3.2209 \(\int \frac{(a+b x)^{5/2} (A+B x)}{\sqrt{d+e x}} \, dx\)

Optimal. Leaf size=246 \[ \frac{5 (b d-a e)^3 (a B e-8 A b e+7 b B d) \tanh ^{-1}\left (\frac{\sqrt{e} \sqrt{a+b x}}{\sqrt{b} \sqrt{d+e x}}\right )}{64 b^{3/2} e^{9/2}}-\frac{5 \sqrt{a+b x} \sqrt{d+e x} (b d-a e)^2 (a B e-8 A b e+7 b B d)}{64 b e^4}+\frac{5 (a+b x)^{3/2} \sqrt{d+e x} (b d-a e) (a B e-8 A b e+7 b B d)}{96 b e^3}-\frac{(a+b x)^{5/2} \sqrt{d+e x} (a B e-8 A b e+7 b B d)}{24 b e^2}+\frac{B (a+b x)^{7/2} \sqrt{d+e x}}{4 b e} \]

[Out]

(-5*(b*d - a*e)^2*(7*b*B*d - 8*A*b*e + a*B*e)*Sqrt[a + b*x]*Sqrt[d + e*x])/(64*b
*e^4) + (5*(b*d - a*e)*(7*b*B*d - 8*A*b*e + a*B*e)*(a + b*x)^(3/2)*Sqrt[d + e*x]
)/(96*b*e^3) - ((7*b*B*d - 8*A*b*e + a*B*e)*(a + b*x)^(5/2)*Sqrt[d + e*x])/(24*b
*e^2) + (B*(a + b*x)^(7/2)*Sqrt[d + e*x])/(4*b*e) + (5*(b*d - a*e)^3*(7*b*B*d -
8*A*b*e + a*B*e)*ArcTanh[(Sqrt[e]*Sqrt[a + b*x])/(Sqrt[b]*Sqrt[d + e*x])])/(64*b
^(3/2)*e^(9/2))

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Rubi [A]  time = 0.504183, antiderivative size = 246, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167 \[ \frac{5 (b d-a e)^3 (a B e-8 A b e+7 b B d) \tanh ^{-1}\left (\frac{\sqrt{e} \sqrt{a+b x}}{\sqrt{b} \sqrt{d+e x}}\right )}{64 b^{3/2} e^{9/2}}-\frac{5 \sqrt{a+b x} \sqrt{d+e x} (b d-a e)^2 (a B e-8 A b e+7 b B d)}{64 b e^4}+\frac{5 (a+b x)^{3/2} \sqrt{d+e x} (b d-a e) (a B e-8 A b e+7 b B d)}{96 b e^3}-\frac{(a+b x)^{5/2} \sqrt{d+e x} (a B e-8 A b e+7 b B d)}{24 b e^2}+\frac{B (a+b x)^{7/2} \sqrt{d+e x}}{4 b e} \]

Antiderivative was successfully verified.

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

[Out]

(-5*(b*d - a*e)^2*(7*b*B*d - 8*A*b*e + a*B*e)*Sqrt[a + b*x]*Sqrt[d + e*x])/(64*b
*e^4) + (5*(b*d - a*e)*(7*b*B*d - 8*A*b*e + a*B*e)*(a + b*x)^(3/2)*Sqrt[d + e*x]
)/(96*b*e^3) - ((7*b*B*d - 8*A*b*e + a*B*e)*(a + b*x)^(5/2)*Sqrt[d + e*x])/(24*b
*e^2) + (B*(a + b*x)^(7/2)*Sqrt[d + e*x])/(4*b*e) + (5*(b*d - a*e)^3*(7*b*B*d -
8*A*b*e + a*B*e)*ArcTanh[(Sqrt[e]*Sqrt[a + b*x])/(Sqrt[b]*Sqrt[d + e*x])])/(64*b
^(3/2)*e^(9/2))

