3.2208 \(\int (a+b x)^{5/2} (A+B x) \sqrt{d+e x} \, dx\)

Optimal. Leaf size=304 \[ \frac{(b d-a e)^4 (3 a B e-10 A b e+7 b B d) \tanh ^{-1}\left (\frac{\sqrt{e} \sqrt{a+b x}}{\sqrt{b} \sqrt{d+e x}}\right )}{128 b^{5/2} e^{9/2}}-\frac{\sqrt{a+b x} \sqrt{d+e x} (b d-a e)^3 (3 a B e-10 A b e+7 b B d)}{128 b^2 e^4}+\frac{(a+b x)^{3/2} \sqrt{d+e x} (b d-a e)^2 (3 a B e-10 A b e+7 b B d)}{192 b^2 e^3}-\frac{(a+b x)^{5/2} \sqrt{d+e x} (b d-a e) (3 a B e-10 A b e+7 b B d)}{240 b^2 e^2}-\frac{(a+b x)^{7/2} \sqrt{d+e x} (3 a B e-10 A b e+7 b B d)}{40 b^2 e}+\frac{B (a+b x)^{7/2} (d+e x)^{3/2}}{5 b e} \]

[Out]

-((b*d - a*e)^3*(7*b*B*d - 10*A*b*e + 3*a*B*e)*Sqrt[a + b*x]*Sqrt[d + e*x])/(128
*b^2*e^4) + ((b*d - a*e)^2*(7*b*B*d - 10*A*b*e + 3*a*B*e)*(a + b*x)^(3/2)*Sqrt[d
+ e*x])/(192*b^2*e^3) - ((b*d - a*e)*(7*b*B*d - 10*A*b*e + 3*a*B*e)*(a + b*x)^(
5/2)*Sqrt[d + e*x])/(240*b^2*e^2) - ((7*b*B*d - 10*A*b*e + 3*a*B*e)*(a + b*x)^(7
/2)*Sqrt[d + e*x])/(40*b^2*e) + (B*(a + b*x)^(7/2)*(d + e*x)^(3/2))/(5*b*e) + ((
b*d - a*e)^4*(7*b*B*d - 10*A*b*e + 3*a*B*e)*ArcTanh[(Sqrt[e]*Sqrt[a + b*x])/(Sqr
t[b]*Sqrt[d + e*x])])/(128*b^(5/2)*e^(9/2))

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Rubi [A]  time = 0.64136, antiderivative size = 304, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167 \[ \frac{(b d-a e)^4 (3 a B e-10 A b e+7 b B d) \tanh ^{-1}\left (\frac{\sqrt{e} \sqrt{a+b x}}{\sqrt{b} \sqrt{d+e x}}\right )}{128 b^{5/2} e^{9/2}}-\frac{\sqrt{a+b x} \sqrt{d+e x} (b d-a e)^3 (3 a B e-10 A b e+7 b B d)}{128 b^2 e^4}+\frac{(a+b x)^{3/2} \sqrt{d+e x} (b d-a e)^2 (3 a B e-10 A b e+7 b B d)}{192 b^2 e^3}-\frac{(a+b x)^{5/2} \sqrt{d+e x} (b d-a e) (3 a B e-10 A b e+7 b B d)}{240 b^2 e^2}-\frac{(a+b x)^{7/2} \sqrt{d+e x} (3 a B e-10 A b e+7 b B d)}{40 b^2 e}+\frac{B (a+b x)^{7/2} (d+e x)^{3/2}}{5 b e} \]

Antiderivative was successfully verified.

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

[Out]

-((b*d - a*e)^3*(7*b*B*d - 10*A*b*e + 3*a*B*e)*Sqrt[a + b*x]*Sqrt[d + e*x])/(128
*b^2*e^4) + ((b*d - a*e)^2*(7*b*B*d - 10*A*b*e + 3*a*B*e)*(a + b*x)^(3/2)*Sqrt[d
+ e*x])/(192*b^2*e^3) - ((b*d - a*e)*(7*b*B*d - 10*A*b*e + 3*a*B*e)*(a + b*x)^(
5/2)*Sqrt[d + e*x])/(240*b^2*e^2) - ((7*b*B*d - 10*A*b*e + 3*a*B*e)*(a + b*x)^(7
/2)*Sqrt[d + e*x])/(40*b^2*e) + (B*(a + b*x)^(7/2)*(d + e*x)^(3/2))/(5*b*e) + ((
b*d - a*e)^4*(7*b*B*d - 10*A*b*e + 3*a*B*e)*ArcTanh[(Sqrt[e]*Sqrt[a + b*x])/(Sqr
t[b]*Sqrt[d + e*x])])/(128*b^(5/2)*e^(9/2))

