3.2211 \(\int \frac{(a+b x)^{5/2} (A+B x)}{(d+e x)^{5/2}} \, dx\)

Optimal. Leaf size=257 \[ \frac{5 \sqrt{b} (b d-a e) (-3 a B e-4 A b e+7 b B d) \tanh ^{-1}\left (\frac{\sqrt{e} \sqrt{a+b x}}{\sqrt{b} \sqrt{d+e x}}\right )}{4 e^{9/2}}-\frac{5 b \sqrt{a+b x} \sqrt{d+e x} (-3 a B e-4 A b e+7 b B d)}{4 e^4}+\frac{5 b (a+b x)^{3/2} \sqrt{d+e x} (-3 a B e-4 A b e+7 b B d)}{6 e^3 (b d-a e)}-\frac{2 (a+b x)^{5/2} (-3 a B e-4 A b e+7 b B d)}{3 e^2 \sqrt{d+e x} (b d-a e)}-\frac{2 (a+b x)^{7/2} (B d-A e)}{3 e (d+e x)^{3/2} (b d-a e)} \]

[Out]

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

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Rubi [A]  time = 0.508585, antiderivative size = 257, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.208 \[ \frac{5 \sqrt{b} (b d-a e) (-3 a B e-4 A b e+7 b B d) \tanh ^{-1}\left (\frac{\sqrt{e} \sqrt{a+b x}}{\sqrt{b} \sqrt{d+e x}}\right )}{4 e^{9/2}}-\frac{5 b \sqrt{a+b x} \sqrt{d+e x} (-3 a B e-4 A b e+7 b B d)}{4 e^4}+\frac{5 b (a+b x)^{3/2} \sqrt{d+e x} (-3 a B e-4 A b e+7 b B d)}{6 e^3 (b d-a e)}-\frac{2 (a+b x)^{5/2} (-3 a B e-4 A b e+7 b B d)}{3 e^2 \sqrt{d+e x} (b d-a e)}-\frac{2 (a+b x)^{7/2} (B d-A e)}{3 e (d+e x)^{3/2} (b d-a e)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 50.2489, size = 253, normalized size = 0.98 \[ \frac{5 \sqrt{b} \left (a e - b d\right ) \left (4 A b e + 3 B a e - 7 B b d\right ) \operatorname{atanh}{\left (\frac{\sqrt{e} \sqrt{a + b x}}{\sqrt{b} \sqrt{d + e x}} \right )}}{4 e^{\frac{9}{2}}} + \frac{5 b \left (a + b x\right )^{\frac{3}{2}} \sqrt{d + e x} \left (4 A b e + 3 B a e - 7 B b d\right )}{6 e^{3} \left (a e - b d\right )} + \frac{5 b \sqrt{a + b x} \sqrt{d + e x} \left (4 A b e + 3 B a e - 7 B b d\right )}{4 e^{4}} - \frac{2 \left (a + b x\right )^{\frac{7}{2}} \left (A e - B d\right )}{3 e \left (d + e x\right )^{\frac{3}{2}} \left (a e - b d\right )} - \frac{2 \left (a + b x\right )^{\frac{5}{2}} \left (4 A b e + 3 B a e - 7 B b d\right )}{3 e^{2} \sqrt{d + e x} \left (a e - b d\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.569231, size = 228, normalized size = 0.89 \[ -\frac{\sqrt{a+b x} \left (8 a^2 e^2 (A e+2 B d+3 B e x)+a b e \left (8 A e (5 d+7 e x)-B \left (115 d^2+158 d e x+27 e^2 x^2\right )\right )+b^2 \left (B \left (105 d^3+140 d^2 e x+21 d e^2 x^2-6 e^3 x^3\right )-4 A e \left (15 d^2+20 d e x+3 e^2 x^2\right )\right )\right )}{12 e^4 (d+e x)^{3/2}}-\frac{5 \sqrt{b} (b d-a e) (3 a B e+4 A b e-7 b B d) \log \left (2 \sqrt{b} \sqrt{e} \sqrt{a+b x} \sqrt{d+e x}+a e+b d+2 b e x\right )}{8 e^{9/2}} \]

Antiderivative was successfully verified.

