3.1725 \(\int \frac{(A+B x) (d+e x)^{7/2}}{a+b x} \, dx\)

Optimal. Leaf size=198 \[ -\frac{2 (A b-a B) (b d-a e)^{7/2} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{b^{11/2}}+\frac{2 \sqrt{d+e x} (A b-a B) (b d-a e)^3}{b^5}+\frac{2 (d+e x)^{3/2} (A b-a B) (b d-a e)^2}{3 b^4}+\frac{2 (d+e x)^{5/2} (A b-a B) (b d-a e)}{5 b^3}+\frac{2 (d+e x)^{7/2} (A b-a B)}{7 b^2}+\frac{2 B (d+e x)^{9/2}}{9 b e} \]

[Out]

(2*(A*b - a*B)*(b*d - a*e)^3*Sqrt[d + e*x])/b^5 + (2*(A*b - a*B)*(b*d - a*e)^2*(
d + e*x)^(3/2))/(3*b^4) + (2*(A*b - a*B)*(b*d - a*e)*(d + e*x)^(5/2))/(5*b^3) +
(2*(A*b - a*B)*(d + e*x)^(7/2))/(7*b^2) + (2*B*(d + e*x)^(9/2))/(9*b*e) - (2*(A*
b - a*B)*(b*d - a*e)^(7/2)*ArcTanh[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d - a*e]])/b^(
11/2)

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Rubi [A]  time = 0.491274, antiderivative size = 198, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 4, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182 \[ -\frac{2 (A b-a B) (b d-a e)^{7/2} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{b^{11/2}}+\frac{2 \sqrt{d+e x} (A b-a B) (b d-a e)^3}{b^5}+\frac{2 (d+e x)^{3/2} (A b-a B) (b d-a e)^2}{3 b^4}+\frac{2 (d+e x)^{5/2} (A b-a B) (b d-a e)}{5 b^3}+\frac{2 (d+e x)^{7/2} (A b-a B)}{7 b^2}+\frac{2 B (d+e x)^{9/2}}{9 b e} \]

Antiderivative was successfully verified.

[In]  Int[((A + B*x)*(d + e*x)^(7/2))/(a + b*x),x]

[Out]

(2*(A*b - a*B)*(b*d - a*e)^3*Sqrt[d + e*x])/b^5 + (2*(A*b - a*B)*(b*d - a*e)^2*(
d + e*x)^(3/2))/(3*b^4) + (2*(A*b - a*B)*(b*d - a*e)*(d + e*x)^(5/2))/(5*b^3) +
(2*(A*b - a*B)*(d + e*x)^(7/2))/(7*b^2) + (2*B*(d + e*x)^(9/2))/(9*b*e) - (2*(A*
b - a*B)*(b*d - a*e)^(7/2)*ArcTanh[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d - a*e]])/b^(
11/2)

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Rubi in Sympy [A]  time = 43.6617, size = 175, normalized size = 0.88 \[ \frac{2 B \left (d + e x\right )^{\frac{9}{2}}}{9 b e} + \frac{2 \left (d + e x\right )^{\frac{7}{2}} \left (A b - B a\right )}{7 b^{2}} - \frac{2 \left (d + e x\right )^{\frac{5}{2}} \left (A b - B a\right ) \left (a e - b d\right )}{5 b^{3}} + \frac{2 \left (d + e x\right )^{\frac{3}{2}} \left (A b - B a\right ) \left (a e - b d\right )^{2}}{3 b^{4}} - \frac{2 \sqrt{d + e x} \left (A b - B a\right ) \left (a e - b d\right )^{3}}{b^{5}} + \frac{2 \left (A b - B a\right ) \left (a e - b d\right )^{\frac{7}{2}} \operatorname{atan}{\left (\frac{\sqrt{b} \sqrt{d + e x}}{\sqrt{a e - b d}} \right )}}{b^{\frac{11}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((B*x+A)*(e*x+d)**(7/2)/(b*x+a),x)

[Out]

2*B*(d + e*x)**(9/2)/(9*b*e) + 2*(d + e*x)**(7/2)*(A*b - B*a)/(7*b**2) - 2*(d +
e*x)**(5/2)*(A*b - B*a)*(a*e - b*d)/(5*b**3) + 2*(d + e*x)**(3/2)*(A*b - B*a)*(a
*e - b*d)**2/(3*b**4) - 2*sqrt(d + e*x)*(A*b - B*a)*(a*e - b*d)**3/b**5 + 2*(A*b
- B*a)*(a*e - b*d)**(7/2)*atan(sqrt(b)*sqrt(d + e*x)/sqrt(a*e - b*d))/b**(11/2)