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Rubi in Sympy [A]  time = 40.4961, size = 238, normalized size = 0.97 \[ \frac{B \left (a + b x\right )^{\frac{7}{2}} \sqrt{d + e x}}{4 b e} + \frac{\left (a + b x\right )^{\frac{5}{2}} \sqrt{d + e x} \left (8 A b e - B a e - 7 B b d\right )}{24 b e^{2}} + \frac{5 \left (a + b x\right )^{\frac{3}{2}} \sqrt{d + e x} \left (a e - b d\right ) \left (8 A b e - B a e - 7 B b d\right )}{96 b e^{3}} + \frac{5 \sqrt{a + b x} \sqrt{d + e x} \left (a e - b d\right )^{2} \left (8 A b e - B a e - 7 B b d\right )}{64 b e^{4}} + \frac{5 \left (a e - b d\right )^{3} \left (8 A b e - B a e - 7 B b d\right ) \operatorname{atanh}{\left (\frac{\sqrt{b} \sqrt{d + e x}}{\sqrt{e} \sqrt{a + b x}} \right )}}{64 b^{\frac{3}{2}} e^{\frac{9}{2}}} \]

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)*sqrt(d + e*x)/(4*b*e) + (a + b*x)**(5/2)*sqrt(d + e*x)*(8*A*b
*e - B*a*e - 7*B*b*d)/(24*b*e**2) + 5*(a + b*x)**(3/2)*sqrt(d + e*x)*(a*e - b*d)
*(8*A*b*e - B*a*e - 7*B*b*d)/(96*b*e**3) + 5*sqrt(a + b*x)*sqrt(d + e*x)*(a*e -
b*d)**2*(8*A*b*e - B*a*e - 7*B*b*d)/(64*b*e**4) + 5*(a*e - b*d)**3*(8*A*b*e - B*
a*e - 7*B*b*d)*atanh(sqrt(b)*sqrt(d + e*x)/(sqrt(e)*sqrt(a + b*x)))/(64*b**(3/2)
*e**(9/2))

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Mathematica [A]  time = 0.355073, size = 243, normalized size = 0.99 \[ \frac{\sqrt{a+b x} \sqrt{d+e x} \left (15 a^3 B e^3+a^2 b e^2 (264 A e-191 B d+118 B e x)+a b^2 e \left (16 A e (13 e x-20 d)+B \left (265 d^2-172 d e x+136 e^2 x^2\right )\right )+b^3 \left (8 A e \left (15 d^2-10 d e x+8 e^2 x^2\right )+B \left (-105 d^3+70 d^2 e x-56 d e^2 x^2+48 e^3 x^3\right )\right )\right )}{192 b e^4}+\frac{5 (b d-a e)^3 (a B e-8 A b e+7 b B d) \log \left (2 \sqrt{b} \sqrt{e} \sqrt{a+b x} \sqrt{d+e x}+a e+b d+2 b e x\right )}{128 b^{3/2} e^{9/2}} \]

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]*(15*a^3*B*e^3 + a^2*b*e^2*(-191*B*d + 264*A*e + 118
*B*e*x) + a*b^2*e*(16*A*e*(-20*d + 13*e*x) + B*(265*d^2 - 172*d*e*x + 136*e^2*x^
2)) + b^3*(8*A*e*(15*d^2 - 10*d*e*x + 8*e^2*x^2) + B*(-105*d^3 + 70*d^2*e*x - 56
*d*e^2*x^2 + 48*e^3*x^3))))/(192*b*e^4) + (5*(b*d - a*e)^3*(7*b*B*d - 8*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]])
/(128*b^(3/2)*e^(9/2))

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Maple [B]  time = 0.036, size = 968, normalized size = 3.9 \[ \text{result too large to display} \]

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]

