3.2210 \(\int \frac{(a+b x)^{5/2} (A+B x)}{(d+e x)^{3/2}} \, dx\)

Optimal. Leaf size=253 \[ -\frac{5 (b d-a e)^2 (-a B e-6 A b e+7 b B d) \tanh ^{-1}\left (\frac{\sqrt{e} \sqrt{a+b x}}{\sqrt{b} \sqrt{d+e x}}\right )}{8 \sqrt{b} e^{9/2}}+\frac{5 \sqrt{a+b x} \sqrt{d+e x} (b d-a e) (-a B e-6 A b e+7 b B d)}{8 e^4}-\frac{5 (a+b x)^{3/2} \sqrt{d+e x} (-a B e-6 A b e+7 b B d)}{12 e^3}+\frac{(a+b x)^{5/2} \sqrt{d+e x} (-a B e-6 A b e+7 b B d)}{3 e^2 (b d-a e)}-\frac{2 (a+b x)^{7/2} (B d-A e)}{e \sqrt{d+e x} (b d-a e)} \]

[Out]

(-2*(B*d - A*e)*(a + b*x)^(7/2))/(e*(b*d - a*e)*Sqrt[d + e*x]) + (5*(b*d - a*e)*
(7*b*B*d - 6*A*b*e - a*B*e)*Sqrt[a + b*x]*Sqrt[d + e*x])/(8*e^4) - (5*(7*b*B*d -
6*A*b*e - a*B*e)*(a + b*x)^(3/2)*Sqrt[d + e*x])/(12*e^3) + ((7*b*B*d - 6*A*b*e
- a*B*e)*(a + b*x)^(5/2)*Sqrt[d + e*x])/(3*e^2*(b*d - a*e)) - (5*(b*d - a*e)^2*(
7*b*B*d - 6*A*b*e - a*B*e)*ArcTanh[(Sqrt[e]*Sqrt[a + b*x])/(Sqrt[b]*Sqrt[d + e*x
])])/(8*Sqrt[b]*e^(9/2))

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Rubi [A]  time = 0.532623, antiderivative size = 253, 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 \[ -\frac{5 (b d-a e)^2 (-a B e-6 A b e+7 b B d) \tanh ^{-1}\left (\frac{\sqrt{e} \sqrt{a+b x}}{\sqrt{b} \sqrt{d+e x}}\right )}{8 \sqrt{b} e^{9/2}}+\frac{5 \sqrt{a+b x} \sqrt{d+e x} (b d-a e) (-a B e-6 A b e+7 b B d)}{8 e^4}-\frac{5 (a+b x)^{3/2} \sqrt{d+e x} (-a B e-6 A b e+7 b B d)}{12 e^3}+\frac{(a+b x)^{5/2} \sqrt{d+e x} (-a B e-6 A b e+7 b B d)}{3 e^2 (b d-a e)}-\frac{2 (a+b x)^{7/2} (B d-A e)}{e \sqrt{d+e x} (b d-a e)} \]

Antiderivative was successfully verified.

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

[Out]

(-2*(B*d - A*e)*(a + b*x)^(7/2))/(e*(b*d - a*e)*Sqrt[d + e*x]) + (5*(b*d - a*e)*
(7*b*B*d - 6*A*b*e - a*B*e)*Sqrt[a + b*x]*Sqrt[d + e*x])/(8*e^4) - (5*(7*b*B*d -
6*A*b*e - a*B*e)*(a + b*x)^(3/2)*Sqrt[d + e*x])/(12*e^3) + ((7*b*B*d - 6*A*b*e
- a*B*e)*(a + b*x)^(5/2)*Sqrt[d + e*x])/(3*e^2*(b*d - a*e)) - (5*(b*d - a*e)^2*(
7*b*B*d - 6*A*b*e - a*B*e)*ArcTanh[(Sqrt[e]*Sqrt[a + b*x])/(Sqrt[b]*Sqrt[d + e*x
])])/(8*Sqrt[b]*e^(9/2))