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Mathematica [A]  time = 0.462303, size = 263, normalized size = 1.33 \[ \frac{2 \sqrt{d+e x} \left (315 a^4 B e^4-105 a^3 b e^3 (3 A e+10 B d+B e x)+21 a^2 b^2 e^2 \left (5 A e (10 d+e x)+B \left (58 d^2+16 d e x+3 e^2 x^2\right )\right )-3 a b^3 e \left (7 A e \left (58 d^2+16 d e x+3 e^2 x^2\right )+B \left (176 d^3+122 d^2 e x+66 d e^2 x^2+15 e^3 x^3\right )\right )+b^4 \left (3 A e \left (176 d^3+122 d^2 e x+66 d e^2 x^2+15 e^3 x^3\right )+35 B (d+e x)^4\right )\right )}{315 b^5 e}-\frac{2 (A b-a B) (b d-a e)^{7/2} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{b^{11/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[((A + B*x)*(d + e*x)^(7/2))/(a + b*x),x]

[Out]

(2*Sqrt[d + e*x]*(315*a^4*B*e^4 - 105*a^3*b*e^3*(10*B*d + 3*A*e + B*e*x) + 21*a^
2*b^2*e^2*(5*A*e*(10*d + e*x) + B*(58*d^2 + 16*d*e*x + 3*e^2*x^2)) - 3*a*b^3*e*(
7*A*e*(58*d^2 + 16*d*e*x + 3*e^2*x^2) + B*(176*d^3 + 122*d^2*e*x + 66*d*e^2*x^2
+ 15*e^3*x^3)) + b^4*(35*B*(d + e*x)^4 + 3*A*e*(176*d^3 + 122*d^2*e*x + 66*d*e^2
*x^2 + 15*e^3*x^3))))/(315*b^5*e) - (2*(A*b - a*B)*(b*d - a*e)^(7/2)*ArcTanh[(Sq
rt[b]*Sqrt[d + e*x])/Sqrt[b*d - a*e]])/b^(11/2)

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Maple [B]  time = 0.025, size = 820, normalized size = 4.1 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x+A)*(e*x+d)^(7/2)/(b*x+a),x)

[Out]

2/((a*e-b*d)*b)^(1/2)*arctan((e*x+d)^(1/2)*b/((a*e-b*d)*b)^(1/2))*A*d^4+2/b*A*d^
3*(e*x+d)^(1/2)+2/3/b*A*(e*x+d)^(3/2)*d^2-2/7/b^2*B*(e*x+d)^(7/2)*a+2/5/b*A*(e*x
+d)^(5/2)*d+2/3*e^2/b^3*A*(e*x+d)^(3/2)*a^2-2/5*e/b^2*A*(e*x+d)^(5/2)*a+2/5*e/b^
3*B*(e*x+d)^(5/2)*a^2-2*e^3/b^4*A*a^3*(e*x+d)^(1/2)-2/3*e^2/b^4*B*(e*x+d)^(3/2)*
a^3-2/3/b^2*B*(e*x+d)^(3/2)*a*d^2+2*e^3/b^5*B*a^4*(e*x+d)^(1/2)-2/b^2*B*a*d^3*(e
*x+d)^(1/2)-2/5/b^2*B*(e*x+d)^(5/2)*a*d-8*e/b/((a*e-b*d)*b)^(1/2)*arctan((e*x+d)
^(1/2)*b/((a*e-b*d)*b)^(1/2))*A*a*d^3+8*e^3/b^4/((a*e-b*d)*b)^(1/2)*arctan((e*x+
d)^(1/2)*b/((a*e-b*d)*b)^(1/2))*B*a^4*d-12*e^2/b^3/((a*e-b*d)*b)^(1/2)*arctan((e
*x+d)^(1/2)*b/((a*e-b*d)*b)^(1/2))*B*a^3*d^2+8*e/b^2/((a*e-b*d)*b)^(1/2)*arctan(
(e*x+d)^(1/2)*b/((a*e-b*d)*b)^(1/2))*B*a^2*d^3-8*e^3/b^3/((a*e-b*d)*b)^(1/2)*arc
tan((e*x+d)^(1/2)*b/((a*e-b*d)*b)^(1/2))*A*a^3*d+12*e^2/b^2/((a*e-b*d)*b)^(1/2)*
arctan((e*x+d)^(1/2)*b/((a*e-b*d)*b)^(1/2))*A*a^2*d^2+2/9*B*(e*x+d)^(9/2)/b/e+4/
3*e/b^3*B*(e*x+d)^(3/2)*a^2*d+2*e^4/b^4/((a*e-b*d)*b)^(1/2)*arctan((e*x+d)^(1/2)
*b/((a*e-b*d)*b)^(1/2))*A*a^4-2*e^4/b^5/((a*e-b*d)*b)^(1/2)*arctan((e*x+d)^(1/2)
*b/((a*e-b*d)*b)^(1/2))*B*a^5+6*e/b^3*B*a^2*d^2*(e*x+d)^(1/2)-4/3*e/b^2*A*(e*x+d
)^(3/2)*a*d-2/b/((a*e-b*d)*b)^(1/2)*arctan((e*x+d)^(1/2)*b/((a*e-b*d)*b)^(1/2))*
B*a*d^4+6*e^2/b^3*A*a^2*d*(e*x+d)^(1/2)-6*e/b^2*A*a*d^2*(e*x+d)^(1/2)-6*e^2/b^4*
B*a^3*d*(e*x+d)^(1/2)+2/7/b*A*(e*x+d)^(7/2)

