3.2200 \(\int \frac{(a+b x)^{3/2} (A+B x)}{(d+e x)^{7/2}} \, dx\)

Optimal. Leaf size=138 \[ -\frac{2 (a+b x)^{5/2} (B d-A e)}{5 e (d+e x)^{5/2} (b d-a e)}+\frac{2 b^{3/2} B \tanh ^{-1}\left (\frac{\sqrt{e} \sqrt{a+b x}}{\sqrt{b} \sqrt{d+e x}}\right )}{e^{7/2}}-\frac{2 b B \sqrt{a+b x}}{e^3 \sqrt{d+e x}}-\frac{2 B (a+b x)^{3/2}}{3 e^2 (d+e x)^{3/2}} \]

[Out]

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

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Rubi [A]  time = 0.218117, antiderivative size = 138, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167 \[ -\frac{2 (a+b x)^{5/2} (B d-A e)}{5 e (d+e x)^{5/2} (b d-a e)}+\frac{2 b^{3/2} B \tanh ^{-1}\left (\frac{\sqrt{e} \sqrt{a+b x}}{\sqrt{b} \sqrt{d+e x}}\right )}{e^{7/2}}-\frac{2 b B \sqrt{a+b x}}{e^3 \sqrt{d+e x}}-\frac{2 B (a+b x)^{3/2}}{3 e^2 (d+e x)^{3/2}} \]

Antiderivative was successfully verified.

[In]  Int[((a + b*x)^(3/2)*(A + B*x))/(d + e*x)^(7/2),x]

[Out]

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

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Rubi in Sympy [A]  time = 21.7614, size = 128, normalized size = 0.93 \[ \frac{2 B b^{\frac{3}{2}} \operatorname{atanh}{\left (\frac{\sqrt{b} \sqrt{d + e x}}{\sqrt{e} \sqrt{a + b x}} \right )}}{e^{\frac{7}{2}}} - \frac{2 B b \sqrt{a + b x}}{e^{3} \sqrt{d + e x}} - \frac{2 B \left (a + b x\right )^{\frac{3}{2}}}{3 e^{2} \left (d + e x\right )^{\frac{3}{2}}} - \frac{2 \left (a + b x\right )^{\frac{5}{2}} \left (A e - B d\right )}{5 e \left (d + e x\right )^{\frac{5}{2}} \left (a e - b d\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)**(3/2)*(B*x+A)/(e*x+d)**(7/2),x)

[Out]

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

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Mathematica [A]  time = 0.484164, size = 158, normalized size = 1.14 \[ \frac{2 \sqrt{a+b x} \left (-\frac{b (d+e x)^2 (20 a B e+3 A b e-23 b B d)}{a e-b d}+(d+e x) (-5 a B e-6 A b e+11 b B d)-3 (b d-a e) (B d-A e)\right )}{15 e^3 (d+e x)^{5/2}}+\frac{b^{3/2} B \log \left (2 \sqrt{b} \sqrt{e} \sqrt{a+b x} \sqrt{d+e x}+a e+b d+2 b e x\right )}{e^{7/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[((a + b*x)^(3/2)*(A + B*x))/(d + e*x)^(7/2),x]

[Out]

(2*Sqrt[a + b*x]*(-3*(b*d - a*e)*(B*d - A*e) + (11*b*B*d - 6*A*b*e - 5*a*B*e)*(d
+ e*x) - (b*(-23*b*B*d + 3*A*b*e + 20*a*B*e)*(d + e*x)^2)/(-(b*d) + a*e)))/(15*
e^3*(d + e*x)^(5/2)) + (b^(3/2)*B*Log[b*d + a*e + 2*b*e*x + 2*Sqrt[b]*Sqrt[e]*Sq
rt[a + b*x]*Sqrt[d + e*x]])/e^(7/2)

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Maple [B]  time = 0.036, size = 780, normalized size = 5.7 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^(3/2)*(B*x+A)/(e*x+d)^(7/2),x)