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Rubi in Sympy [A]  time = 57.9929, size = 296, normalized size = 0.97 \[ \frac{B \left (a + b x\right )^{\frac{7}{2}} \left (d + e x\right )^{\frac{3}{2}}}{5 b e} + \frac{\left (a + b x\right )^{\frac{5}{2}} \left (d + e x\right )^{\frac{3}{2}} \left (10 A b e - 3 B a e - 7 B b d\right )}{40 b e^{2}} + \frac{\left (a + b x\right )^{\frac{3}{2}} \left (d + e x\right )^{\frac{3}{2}} \left (a e - b d\right ) \left (10 A b e - 3 B a e - 7 B b d\right )}{48 b e^{3}} + \frac{\left (a + b x\right )^{\frac{3}{2}} \sqrt{d + e x} \left (a e - b d\right )^{2} \left (10 A b e - 3 B a e - 7 B b d\right )}{64 b^{2} e^{3}} - \frac{\sqrt{a + b x} \sqrt{d + e x} \left (a e - b d\right )^{3} \left (10 A b e - 3 B a e - 7 B b d\right )}{128 b^{2} e^{4}} - \frac{\left (a e - b d\right )^{4} \left (10 A b e - 3 B a e - 7 B b d\right ) \operatorname{atanh}{\left (\frac{\sqrt{b} \sqrt{d + e x}}{\sqrt{e} \sqrt{a + b x}} \right )}}{128 b^{\frac{5}{2}} e^{\frac{9}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)**(5/2)*(B*x+A)*(e*x+d)**(1/2),x)

[Out]

B*(a + b*x)**(7/2)*(d + e*x)**(3/2)/(5*b*e) + (a + b*x)**(5/2)*(d + e*x)**(3/2)*
(10*A*b*e - 3*B*a*e - 7*B*b*d)/(40*b*e**2) + (a + b*x)**(3/2)*(d + e*x)**(3/2)*(
a*e - b*d)*(10*A*b*e - 3*B*a*e - 7*B*b*d)/(48*b*e**3) + (a + b*x)**(3/2)*sqrt(d
+ e*x)*(a*e - b*d)**2*(10*A*b*e - 3*B*a*e - 7*B*b*d)/(64*b**2*e**3) - sqrt(a + b
*x)*sqrt(d + e*x)*(a*e - b*d)**3*(10*A*b*e - 3*B*a*e - 7*B*b*d)/(128*b**2*e**4)
- (a*e - b*d)**4*(10*A*b*e - 3*B*a*e - 7*B*b*d)*atanh(sqrt(b)*sqrt(d + e*x)/(sqr
t(e)*sqrt(a + b*x)))/(128*b**(5/2)*e**(9/2))

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Mathematica [A]  time = 0.510838, size = 334, normalized size = 1.1 \[ \frac{\sqrt{a+b x} \sqrt{d+e x} \left (-45 a^4 B e^4+30 a^3 b e^3 (5 A e+2 B d+B e x)+2 a^2 b^2 e^2 \left (5 A e (73 d+118 e x)+B \left (-173 d^2+109 d e x+372 e^2 x^2\right )\right )+2 a b^3 e \left (5 A e \left (-55 d^2+36 d e x+136 e^2 x^2\right )+B \left (170 d^3-111 d^2 e x+88 d e^2 x^2+504 e^3 x^3\right )\right )+b^4 \left (10 A e \left (15 d^3-10 d^2 e x+8 d e^2 x^2+48 e^3 x^3\right )+B \left (-105 d^4+70 d^3 e x-56 d^2 e^2 x^2+48 d e^3 x^3+384 e^4 x^4\right )\right )\right )}{1920 b^2 e^4}+\frac{(b d-a e)^4 (3 a B e-10 A b e+7 b B d) \log \left (2 \sqrt{b} \sqrt{e} \sqrt{a+b x} \sqrt{d+e x}+a e+b d+2 b e x\right )}{256 b^{5/2} e^{9/2}} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[a + b*x]*Sqrt[d + e*x]*(-45*a^4*B*e^4 + 30*a^3*b*e^3*(2*B*d + 5*A*e + B*e*
x) + 2*a^2*b^2*e^2*(5*A*e*(73*d + 118*e*x) + B*(-173*d^2 + 109*d*e*x + 372*e^2*x
^2)) + 2*a*b^3*e*(5*A*e*(-55*d^2 + 36*d*e*x + 136*e^2*x^2) + B*(170*d^3 - 111*d^
2*e*x + 88*d*e^2*x^2 + 504*e^3*x^3)) + b^4*(10*A*e*(15*d^3 - 10*d^2*e*x + 8*d*e^
2*x^2 + 48*e^3*x^3) + B*(-105*d^4 + 70*d^3*e*x - 56*d^2*e^2*x^2 + 48*d*e^3*x^3 +
384*e^4*x^4))))/(1920*b^2*e^4) + ((b*d - a*e)^4*(7*b*B*d - 10*A*b*e + 3*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]])/(256*b
^(5/2)*e^(9/2))