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Rubi in Sympy [A]  time = 50.4226, size = 241, normalized size = 0.95 \[ - \frac{2 \left (a + b x\right )^{\frac{7}{2}} \left (A e - B d\right )}{e \sqrt{d + e x} \left (a e - b d\right )} + \frac{\left (a + b x\right )^{\frac{5}{2}} \sqrt{d + e x} \left (6 A b e + B a e - 7 B b d\right )}{3 e^{2} \left (a e - b d\right )} + \frac{5 \left (a + b x\right )^{\frac{3}{2}} \sqrt{d + e x} \left (6 A b e + B a e - 7 B b d\right )}{12 e^{3}} + \frac{5 \sqrt{a + b x} \sqrt{d + e x} \left (a e - b d\right ) \left (6 A b e + B a e - 7 B b d\right )}{8 e^{4}} + \frac{5 \left (a e - b d\right )^{2} \left (6 A b e + B a e - 7 B b d\right ) \operatorname{atanh}{\left (\frac{\sqrt{e} \sqrt{a + b x}}{\sqrt{b} \sqrt{d + e x}} \right )}}{8 \sqrt{b} e^{\frac{9}{2}}} \]

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]

-2*(a + b*x)**(7/2)*(A*e - B*d)/(e*sqrt(d + e*x)*(a*e - b*d)) + (a + b*x)**(5/2)
*sqrt(d + e*x)*(6*A*b*e + B*a*e - 7*B*b*d)/(3*e**2*(a*e - b*d)) + 5*(a + b*x)**(
3/2)*sqrt(d + e*x)*(6*A*b*e + B*a*e - 7*B*b*d)/(12*e**3) + 5*sqrt(a + b*x)*sqrt(
d + e*x)*(a*e - b*d)*(6*A*b*e + B*a*e - 7*B*b*d)/(8*e**4) + 5*(a*e - b*d)**2*(6*
A*b*e + B*a*e - 7*B*b*d)*atanh(sqrt(e)*sqrt(a + b*x)/(sqrt(b)*sqrt(d + e*x)))/(8
*sqrt(b)*e**(9/2))

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Mathematica [A]  time = 0.430037, size = 230, normalized size = 0.91 \[ \frac{\sqrt{a+b x} \left (3 a^2 e^2 (-16 A e+27 B d+11 B e x)+2 a b e \left (3 A e (25 d+9 e x)+B \left (-95 d^2-34 d e x+13 e^2 x^2\right )\right )+b^2 \left (6 A e \left (-15 d^2-5 d e x+2 e^2 x^2\right )+B \left (105 d^3+35 d^2 e x-14 d e^2 x^2+8 e^3 x^3\right )\right )\right )}{24 e^4 \sqrt{d+e x}}+\frac{5 (b d-a e)^2 (a B e+6 A b e-7 b B d) \log \left (2 \sqrt{b} \sqrt{e} \sqrt{a+b x} \sqrt{d+e x}+a e+b d+2 b e x\right )}{16 \sqrt{b} e^{9/2}} \]

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]*(3*a^2*e^2*(27*B*d - 16*A*e + 11*B*e*x) + 2*a*b*e*(3*A*e*(25*d +
9*e*x) + B*(-95*d^2 - 34*d*e*x + 13*e^2*x^2)) + b^2*(6*A*e*(-15*d^2 - 5*d*e*x +
2*e^2*x^2) + B*(105*d^3 + 35*d^2*e*x - 14*d*e^2*x^2 + 8*e^3*x^3))))/(24*e^4*Sqrt
[d + e*x]) + (5*(b*d - a*e)^2*(-7*b*B*d + 6*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]])/(16*Sqrt[b]*e^(9/2))

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Maple [B]  time = 0.046, size = 1184, normalized size = 4.7 \[ \text{result too large to display} \]

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]

