a*b^4 + 36*A*b^5)*d^4*e - 42*(13*B*a^2*b^3 + 20*A*a*b^4)*d^3*e^2 + 6*(2
5*B*a^3*b^2 + 256*A*a^2*b^3)*d^2*e^3 - 35*(7*B*a^4*b - 24*A*a^3*b^2)*d*e^4 + 15*
(5*B*a^5 - 12*A*a^4*b)*e^5 + 128*(13*B*b^5*d*e^4 + (25*B*a*b^4 + 12*A*b^5)*e^5)*
x^4 + 16*(3*B*b^5*d^2*e^3 + 2*(139*B*a*b^4 + 66*A*b^5)*d*e^4 + 9*(15*B*a^2*b^3 +
28*A*a*b^4)*e^5)*x^3 - 8*(7*B*b^5*d^3*e^2 - 3*(9*B*a*b^4 + 4*A*b^5)*d^2*e^3 - 3
*(141*B*a^2*b^3 + 256*A*a*b^4)*d*e^4 - (5*B*a^3*b^2 + 372*A*a^2*b^3)*e^5)*x^2 +
2*(35*B*b^5*d^4*e - 4*(34*B*a*b^4 + 15*A*b^5)*d^3*e^2 + 6*(29*B*a^2*b^3 + 46*A*a
*b^4)*d^2*e^3 + 4*(20*B*a^3*b^2 + 699*A*a^2*b^3)*d*e^4 - 5*(5*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*(7*B*b^6*d^6 - 6*(5*B*
a*b^5 + 2*A*b^6)*d^5*e + 15*(3*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*(B*a^4*b^2 - 8*A*a^3*b^3)*d^2*e^4 + 6*(3*B*a^5*b - 10*
A*a^4*b^2)*d*e^5 - (5*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^3*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)**(3/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.52859, 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)^(3/2),x, algorithm="giac")

[Out]

Done