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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)*(e*x + d)^(7/2)/(b*x + a),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.251215, size = 1, normalized size = 0.01 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*(e*x + d)^(7/2)/(b*x + a),x, algorithm="fricas")

[Out]

[-1/315*(315*((B*a*b^3 - A*b^4)*d^3*e - 3*(B*a^2*b^2 - A*a*b^3)*d^2*e^2 + 3*(B*a
^3*b - A*a^2*b^2)*d*e^3 - (B*a^4 - A*a^3*b)*e^4)*sqrt((b*d - a*e)/b)*log((b*e*x
+ 2*b*d - a*e - 2*sqrt(e*x + d)*b*sqrt((b*d - a*e)/b))/(b*x + a)) - 2*(35*B*b^4*
e^4*x^4 + 35*B*b^4*d^4 - 528*(B*a*b^3 - A*b^4)*d^3*e + 1218*(B*a^2*b^2 - A*a*b^3
)*d^2*e^2 - 1050*(B*a^3*b - A*a^2*b^2)*d*e^3 + 315*(B*a^4 - A*a^3*b)*e^4 + 5*(28
*B*b^4*d*e^3 - 9*(B*a*b^3 - A*b^4)*e^4)*x^3 + 3*(70*B*b^4*d^2*e^2 - 66*(B*a*b^3
- A*b^4)*d*e^3 + 21*(B*a^2*b^2 - A*a*b^3)*e^4)*x^2 + (140*B*b^4*d^3*e - 366*(B*a
*b^3 - A*b^4)*d^2*e^2 + 336*(B*a^2*b^2 - A*a*b^3)*d*e^3 - 105*(B*a^3*b - A*a^2*b
^2)*e^4)*x)*sqrt(e*x + d))/(b^5*e), 2/315*(315*((B*a*b^3 - A*b^4)*d^3*e - 3*(B*a
^2*b^2 - A*a*b^3)*d^2*e^2 + 3*(B*a^3*b - A*a^2*b^2)*d*e^3 - (B*a^4 - A*a^3*b)*e^
4)*sqrt(-(b*d - a*e)/b)*arctan(sqrt(e*x + d)/sqrt(-(b*d - a*e)/b)) + (35*B*b^4*e
^4*x^4 + 35*B*b^4*d^4 - 528*(B*a*b^3 - A*b^4)*d^3*e + 1218*(B*a^2*b^2 - A*a*b^3)
*d^2*e^2 - 1050*(B*a^3*b - A*a^2*b^2)*d*e^3 + 315*(B*a^4 - A*a^3*b)*e^4 + 5*(28*
B*b^4*d*e^3 - 9*(B*a*b^3 - A*b^4)*e^4)*x^3 + 3*(70*B*b^4*d^2*e^2 - 66*(B*a*b^3 -
A*b^4)*d*e^3 + 21*(B*a^2*b^2 - A*a*b^3)*e^4)*x^2 + (140*B*b^4*d^3*e - 366*(B*a*
b^3 - A*b^4)*d^2*e^2 + 336*(B*a^2*b^2 - A*a*b^3)*d*e^3 - 105*(B*a^3*b - A*a^2*b^
2)*e^4)*x)*sqrt(e*x + d))/(b^5*e)]