1/48*(b*x+a)^(1/2)*(24*A*x^2*b^2*e^3*(b*e)^(1/2)*((b*x+a)*(e*x+d))^(1/2)-105*B*l
n(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))*x*b^3
*d^3*e+225*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*b^2*d^3*e+66*B*x*a^2*e^3*(b*e)^(1/2)*((b*x+a)*(e*x+d))^(1/2)+162*B*a^
2*d*e^2*(b*e)^(1/2)*((b*x+a)*(e*x+d))^(1/2)+210*B*b^2*d^3*(b*e)^(1/2)*((b*x+a)*(
e*x+d))^(1/2)+225*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))*x*a*b^2*d^2*e^2+52*B*x^2*a*b*e^3*(b*e)^(1/2)*((b*x+a)*(e*x+d))^(
1/2)-28*B*x^2*b^2*d*e^2*(b*e)^(1/2)*((b*x+a)*(e*x+d))^(1/2)+108*A*x*a*b*e^3*(b*e
)^(1/2)*((b*x+a)*(e*x+d))^(1/2)+70*B*x*b^2*d^2*e*(b*e)^(1/2)*((b*x+a)*(e*x+d))^(
1/2)-380*B*a*b*d^2*e*(b*e)^(1/2)*((b*x+a)*(e*x+d))^(1/2)-136*B*x*a*b*d*e^2*(b*e)
^(1/2)*((b*x+a)*(e*x+d))^(1/2)-105*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))*b^3*d^4-96*A*a^2*e^3*(b*e)^(1/2)*((b*x+a)*(e*x+
d))^(1/2)-180*A*b^2*d^2*e*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)+90*A*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))*x*a^2*b*e^4+90*A*
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))*x*b^
3*d^2*e^2+90*A*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*d*e^3-180*A*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^2*d^2*e^2-135*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^2*b*d^2*e^2+16*B*x^3*b^2*e^3*((b*x+a)
*(e*x+d))^(1/2)*(b*e)^(1/2)-60*A*x*b^2*d*e^2*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)
+300*A*a*b*d*e^2*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)-180*A*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))*x*a*b^2*d*e^3-135*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))*x*a^2*b*d
*e^3+15*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))*x*a^3*e^4+90*A*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^3*d^3*e+15*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^3*d*e^3)/((b*x+a)*(e*x+d))^(1/2)/(b*e)^(1/2)/(e*x+
d)^(1/2)/e^4

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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 = 1.01544, 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]

[1/96*(4*(8*B*b^2*e^3*x^3 + 105*B*b^2*d^3 - 48*A*a^2*e^3 - 10*(19*B*a*b + 9*A*b^
2)*d^2*e + 3*(27*B*a^2 + 50*A*a*b)*d*e^2 - 2*(7*B*b^2*d*e^2 - (13*B*a*b + 6*A*b^
2)*e^3)*x^2 + (35*B*b^2*d^2*e - 2*(34*B*a*b + 15*A*b^2)*d*e^2 + 3*(11*B*a^2 + 18
*A*a*b)*e^3)*x)*sqrt(b*e)*sqrt(b*x + a)*sqrt(e*x + d) - 15*(7*B*b^3*d^4 - 3*(5*B
*a*b^2 + 2*A*b^3)*d^3*e + 3*(3*B*a^2*b + 4*A*a*b^2)*d^2*e^2 - (B*a^3 + 6*A*a^2*b
)*d*e^3 + (7*B*b^3*d^3*e - 3*(5*B*a*b^2 + 2*A*b^3)*d^2*e^2 + 3*(3*B*a^2*b + 4*A*
a*b^2)*d*e^3 - (B*a^3 + 6*A*a^2*b)*e^4)*x)*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)))/((e^5*x + d*e^4)*sqrt(b*e)), 1/48*(2*(8*B
*b^2*e^3*x^3 + 105*B*b^2*d^3 - 48*A*a^2*e^3 - 10*(19*B*a*b + 9*A*b^2)*d^2*e + 3*
(27*B*a^2 + 50*A*a*b)*d*e^2 - 2*(7*B*b^2*d*e^2 - (13*B*a*b + 6*A*b^2)*e^3)*x^2 +
(35*B*b^2*d^2*e - 2*(34*B*a*b + 15*A*b^2)*d*e^2 + 3*(11*B*a^2 + 18*A*a*b)*e^3)*
x)*sqrt(-b*e)*sqrt(b*x + a)*sqrt(e*x + d) - 15*(7*B*b^3*d^4 - 3*(5*B*a*b^2 + 2*A
*b^3)*d^3*e + 3*(3*B*a^2*b + 4*A*a*b^2)*d^2*e^2 - (B*a^3 + 6*A*a^2*b)*d*e^3 + (7
*B*b^3*d^3*e - 3*(5*B*a*b^2 + 2*A*b^3)*d^2*e^2 + 3*(3*B*a^2*b + 4*A*a*b^2)*d*e^3
- (B*a^3 + 6*A*a^2*b)*e^4)*x)*arctan(1/2*(2*b*e*x + b*d + a*e)*sqrt(-b*e)/(sqrt
(b*x + a)*sqrt(e*x + d)*b*e)))/((e^5*x + d*e^4)*sqrt(-b*e))]