1/384*(b*x+a)^(1/2)*(e*x+d)^(1/2)*(-15*e^4*B*ln(1/2*(2*b*x*e+2*((b*x+a)*(e*x+d))
^(1/2)*(b*e)^(1/2)+a*e+b*d)/(b*e)^(1/2))*a^4+105*ln(1/2*(2*b*x*e+2*((b*x+a)*(e*x
+d))^(1/2)*(b*e)^(1/2)+a*e+b*d)/(b*e)^(1/2))*b^4*d^4*B-360*ln(1/2*(2*b*x*e+2*((b
*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)+a*e+b*d)/(b*e)^(1/2))*a^2*A*b^2*d*e^3-344*((b*x
+a)*(e*x+d))^(1/2)*x*a*d*B*b^2*e^2*(b*e)^(1/2)-120*ln(1/2*(2*b*x*e+2*((b*x+a)*(e
*x+d))^(1/2)*(b*e)^(1/2)+a*e+b*d)/(b*e)^(1/2))*b^4*d^3*A*e-210*((b*x+a)*(e*x+d))
^(1/2)*B*b^3*d^3*(b*e)^(1/2)+120*ln(1/2*(2*b*x*e+2*((b*x+a)*(e*x+d))^(1/2)*(b*e)
^(1/2)+a*e+b*d)/(b*e)^(1/2))*a^3*A*e^4*b+30*e^3*B*((b*x+a)*(e*x+d))^(1/2)*a^3*(b
*e)^(1/2)+360*ln(1/2*(2*b*x*e+2*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)+a*e+b*d)/(b*
e)^(1/2))*a*A*b^3*d^2*e^2-60*ln(1/2*(2*b*x*e+2*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/
2)+a*e+b*d)/(b*e)^(1/2))*a^3*B*d*e^3*b+270*ln(1/2*(2*b*x*e+2*((b*x+a)*(e*x+d))^(
1/2)*(b*e)^(1/2)+a*e+b*d)/(b*e)^(1/2))*a^2*B*b^2*d^2*e^2-300*ln(1/2*(2*b*x*e+2*(
(b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)+a*e+b*d)/(b*e)^(1/2))*a*B*b^3*d^3*e+528*((b*x
+a)*(e*x+d))^(1/2)*A*a^2*e^3*(b*e)^(1/2)*b+240*((b*x+a)*(e*x+d))^(1/2)*A*b^3*d^2
*e*(b*e)^(1/2)+96*B*x^3*b^3*e^3*(b*e)^(1/2)*((b*x+a)*(e*x+d))^(1/2)+128*A*x^2*b^
3*e^3*(b*e)^(1/2)*((b*x+a)*(e*x+d))^(1/2)+236*e^3*B*((b*x+a)*(e*x+d))^(1/2)*x*a^
2*b*(b*e)^(1/2)+140*((b*x+a)*(e*x+d))^(1/2)*x*d^2*B*b^3*e*(b*e)^(1/2)+416*((b*x+
a)*(e*x+d))^(1/2)*x*a*A*e^3*b^2*(b*e)^(1/2)-160*d*A*((b*x+a)*(e*x+d))^(1/2)*x*b^
3*e^2*(b*e)^(1/2)+272*B*x^2*a*b^2*e^3*(b*e)^(1/2)*((b*x+a)*(e*x+d))^(1/2)-112*B*
x^2*b^3*d*e^2*(b*e)^(1/2)*((b*x+a)*(e*x+d))^(1/2)-640*((b*x+a)*(e*x+d))^(1/2)*A*
a*b^2*d*e^2*(b*e)^(1/2)-382*((b*x+a)*(e*x+d))^(1/2)*B*a^2*d*e^2*(b*e)^(1/2)*b+53
0*((b*x+a)*(e*x+d))^(1/2)*B*a*b^2*d^2*e*(b*e)^(1/2))/((b*x+a)*(e*x+d))^(1/2)/e^4
/(b*e)^(1/2)/b

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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 = 0.620194, 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]