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Sympy [A]  time = 119.826, size = 456, normalized size = 2.3 \[ \frac{2 B \left (d + e x\right )^{\frac{9}{2}}}{9 b e} + \frac{\left (d + e x\right )^{\frac{7}{2}} \left (2 A b - 2 B a\right )}{7 b^{2}} + \frac{\left (d + e x\right )^{\frac{5}{2}} \left (- 2 A a b e + 2 A b^{2} d + 2 B a^{2} e - 2 B a b d\right )}{5 b^{3}} + \frac{\left (d + e x\right )^{\frac{3}{2}} \left (2 A a^{2} b e^{2} - 4 A a b^{2} d e + 2 A b^{3} d^{2} - 2 B a^{3} e^{2} + 4 B a^{2} b d e - 2 B a b^{2} d^{2}\right )}{3 b^{4}} + \frac{\sqrt{d + e x} \left (- 2 A a^{3} b e^{3} + 6 A a^{2} b^{2} d e^{2} - 6 A a b^{3} d^{2} e + 2 A b^{4} d^{3} + 2 B a^{4} e^{3} - 6 B a^{3} b d e^{2} + 6 B a^{2} b^{2} d^{2} e - 2 B a b^{3} d^{3}\right )}{b^{5}} - \frac{2 \left (- A b + B a\right ) \left (a e - b d\right )^{4} \left (\begin{cases} \frac{\operatorname{atan}{\left (\frac{\sqrt{d + e x}}{\sqrt{\frac{a e - b d}{b}}} \right )}}{b \sqrt{\frac{a e - b d}{b}}} & \text{for}\: \frac{a e - b d}{b} > 0 \\- \frac{\operatorname{acoth}{\left (\frac{\sqrt{d + e x}}{\sqrt{\frac{- a e + b d}{b}}} \right )}}{b \sqrt{\frac{- a e + b d}{b}}} & \text{for}\: d + e x > \frac{- a e + b d}{b} \wedge \frac{a e - b d}{b} < 0 \\- \frac{\operatorname{atanh}{\left (\frac{\sqrt{d + e x}}{\sqrt{\frac{- a e + b d}{b}}} \right )}}{b \sqrt{\frac{- a e + b d}{b}}} & \text{for}\: \frac{a e - b d}{b} < 0 \wedge d + e x < \frac{- a e + b d}{b} \end{cases}\right )}{b^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x+A)*(e*x+d)**(7/2)/(b*x+a),x)

[Out]

2*B*(d + e*x)**(9/2)/(9*b*e) + (d + e*x)**(7/2)*(2*A*b - 2*B*a)/(7*b**2) + (d +
e*x)**(5/2)*(-2*A*a*b*e + 2*A*b**2*d + 2*B*a**2*e - 2*B*a*b*d)/(5*b**3) + (d + e
*x)**(3/2)*(2*A*a**2*b*e**2 - 4*A*a*b**2*d*e + 2*A*b**3*d**2 - 2*B*a**3*e**2 + 4
*B*a**2*b*d*e - 2*B*a*b**2*d**2)/(3*b**4) + sqrt(d + e*x)*(-2*A*a**3*b*e**3 + 6*
A*a**2*b**2*d*e**2 - 6*A*a*b**3*d**2*e + 2*A*b**4*d**3 + 2*B*a**4*e**3 - 6*B*a**
3*b*d*e**2 + 6*B*a**2*b**2*d**2*e - 2*B*a*b**3*d**3)/b**5 - 2*(-A*b + B*a)*(a*e
- b*d)**4*Piecewise((atan(sqrt(d + e*x)/sqrt((a*e - b*d)/b))/(b*sqrt((a*e - b*d)
/b)), (a*e - b*d)/b > 0), (-acoth(sqrt(d + e*x)/sqrt((-a*e + b*d)/b))/(b*sqrt((-
a*e + b*d)/b)), ((a*e - b*d)/b < 0) & (d + e*x > (-a*e + b*d)/b)), (-atanh(sqrt(
d + e*x)/sqrt((-a*e + b*d)/b))/(b*sqrt((-a*e + b*d)/b)), ((a*e - b*d)/b < 0) & (
d + e*x < (-a*e + b*d)/b)))/b**5