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

[Out]

-(Sqrt[a + b*x]*(8*a^2*e^2*(2*B*d + A*e + 3*B*e*x) + a*b*e*(8*A*e*(5*d + 7*e*x)
- B*(115*d^2 + 158*d*e*x + 27*e^2*x^2)) + b^2*(-4*A*e*(15*d^2 + 20*d*e*x + 3*e^2
*x^2) + B*(105*d^3 + 140*d^2*e*x + 21*d*e^2*x^2 - 6*e^3*x^3))))/(12*e^4*(d + e*x
)^(3/2)) - (5*Sqrt[b]*(b*d - a*e)*(-7*b*B*d + 4*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]])/(8*e^(9/2))

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Maple [B]  time = 0.038, size = 1250, normalized size = 4.9 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/24*(b*x+a)^(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))*x^2*b^3*d^2*e^2+24*A*x^2*b^2*e^3*(b*e)^(1/2)*((b*x+a)*(e*x
+d))^(1/2)+210*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*b^3*d^3*e-150*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-48*B*x*a^2*e^3*(b*e)^(1/2)*((b*x+a)*(e*x+
d))^(1/2)-32*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)-150*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^2*a*b^2*d*e^3-300*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+54*B*x^2*a*b*
e^3*(b*e)^(1/2)*((b*x+a)*(e*x+d))^(1/2)-42*B*x^2*b^2*d*e^2*(b*e)^(1/2)*((b*x+a)*
(e*x+d))^(1/2)-112*A*x*a*b*e^3*(b*e)^(1/2)*((b*x+a)*(e*x+d))^(1/2)-280*B*x*b^2*d
^2*e*(b*e)^(1/2)*((b*x+a)*(e*x+d))^(1/2)+230*B*a*b*d^2*e*(b*e)^(1/2)*((b*x+a)*(e
*x+d))^(1/2)+316*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^4-16*
A*a^2*e^3*(b*e)^(1/2)*((b*x+a)*(e*x+d))^(1/2)+120*A*b^2*d^2*e*((b*x+a)*(e*x+d))^
(1/2)*(b*e)^(1/2)-120*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+60*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+45*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+12*B*x^3*b^2*e^3
*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)-60*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^2*b^3*d*e^3+45*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^2*a^2*b*e^4+60*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^2*a*b^2*e^
4+160*A*x*b^2*d*e^2*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)-80*A*a*b*d*e^2*((b*x+a)*
(e*x+d))^(1/2)*(b*e)^(1/2)+120*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+90*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^3-60*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)/((b*x+a)*(
e*x+d))^(1/2)/(b*e)^(1/2)/(e*x+d)^(3/2)/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)^(5/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.12278, 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)^(5/2),x, algorithm="fricas")

[Out]