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Maple [B]  time = 0.027, size = 1631, normalized size = 5.4 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^(5/2)*(B*x+A)*(e*x+d)^(1/2),x)

[Out]

-1/3840*(b*x+a)^(1/2)*(e*x+d)^(1/2)*(-960*A*x^3*b^4*e^4*(b*e)^(1/2)*(b*e*x^2+a*e
*x+b*d*x+a*d)^(1/2)-680*(b*e*x^2+a*e*x+b*d*x+a*d)^(1/2)*a*d^3*B*b^3*(b*e)^(1/2)*
e-2016*B*x^3*a*b^3*e^4*(b*e)^(1/2)*(b*e*x^2+a*e*x+b*d*x+a*d)^(1/2)-96*B*x^3*b^4*
d*e^3*(b*e)^(1/2)*(b*e*x^2+a*e*x+b*d*x+a*d)^(1/2)-2720*A*x^2*a*b^3*e^4*(b*e)^(1/
2)*(b*e*x^2+a*e*x+b*d*x+a*d)^(1/2)-160*A*x^2*b^4*d*e^3*(b*e)^(1/2)*(b*e*x^2+a*e*
x+b*d*x+a*d)^(1/2)-1488*B*x^2*a^2*b^2*e^4*(b*e)^(1/2)*(b*e*x^2+a*e*x+b*d*x+a*d)^
(1/2)+112*B*x^2*b^4*d^2*e^2*(b*e)^(1/2)*(b*e*x^2+a*e*x+b*d*x+a*d)^(1/2)+444*(b*e
*x^2+a*e*x+b*d*x+a*d)^(1/2)*x*a*d^2*B*b^3*(b*e)^(1/2)*e^2-105*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-45*e
^5*B*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^5-1460*e^3*(b*e*x^2+a*e*x+b*d*x+a*d)^(1/2)*a^2*d*A*b^2*(b*e)^(1/2)-23
60*(b*e*x^2+a*e*x+b*d*x+a*d)^(1/2)*x*a^2*e^4*A*b^2*(b*e)^(1/2)+200*d^2*A*(b*e*x^
2+a*e*x+b*d*x+a*d)^(1/2)*x*b^4*(b*e)^(1/2)*e^2-60*e^4*B*(b*e*x^2+a*e*x+b*d*x+a*d
)^(1/2)*x*a^3*b*(b*e)^(1/2)-140*(b*e*x^2+a*e*x+b*d*x+a*d)^(1/2)*x*d^3*B*b^4*(b*e
)^(1/2)*e+150*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*A*b-436*(b*e*x^2+a*e*x+b*d*x+a*d)^(1/2)*x*a^2*d*e^3*B*
b^2*(b*e)^(1/2)-720*e^3*(b*e*x^2+a*e*x+b*d*x+a*d)^(1/2)*x*a*d*A*b^3*(b*e)^(1/2)-
352*B*x^2*a*b^3*d*e^3*(b*e)^(1/2)*(b*e*x^2+a*e*x+b*d*x+a*d)^(1/2)+150*d^4*A*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))*e+90*e^4*B*(b*e*x^2+a*e*x+b*d*x+a*d)^(1/2)*a^4*(b*e)^(1/2)+210*(b*e*x^2+a*e*
x+b*d*x+a*d)^(1/2)*d^4*B*b^4*(b*e)^(1/2)-768*B*x^4*b^4*e^4*(b*e)^(1/2)*(b*e*x^2+
a*e*x+b*d*x+a*d)^(1/2)-600*a^3*d*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))*e^4*A*b^2+900*d^2*A*e^3*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*b^3-600*d^3
*A*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*b^4*e^2+75*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^4*d*B*b+150*a^3*d^2*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))*e^3*B*b^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^2*d^3*
B*b^3*e^2+375*a*d^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))*B*b^4*e-300*(b*e*x^2+a*e*x+b*d*x+a*d)^(1/2)*a^3*e^4*A*b*
(b*e)^(1/2)-300*d^3*A*(b*e*x^2+a*e*x+b*d*x+a*d)^(1/2)*b^4*(b*e)^(1/2)*e+692*(b*e
*x^2+a*e*x+b*d*x+a*d)^(1/2)*a^2*d^2*B*b^2*(b*e)^(1/2)*e^2+1100*d^2*A*(b*e*x^2+a*
e*x+b*d*x+a*d)^(1/2)*a*b^3*(b*e)^(1/2)*e^2-120*(b*e*x^2+a*e*x+b*d*x+a*d)^(1/2)*a
^3*d*e^3*B*b*(b*e)^(1/2))/(b*e*x^2+a*e*x+b*d*x+a*d)^(1/2)/(b*e)^(1/2)/e^4/b^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)*sqrt(e*x + d),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.294942, 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)*sqrt(e*x + d),x, algorithm="fricas")