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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 = 0.265439, size = 564, normalized size = 2.23 \[ \frac{{\left ({\left (2 \,{\left (\frac{4 \,{\left (b x + a\right )} B b{\left | b \right |} e^{6}}{b^{10} d e^{8} - a b^{9} e^{9}} - \frac{7 \, B b^{2} d{\left | b \right |} e^{5} - B a b{\left | b \right |} e^{6} - 6 \, A b^{2}{\left | b \right |} e^{6}}{b^{10} d e^{8} - a b^{9} e^{9}}\right )}{\left (b x + a\right )} + \frac{5 \,{\left (7 \, B b^{3} d^{2}{\left | b \right |} e^{4} - 8 \, B a b^{2} d{\left | b \right |} e^{5} - 6 \, A b^{3} d{\left | b \right |} e^{5} + B a^{2} b{\left | b \right |} e^{6} + 6 \, A a b^{2}{\left | b \right |} e^{6}\right )}}{b^{10} d e^{8} - a b^{9} e^{9}}\right )}{\left (b x + a\right )} + \frac{15 \,{\left (7 \, B b^{4} d^{3}{\left | b \right |} e^{3} - 15 \, B a b^{3} d^{2}{\left | b \right |} e^{4} - 6 \, A b^{4} d^{2}{\left | b \right |} e^{4} + 9 \, B a^{2} b^{2} d{\left | b \right |} e^{5} + 12 \, A a b^{3} d{\left | b \right |} e^{5} - B a^{3} b{\left | b \right |} e^{6} - 6 \, A a^{2} b^{2}{\left | b \right |} e^{6}\right )}}{b^{10} d e^{8} - a b^{9} e^{9}}\right )} \sqrt{b x + a}}{184320 \, \sqrt{b^{2} d +{\left (b x + a\right )} b e - a b e}} + \frac{{\left (7 \, B b^{2} d^{2}{\left | b \right |} - 8 \, B a b d{\left | b \right |} e - 6 \, A b^{2} d{\left | b \right |} e + B a^{2}{\left | b \right |} e^{2} + 6 \, A a b{\left | b \right |} e^{2}\right )} e^{\left (-\frac{11}{2}\right )}{\rm ln}\left ({\left | -\sqrt{b x + a} \sqrt{b} e^{\frac{1}{2}} + \sqrt{b^{2} d +{\left (b x + a\right )} b e - a b e} \right |}\right )}{12288 \, b^{\frac{17}{2}}} \]

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]

1/184320*((2*(4*(b*x + a)*B*b*abs(b)*e^6/(b^10*d*e^8 - a*b^9*e^9) - (7*B*b^2*d*a
bs(b)*e^5 - B*a*b*abs(b)*e^6 - 6*A*b^2*abs(b)*e^6)/(b^10*d*e^8 - a*b^9*e^9))*(b*
x + a) + 5*(7*B*b^3*d^2*abs(b)*e^4 - 8*B*a*b^2*d*abs(b)*e^5 - 6*A*b^3*d*abs(b)*e
^5 + B*a^2*b*abs(b)*e^6 + 6*A*a*b^2*abs(b)*e^6)/(b^10*d*e^8 - a*b^9*e^9))*(b*x +
a) + 15*(7*B*b^4*d^3*abs(b)*e^3 - 15*B*a*b^3*d^2*abs(b)*e^4 - 6*A*b^4*d^2*abs(b
)*e^4 + 9*B*a^2*b^2*d*abs(b)*e^5 + 12*A*a*b^3*d*abs(b)*e^5 - B*a^3*b*abs(b)*e^6
- 6*A*a^2*b^2*abs(b)*e^6)/(b^10*d*e^8 - a*b^9*e^9))*sqrt(b*x + a)/sqrt(b^2*d + (
b*x + a)*b*e - a*b*e) + 1/12288*(7*B*b^2*d^2*abs(b) - 8*B*a*b*d*abs(b)*e - 6*A*b
^2*d*abs(b)*e + B*a^2*abs(b)*e^2 + 6*A*a*b*abs(b)*e^2)*e^(-11/2)*ln(abs(-sqrt(b*
x + a)*sqrt(b)*e^(1/2) + sqrt(b^2*d + (b*x + a)*b*e - a*b*e)))/b^(17/2)