[1/48*(15*(7*B*b^2*d^4 - 2*(5*B*a*b + 2*A*b^2)*d^3*e + (3*B*a^2 + 4*A*a*b)*d^2*e
^2 + (7*B*b^2*d^2*e^2 - 2*(5*B*a*b + 2*A*b^2)*d*e^3 + (3*B*a^2 + 4*A*a*b)*e^4)*x
^2 + 2*(7*B*b^2*d^3*e - 2*(5*B*a*b + 2*A*b^2)*d^2*e^2 + (3*B*a^2 + 4*A*a*b)*d*e^
3)*x)*sqrt(b/e)*log(8*b^2*e^2*x^2 + b^2*d^2 + 6*a*b*d*e + a^2*e^2 + 4*(2*b*e^2*x
+ b*d*e + a*e^2)*sqrt(b*x + a)*sqrt(e*x + d)*sqrt(b/e) + 8*(b^2*d*e + a*b*e^2)*
x) + 4*(6*B*b^2*e^3*x^3 - 105*B*b^2*d^3 - 8*A*a^2*e^3 + 5*(23*B*a*b + 12*A*b^2)*
d^2*e - 8*(2*B*a^2 + 5*A*a*b)*d*e^2 - 3*(7*B*b^2*d*e^2 - (9*B*a*b + 4*A*b^2)*e^3
)*x^2 - 2*(70*B*b^2*d^2*e - (79*B*a*b + 40*A*b^2)*d*e^2 + 4*(3*B*a^2 + 7*A*a*b)*
e^3)*x)*sqrt(b*x + a)*sqrt(e*x + d))/(e^6*x^2 + 2*d*e^5*x + d^2*e^4), 1/24*(15*(
7*B*b^2*d^4 - 2*(5*B*a*b + 2*A*b^2)*d^3*e + (3*B*a^2 + 4*A*a*b)*d^2*e^2 + (7*B*b
^2*d^2*e^2 - 2*(5*B*a*b + 2*A*b^2)*d*e^3 + (3*B*a^2 + 4*A*a*b)*e^4)*x^2 + 2*(7*B
*b^2*d^3*e - 2*(5*B*a*b + 2*A*b^2)*d^2*e^2 + (3*B*a^2 + 4*A*a*b)*d*e^3)*x)*sqrt(
-b/e)*arctan(1/2*(2*b*e*x + b*d + a*e)/(sqrt(b*x + a)*sqrt(e*x + d)*e*sqrt(-b/e)
)) + 2*(6*B*b^2*e^3*x^3 - 105*B*b^2*d^3 - 8*A*a^2*e^3 + 5*(23*B*a*b + 12*A*b^2)*
d^2*e - 8*(2*B*a^2 + 5*A*a*b)*d*e^2 - 3*(7*B*b^2*d*e^2 - (9*B*a*b + 4*A*b^2)*e^3
)*x^2 - 2*(70*B*b^2*d^2*e - (79*B*a*b + 40*A*b^2)*d*e^2 + 4*(3*B*a^2 + 7*A*a*b)*
e^3)*x)*sqrt(b*x + a)*sqrt(e*x + d))/(e^6*x^2 + 2*d*e^5*x + d^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)**(5/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.293697, size = 721, normalized size = 2.81 \[ -\frac{5 \,{\left (7 \, B b^{2} d^{2}{\left | b \right |} - 10 \, B a b d{\left | b \right |} e - 4 \, A b^{2} d{\left | b \right |} e + 3 \, B a^{2}{\left | b \right |} e^{2} + 4 \, A a b{\left | b \right |} e^{2}\right )} e^{\left (-\frac{9}{2}\right )}{\rm ln}\left ({\left | -\sqrt{b x + a} \sqrt{b} e^{\frac{1}{2}} + \sqrt{b^{2} d +{\left (b x + a\right )} b e - a b e} \right |}\right )}{4 \, \sqrt{b}} + \frac{{\left ({\left (3 \,{\left (b x + a\right )}{\left (\frac{2 \,{\left (B b^{5} d{\left | b \right |} e^{6} - B a b^{4}{\left | b \right |} e^{7}\right )}{\left (b x + a\right )}}{b^{4} d e^{7} - a b^{3} e^{8}} - \frac{7 \, B b^{6} d^{2}{\left | b \right |} e^{5} - 10 \, B a b^{5} d{\left | b \right |} e^{6} - 4 \, A b^{6} d{\left | b \right |} e^{6} + 3 \, B a^{2} b^{4}{\left | b \right |} e^{7} + 4 \, A a b^{5}{\left | b \right |} e^{7}}{b^{4} d e^{7} - a b^{3} e^{8}}\right )} - \frac{20 \,{\left (7 \, B b^{7} d^{3}{\left | b \right |} e^{4} - 17 \, B a b^{6} d^{2}{\left | b \right |} e^{5} - 4 \, A b^{7} d^{2}{\left | b \right |} e^{5} + 13 \, B a^{2} b^{5} d{\left | b \right |} e^{6} + 8 \, A a b^{6} d{\left | b \right |} e^{6} - 3 \, B a^{3} b^{4}{\left | b \right |} e^{7} - 4 \, A a^{2} b^{5}{\left | b \right |} e^{7}\right )}}{b^{4} d e^{7} - a b^{3} e^{8}}\right )}{\left (b x + a\right )} - \frac{15 \,{\left (7 \, B b^{8} d^{4}{\left | b \right |} e^{3} - 24 \, B a b^{7} d^{3}{\left | b \right |} e^{4} - 4 \, A b^{8} d^{3}{\left | b \right |} e^{4} + 30 \, B a^{2} b^{6} d^{2}{\left | b \right |} e^{5} + 12 \, A a b^{7} d^{2}{\left | b \right |} e^{5} - 16 \, B a^{3} b^{5} d{\left | b \right |} e^{6} - 12 \, A a^{2} b^{6} d{\left | b \right |} e^{6} + 3 \, B a^{4} b^{4}{\left | b \right |} e^{7} + 4 \, A a^{3} b^{5}{\left | b \right |} e^{7}\right )}}{b^{4} d e^{7} - a b^{3} e^{8}}\right )} \sqrt{b x + a}}{12 \,{\left (b^{2} d +{\left (b x + a\right )} b e - a b e\right )}^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*(b*x + a)^(5/2)/(e*x + d)^(5/2),x, algorithm="giac")