[Out]

-1/15*(b*x+a)^(1/2)*(-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^3*a*b^2*e^4+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^3*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*b^2*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*b^3*d^
2*e^2+6*A*x^2*b^2*e^3*(b*e)^(1/2)*((b*x+a)*(e*x+d))^(1/2)-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))*x*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))*x*b^
3*d^3*e+40*B*x^2*a*b*e^3*(b*e)^(1/2)*((b*x+a)*(e*x+d))^(1/2)-46*B*x^2*b^2*d*e^2*
(b*e)^(1/2)*((b*x+a)*(e*x+d))^(1/2)+12*A*x*a*b*e^3*(b*e)^(1/2)*((b*x+a)*(e*x+d))
^(1/2)-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*b^2*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))*b^3*d^4+10*B*x*a^2*e^3*(b*e)^(1/2)*((b*x+a)*(e*x+d))^(1/2)+
48*B*x*a*b*d*e^2*(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)+6*A*a^2*e^3*(b*e)^(1/2)*((b*x+a)*(e*x+d))^(1/2)+4*B*a^
2*d*e^2*(b*e)^(1/2)*((b*x+a)*(e*x+d))^(1/2)+20*B*a*b*d^2*e*(b*e)^(1/2)*((b*x+a)*
(e*x+d))^(1/2)-30*B*b^2*d^3*(b*e)^(1/2)*((b*x+a)*(e*x+d))^(1/2))/((b*x+a)*(e*x+d
))^(1/2)/(a*e-b*d)/(b*e)^(1/2)/(e*x+d)^(5/2)/e^3

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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)^(3/2)/(e*x + d)^(7/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.870137, size = 1, normalized size = 0.01 \[ \left [\frac{15 \,{\left (B b^{2} d^{4} - B a b d^{3} e +{\left (B b^{2} d e^{3} - B a b e^{4}\right )} x^{3} + 3 \,{\left (B b^{2} d^{2} e^{2} - B a b d e^{3}\right )} x^{2} + 3 \,{\left (B b^{2} d^{3} e - B a b d^{2} e^{2}\right )} x\right )} \sqrt{\frac{b}{e}} \log \left (8 \, b^{2} e^{2} x^{2} + b^{2} d^{2} + 6 \, a b d e + a^{2} e^{2} + 4 \,{\left (2 \, b e^{2} x + b d e + a e^{2}\right )} \sqrt{b x + a} \sqrt{e x + d} \sqrt{\frac{b}{e}} + 8 \,{\left (b^{2} d e + a b e^{2}\right )} x\right ) - 4 \,{\left (15 \, B b^{2} d^{3} - 10 \, B a b d^{2} e - 2 \, B a^{2} d e^{2} - 3 \, A a^{2} e^{3} +{\left (23 \, B b^{2} d e^{2} -{\left (20 \, B a b + 3 \, A b^{2}\right )} e^{3}\right )} x^{2} +{\left (35 \, B b^{2} d^{2} e - 24 \, B a b d e^{2} -{\left (5 \, B a^{2} + 6 \, A a b\right )} e^{3}\right )} x\right )} \sqrt{b x + a} \sqrt{e x + d}}{30 \,{\left (b d^{4} e^{3} - a d^{3} e^{4} +{\left (b d e^{6} - a e^{7}\right )} x^{3} + 3 \,{\left (b d^{2} e^{5} - a d e^{6}\right )} x^{2} + 3 \,{\left (b d^{3} e^{4} - a d^{2} e^{5}\right )} x\right )}}, \frac{15 \,{\left (B b^{2} d^{4} - B a b d^{3} e +{\left (B b^{2} d e^{3} - B a b e^{4}\right )} x^{3} + 3 \,{\left (B b^{2} d^{2} e^{2} - B a b d e^{3}\right )} x^{2} + 3 \,{\left (B b^{2} d^{3} e - B a b d^{2} e^{2}\right )} x\right )} \sqrt{-\frac{b}{e}} \arctan \left (\frac{2 \, b e x + b d + a e}{2 \, \sqrt{b x + a} \sqrt{e x + d} e \sqrt{-\frac{b}{e}}}\right ) - 2 \,{\left (15 \, B b^{2} d^{3} - 10 \, B a b d^{2} e - 2 \, B a^{2} d e^{2} - 3 \, A a^{2} e^{3} +{\left (23 \, B b^{2} d e^{2} -{\left (20 \, B a b + 3 \, A b^{2}\right )} e^{3}\right )} x^{2} +{\left (35 \, B b^{2} d^{2} e - 24 \, B a b d e^{2} -{\left (5 \, B a^{2} + 6 \, A a b\right )} e^{3}\right )} x\right )} \sqrt{b x + a} \sqrt{e x + d}}{15 \,{\left (b d^{4} e^{3} - a d^{3} e^{4} +{\left (b d e^{6} - a e^{7}\right )} x^{3} + 3 \,{\left (b d^{2} e^{5} - a d e^{6}\right )} x^{2} + 3 \,{\left (b d^{3} e^{4} - a d^{2} e^{5}\right )} x\right )}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*(b*x + a)^(3/2)/(e*x + d)^(7/2),x, algorithm="fricas")