[1/768*(4*(48*B*b^3*e^3*x^3 - 105*B*b^3*d^3 + 5*(53*B*a*b^2 + 24*A*b^3)*d^2*e -
(191*B*a^2*b + 320*A*a*b^2)*d*e^2 + 3*(5*B*a^3 + 88*A*a^2*b)*e^3 - 8*(7*B*b^3*d*
e^2 - (17*B*a*b^2 + 8*A*b^3)*e^3)*x^2 + 2*(35*B*b^3*d^2*e - 2*(43*B*a*b^2 + 20*A
*b^3)*d*e^2 + (59*B*a^2*b + 104*A*a*b^2)*e^3)*x)*sqrt(b*e)*sqrt(b*x + a)*sqrt(e*
x + d) - 15*(7*B*b^4*d^4 - 4*(5*B*a*b^3 + 2*A*b^4)*d^3*e + 6*(3*B*a^2*b^2 + 4*A*
a*b^3)*d^2*e^2 - 4*(B*a^3*b + 6*A*a^2*b^2)*d*e^3 - (B*a^4 - 8*A*a^3*b)*e^4)*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*e^4), 1/384*(2*(48*B*b^3*e^3*x^3 - 105*B*b^3*d^3 + 5*(53*B*a*b^2 + 24*A*b^
3)*d^2*e - (191*B*a^2*b + 320*A*a*b^2)*d*e^2 + 3*(5*B*a^3 + 88*A*a^2*b)*e^3 - 8*
(7*B*b^3*d*e^2 - (17*B*a*b^2 + 8*A*b^3)*e^3)*x^2 + 2*(35*B*b^3*d^2*e - 2*(43*B*a
*b^2 + 20*A*b^3)*d*e^2 + (59*B*a^2*b + 104*A*a*b^2)*e^3)*x)*sqrt(-b*e)*sqrt(b*x
+ a)*sqrt(e*x + d) + 15*(7*B*b^4*d^4 - 4*(5*B*a*b^3 + 2*A*b^4)*d^3*e + 6*(3*B*a^
2*b^2 + 4*A*a*b^3)*d^2*e^2 - 4*(B*a^3*b + 6*A*a^2*b^2)*d*e^3 - (B*a^4 - 8*A*a^3*
b)*e^4)*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*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 = 0.246939, size = 527, normalized size = 2.14 \[ \frac{{\left (\sqrt{b^{2} d +{\left (b x + a\right )} b e - a b e}{\left (2 \,{\left (b x + a\right )}{\left (4 \,{\left (b x + a\right )}{\left (\frac{6 \,{\left (b x + a\right )} B e^{\left (-1\right )}}{b^{2}} - \frac{{\left (7 \, B b^{3} d e^{5} + B a b^{2} e^{6} - 8 \, A b^{3} e^{6}\right )} e^{\left (-7\right )}}{b^{4}}\right )} + \frac{5 \,{\left (7 \, B b^{4} d^{2} e^{4} - 6 \, B a b^{3} d e^{5} - 8 \, A b^{4} d e^{5} - B a^{2} b^{2} e^{6} + 8 \, A a b^{3} e^{6}\right )} e^{\left (-7\right )}}{b^{4}}\right )} - \frac{15 \,{\left (7 \, B b^{5} d^{3} e^{3} - 13 \, B a b^{4} d^{2} e^{4} - 8 \, A b^{5} d^{2} e^{4} + 5 \, B a^{2} b^{3} d e^{5} + 16 \, A a b^{4} d e^{5} + B a^{3} b^{2} e^{6} - 8 \, A a^{2} b^{3} e^{6}\right )} e^{\left (-7\right )}}{b^{4}}\right )} \sqrt{b x + a} - \frac{15 \,{\left (7 \, B b^{4} d^{4} - 20 \, B a b^{3} d^{3} e - 8 \, A b^{4} d^{3} e + 18 \, B a^{2} b^{2} d^{2} e^{2} + 24 \, A a b^{3} d^{2} e^{2} - 4 \, B a^{3} b d e^{3} - 24 \, A a^{2} b^{2} d e^{3} - B a^{4} e^{4} + 8 \, A a^{3} b e^{4}\right )} e^{\left (-\frac{9}{2}\right )}{\rm ln}\left ({\left | -\sqrt{b x + a} \sqrt{b} e^{\frac{1}{2}} + \sqrt{b^{2} d +{\left (b x + a\right )} b e - a b e} \right |}\right )}{b^{\frac{3}{2}}}\right )} b}{192 \,{\left | b \right |}} \]

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]

1/192*(sqrt(b^2*d + (b*x + a)*b*e - a*b*e)*(2*(b*x + a)*(4*(b*x + a)*(6*(b*x + a
)*B*e^(-1)/b^2 - (7*B*b^3*d*e^5 + B*a*b^2*e^6 - 8*A*b^3*e^6)*e^(-7)/b^4) + 5*(7*
B*b^4*d^2*e^4 - 6*B*a*b^3*d*e^5 - 8*A*b^4*d*e^5 - B*a^2*b^2*e^6 + 8*A*a*b^3*e^6)
*e^(-7)/b^4) - 15*(7*B*b^5*d^3*e^3 - 13*B*a*b^4*d^2*e^4 - 8*A*b^5*d^2*e^4 + 5*B*
a^2*b^3*d*e^5 + 16*A*a*b^4*d*e^5 + B*a^3*b^2*e^6 - 8*A*a^2*b^3*e^6)*e^(-7)/b^4)*
sqrt(b*x + a) - 15*(7*B*b^4*d^4 - 20*B*a*b^3*d^3*e - 8*A*b^4*d^3*e + 18*B*a^2*b^
2*d^2*e^2 + 24*A*a*b^3*d^2*e^2 - 4*B*a^3*b*d*e^3 - 24*A*a^2*b^2*d*e^3 - B*a^4*e^
4 + 8*A*a^3*b*e^4)*e^(-9/2)*ln(abs(-sqrt(b*x + a)*sqrt(b)*e^(1/2) + sqrt(b^2*d +
(b*x + a)*b*e - a*b*e)))/b^(3/2))*b/abs(b)