[Out]

-5/4*(7*B*b^2*d^2*abs(b) - 10*B*a*b*d*abs(b)*e - 4*A*b^2*d*abs(b)*e + 3*B*a^2*ab
s(b)*e^2 + 4*A*a*b*abs(b)*e^2)*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)))/sqrt(b) + 1/12*((3*(b*x + a)*(2*(B*b^5*d*a
bs(b)*e^6 - B*a*b^4*abs(b)*e^7)*(b*x + a)/(b^4*d*e^7 - a*b^3*e^8) - (7*B*b^6*d^2
*abs(b)*e^5 - 10*B*a*b^5*d*abs(b)*e^6 - 4*A*b^6*d*abs(b)*e^6 + 3*B*a^2*b^4*abs(b
)*e^7 + 4*A*a*b^5*abs(b)*e^7)/(b^4*d*e^7 - a*b^3*e^8)) - 20*(7*B*b^7*d^3*abs(b)*
e^4 - 17*B*a*b^6*d^2*abs(b)*e^5 - 4*A*b^7*d^2*abs(b)*e^5 + 13*B*a^2*b^5*d*abs(b)
*e^6 + 8*A*a*b^6*d*abs(b)*e^6 - 3*B*a^3*b^4*abs(b)*e^7 - 4*A*a^2*b^5*abs(b)*e^7)
/(b^4*d*e^7 - a*b^3*e^8))*(b*x + a) - 15*(7*B*b^8*d^4*abs(b)*e^3 - 24*B*a*b^7*d^
3*abs(b)*e^4 - 4*A*b^8*d^3*abs(b)*e^4 + 30*B*a^2*b^6*d^2*abs(b)*e^5 + 12*A*a*b^7
*d^2*abs(b)*e^5 - 16*B*a^3*b^5*d*abs(b)*e^6 - 12*A*a^2*b^6*d*abs(b)*e^6 + 3*B*a^
4*b^4*abs(b)*e^7 + 4*A*a^3*b^5*abs(b)*e^7)/(b^4*d*e^7 - a*b^3*e^8))*sqrt(b*x + a
)/(b^2*d + (b*x + a)*b*e - a*b*e)^(3/2)