[Out]

[1/30*(15*(B*b^2*d^4 - B*a*b*d^3*e + (B*b^2*d*e^3 - B*a*b*e^4)*x^3 + 3*(B*b^2*d^
2*e^2 - B*a*b*d*e^3)*x^2 + 3*(B*b^2*d^3*e - B*a*b*d^2*e^2)*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*(15*B*b^2*d^3 - 1
0*B*a*b*d^2*e - 2*B*a^2*d*e^2 - 3*A*a^2*e^3 + (23*B*b^2*d*e^2 - (20*B*a*b + 3*A*
b^2)*e^3)*x^2 + (35*B*b^2*d^2*e - 24*B*a*b*d*e^2 - (5*B*a^2 + 6*A*a*b)*e^3)*x)*s
qrt(b*x + a)*sqrt(e*x + d))/(b*d^4*e^3 - a*d^3*e^4 + (b*d*e^6 - a*e^7)*x^3 + 3*(
b*d^2*e^5 - a*d*e^6)*x^2 + 3*(b*d^3*e^4 - a*d^2*e^5)*x), 1/15*(15*(B*b^2*d^4 - B
*a*b*d^3*e + (B*b^2*d*e^3 - B*a*b*e^4)*x^3 + 3*(B*b^2*d^2*e^2 - B*a*b*d*e^3)*x^2
+ 3*(B*b^2*d^3*e - B*a*b*d^2*e^2)*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*(15*B*b^2*d^3 - 10*B*a*b*d^2*e
- 2*B*a^2*d*e^2 - 3*A*a^2*e^3 + (23*B*b^2*d*e^2 - (20*B*a*b + 3*A*b^2)*e^3)*x^2
+ (35*B*b^2*d^2*e - 24*B*a*b*d*e^2 - (5*B*a^2 + 6*A*a*b)*e^3)*x)*sqrt(b*x + a)*
sqrt(e*x + d))/(b*d^4*e^3 - a*d^3*e^4 + (b*d*e^6 - a*e^7)*x^3 + 3*(b*d^2*e^5 - a
*d*e^6)*x^2 + 3*(b*d^3*e^4 - a*d^2*e^5)*x)]