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GIAC/XCAS [A]  time = 0.226874, size = 753, normalized size = 3.8 \[ -\frac{2 \,{\left (B a b^{4} d^{4} - A b^{5} d^{4} - 4 \, B a^{2} b^{3} d^{3} e + 4 \, A a b^{4} d^{3} e + 6 \, B a^{3} b^{2} d^{2} e^{2} - 6 \, A a^{2} b^{3} d^{2} e^{2} - 4 \, B a^{4} b d e^{3} + 4 \, A a^{3} b^{2} d e^{3} + B a^{5} e^{4} - A a^{4} b e^{4}\right )} \arctan \left (\frac{\sqrt{x e + d} b}{\sqrt{-b^{2} d + a b e}}\right )}{\sqrt{-b^{2} d + a b e} b^{5}} + \frac{2 \,{\left (35 \,{\left (x e + d\right )}^{\frac{9}{2}} B b^{8} e^{8} - 45 \,{\left (x e + d\right )}^{\frac{7}{2}} B a b^{7} e^{9} + 45 \,{\left (x e + d\right )}^{\frac{7}{2}} A b^{8} e^{9} - 63 \,{\left (x e + d\right )}^{\frac{5}{2}} B a b^{7} d e^{9} + 63 \,{\left (x e + d\right )}^{\frac{5}{2}} A b^{8} d e^{9} - 105 \,{\left (x e + d\right )}^{\frac{3}{2}} B a b^{7} d^{2} e^{9} + 105 \,{\left (x e + d\right )}^{\frac{3}{2}} A b^{8} d^{2} e^{9} - 315 \, \sqrt{x e + d} B a b^{7} d^{3} e^{9} + 315 \, \sqrt{x e + d} A b^{8} d^{3} e^{9} + 63 \,{\left (x e + d\right )}^{\frac{5}{2}} B a^{2} b^{6} e^{10} - 63 \,{\left (x e + d\right )}^{\frac{5}{2}} A a b^{7} e^{10} + 210 \,{\left (x e + d\right )}^{\frac{3}{2}} B a^{2} b^{6} d e^{10} - 210 \,{\left (x e + d\right )}^{\frac{3}{2}} A a b^{7} d e^{10} + 945 \, \sqrt{x e + d} B a^{2} b^{6} d^{2} e^{10} - 945 \, \sqrt{x e + d} A a b^{7} d^{2} e^{10} - 105 \,{\left (x e + d\right )}^{\frac{3}{2}} B a^{3} b^{5} e^{11} + 105 \,{\left (x e + d\right )}^{\frac{3}{2}} A a^{2} b^{6} e^{11} - 945 \, \sqrt{x e + d} B a^{3} b^{5} d e^{11} + 945 \, \sqrt{x e + d} A a^{2} b^{6} d e^{11} + 315 \, \sqrt{x e + d} B a^{4} b^{4} e^{12} - 315 \, \sqrt{x e + d} A a^{3} b^{5} e^{12}\right )} e^{\left (-9\right )}}{315 \, b^{9}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*(e*x + d)^(7/2)/(b*x + a),x, algorithm="giac")

[Out]

-2*(B*a*b^4*d^4 - A*b^5*d^4 - 4*B*a^2*b^3*d^3*e + 4*A*a*b^4*d^3*e + 6*B*a^3*b^2*
d^2*e^2 - 6*A*a^2*b^3*d^2*e^2 - 4*B*a^4*b*d*e^3 + 4*A*a^3*b^2*d*e^3 + B*a^5*e^4
- A*a^4*b*e^4)*arctan(sqrt(x*e + d)*b/sqrt(-b^2*d + a*b*e))/(sqrt(-b^2*d + a*b*e
)*b^5) + 2/315*(35*(x*e + d)^(9/2)*B*b^8*e^8 - 45*(x*e + d)^(7/2)*B*a*b^7*e^9 +
45*(x*e + d)^(7/2)*A*b^8*e^9 - 63*(x*e + d)^(5/2)*B*a*b^7*d*e^9 + 63*(x*e + d)^(
5/2)*A*b^8*d*e^9 - 105*(x*e + d)^(3/2)*B*a*b^7*d^2*e^9 + 105*(x*e + d)^(3/2)*A*b
^8*d^2*e^9 - 315*sqrt(x*e + d)*B*a*b^7*d^3*e^9 + 315*sqrt(x*e + d)*A*b^8*d^3*e^9
+ 63*(x*e + d)^(5/2)*B*a^2*b^6*e^10 - 63*(x*e + d)^(5/2)*A*a*b^7*e^10 + 210*(x*
e + d)^(3/2)*B*a^2*b^6*d*e^10 - 210*(x*e + d)^(3/2)*A*a*b^7*d*e^10 + 945*sqrt(x*
e + d)*B*a^2*b^6*d^2*e^10 - 945*sqrt(x*e + d)*A*a*b^7*d^2*e^10 - 105*(x*e + d)^(
3/2)*B*a^3*b^5*e^11 + 105*(x*e + d)^(3/2)*A*a^2*b^6*e^11 - 945*sqrt(x*e + d)*B*a
^3*b^5*d*e^11 + 945*sqrt(x*e + d)*A*a^2*b^6*d*e^11 + 315*sqrt(x*e + d)*B*a^4*b^4
*e^12 - 315*sqrt(x*e + d)*A*a^3*b^5*e^12)*e^(-9)/b^9