[Out]

[1/7680*(4*(384*B*b^4*e^4*x^4 - 105*B*b^4*d^4 + 10*(34*B*a*b^3 + 15*A*b^4)*d^3*e
- 2*(173*B*a^2*b^2 + 275*A*a*b^3)*d^2*e^2 + 10*(6*B*a^3*b + 73*A*a^2*b^2)*d*e^3
- 15*(3*B*a^4 - 10*A*a^3*b)*e^4 + 48*(B*b^4*d*e^3 + (21*B*a*b^3 + 10*A*b^4)*e^4
)*x^3 - 8*(7*B*b^4*d^2*e^2 - 2*(11*B*a*b^3 + 5*A*b^4)*d*e^3 - (93*B*a^2*b^2 + 17
0*A*a*b^3)*e^4)*x^2 + 2*(35*B*b^4*d^3*e - (111*B*a*b^3 + 50*A*b^4)*d^2*e^2 + (10
9*B*a^2*b^2 + 180*A*a*b^3)*d*e^3 + 5*(3*B*a^3*b + 118*A*a^2*b^2)*e^4)*x)*sqrt(b*
e)*sqrt(b*x + a)*sqrt(e*x + d) - 15*(7*B*b^5*d^5 - 5*(5*B*a*b^4 + 2*A*b^5)*d^4*e
+ 10*(3*B*a^2*b^3 + 4*A*a*b^4)*d^3*e^2 - 10*(B*a^3*b^2 + 6*A*a^2*b^3)*d^2*e^3 -
5*(B*a^4*b - 8*A*a^3*b^2)*d*e^4 + (3*B*a^5 - 10*A*a^4*b)*e^5)*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^2*e^4),
1/3840*(2*(384*B*b^4*e^4*x^4 - 105*B*b^4*d^4 + 10*(34*B*a*b^3 + 15*A*b^4)*d^3*e
- 2*(173*B*a^2*b^2 + 275*A*a*b^3)*d^2*e^2 + 10*(6*B*a^3*b + 73*A*a^2*b^2)*d*e^3
- 15*(3*B*a^4 - 10*A*a^3*b)*e^4 + 48*(B*b^4*d*e^3 + (21*B*a*b^3 + 10*A*b^4)*e^4
)*x^3 - 8*(7*B*b^4*d^2*e^2 - 2*(11*B*a*b^3 + 5*A*b^4)*d*e^3 - (93*B*a^2*b^2 + 17
0*A*a*b^3)*e^4)*x^2 + 2*(35*B*b^4*d^3*e - (111*B*a*b^3 + 50*A*b^4)*d^2*e^2 + (10
9*B*a^2*b^2 + 180*A*a*b^3)*d*e^3 + 5*(3*B*a^3*b + 118*A*a^2*b^2)*e^4)*x)*sqrt(-b
*e)*sqrt(b*x + a)*sqrt(e*x + d) + 15*(7*B*b^5*d^5 - 5*(5*B*a*b^4 + 2*A*b^5)*d^4*
e + 10*(3*B*a^2*b^3 + 4*A*a*b^4)*d^3*e^2 - 10*(B*a^3*b^2 + 6*A*a^2*b^3)*d^2*e^3
- 5*(B*a^4*b - 8*A*a^3*b^2)*d*e^4 + (3*B*a^5 - 10*A*a^4*b)*e^5)*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^2*
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)**(1/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.358789, 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)*sqrt(e*x + d),x, algorithm="giac")

[Out]

Done