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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)**(3/2)*(B*x+A)/(e*x+d)**(7/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.285742, size = 587, normalized size = 4.25 \[ \frac{B \sqrt{b}{\left | b \right |} e^{\frac{1}{2}}{\rm ln}\left ({\left | -\sqrt{b x + a} \sqrt{b} e^{\frac{1}{2}} + \sqrt{b^{2} d +{\left (b x + a\right )} b e - a b e} \right |}\right )}{64 \,{\left (b^{8} d e^{5} - a b^{7} e^{6}\right )}} + \frac{{\left ({\left (b x + a\right )}{\left (\frac{{\left (23 \, B b^{7} d^{2}{\left | b \right |} e^{4} - 43 \, B a b^{6} d{\left | b \right |} e^{5} - 3 \, A b^{7} d{\left | b \right |} e^{5} + 20 \, B a^{2} b^{5}{\left | b \right |} e^{6} + 3 \, A a b^{6}{\left | b \right |} e^{6}\right )}{\left (b x + a\right )}}{b^{12} d^{3} e^{6} - 3 \, a b^{11} d^{2} e^{7} + 3 \, a^{2} b^{10} d e^{8} - a^{3} b^{9} e^{9}} + \frac{35 \,{\left (B b^{8} d^{3}{\left | b \right |} e^{3} - 3 \, B a b^{7} d^{2}{\left | b \right |} e^{4} + 3 \, B a^{2} b^{6} d{\left | b \right |} e^{5} - B a^{3} b^{5}{\left | b \right |} e^{6}\right )}}{b^{12} d^{3} e^{6} - 3 \, a b^{11} d^{2} e^{7} + 3 \, a^{2} b^{10} d e^{8} - a^{3} b^{9} e^{9}}\right )} + \frac{15 \,{\left (B b^{9} d^{4}{\left | b \right |} e^{2} - 4 \, B a b^{8} d^{3}{\left | b \right |} e^{3} + 6 \, B a^{2} b^{7} d^{2}{\left | b \right |} e^{4} - 4 \, B a^{3} b^{6} d{\left | b \right |} e^{5} + B a^{4} b^{5}{\left | b \right |} e^{6}\right )}}{b^{12} d^{3} e^{6} - 3 \, a b^{11} d^{2} e^{7} + 3 \, a^{2} b^{10} d e^{8} - a^{3} b^{9} e^{9}}\right )} \sqrt{b x + a}}{960 \,{\left (b^{2} d +{\left (b x + a\right )} b e - a b e\right )}^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*(b*x + a)^(3/2)/(e*x + d)^(7/2),x, algorithm="giac")

[Out]

1/64*B*sqrt(b)*abs(b)*e^(1/2)*ln(abs(-sqrt(b*x + a)*sqrt(b)*e^(1/2) + sqrt(b^2*d
+ (b*x + a)*b*e - a*b*e)))/(b^8*d*e^5 - a*b^7*e^6) + 1/960*((b*x + a)*((23*B*b^
7*d^2*abs(b)*e^4 - 43*B*a*b^6*d*abs(b)*e^5 - 3*A*b^7*d*abs(b)*e^5 + 20*B*a^2*b^5
*abs(b)*e^6 + 3*A*a*b^6*abs(b)*e^6)*(b*x + a)/(b^12*d^3*e^6 - 3*a*b^11*d^2*e^7 +
3*a^2*b^10*d*e^8 - a^3*b^9*e^9) + 35*(B*b^8*d^3*abs(b)*e^3 - 3*B*a*b^7*d^2*abs(
b)*e^4 + 3*B*a^2*b^6*d*abs(b)*e^5 - B*a^3*b^5*abs(b)*e^6)/(b^12*d^3*e^6 - 3*a*b^
11*d^2*e^7 + 3*a^2*b^10*d*e^8 - a^3*b^9*e^9)) + 15*(B*b^9*d^4*abs(b)*e^2 - 4*B*a
*b^8*d^3*abs(b)*e^3 + 6*B*a^2*b^7*d^2*abs(b)*e^4 - 4*B*a^3*b^6*d*abs(b)*e^5 + B*
a^4*b^5*abs(b)*e^6)/(b^12*d^3*e^6 - 3*a*b^11*d^2*e^7 + 3*a^2*b^10*d*e^8 - a^3*b^
9*e^9))*sqrt(b*x + a)/(b^2*d + (b*x + a)*b*e - a*b*e)^(5/2)