3.2202 \(\int \frac{(a+b x)^{3/2} (A+B x)}{(d+e x)^{11/2}} \, dx\)

Optimal. Leaf size=147 \[ \frac{4 b (a+b x)^{5/2} (-9 a B e+4 A b e+5 b B d)}{315 e (d+e x)^{5/2} (b d-a e)^3}+\frac{2 (a+b x)^{5/2} (-9 a B e+4 A b e+5 b B d)}{63 e (d+e x)^{7/2} (b d-a e)^2}-\frac{2 (a+b x)^{5/2} (B d-A e)}{9 e (d+e x)^{9/2} (b d-a e)} \]

[Out]

(-2*(B*d - A*e)*(a + b*x)^(5/2))/(9*e*(b*d - a*e)*(d + e*x)^(9/2)) + (2*(5*b*B*d
+ 4*A*b*e - 9*a*B*e)*(a + b*x)^(5/2))/(63*e*(b*d - a*e)^2*(d + e*x)^(7/2)) + (4
*b*(5*b*B*d + 4*A*b*e - 9*a*B*e)*(a + b*x)^(5/2))/(315*e*(b*d - a*e)^3*(d + e*x)
^(5/2))

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Rubi [A]  time = 0.269045, antiderivative size = 147, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125 \[ \frac{4 b (a+b x)^{5/2} (-9 a B e+4 A b e+5 b B d)}{315 e (d+e x)^{5/2} (b d-a e)^3}+\frac{2 (a+b x)^{5/2} (-9 a B e+4 A b e+5 b B d)}{63 e (d+e x)^{7/2} (b d-a e)^2}-\frac{2 (a+b x)^{5/2} (B d-A e)}{9 e (d+e x)^{9/2} (b d-a e)} \]

Antiderivative was successfully verified.

[In]  Int[((a + b*x)^(3/2)*(A + B*x))/(d + e*x)^(11/2),x]

[Out]

(-2*(B*d - A*e)*(a + b*x)^(5/2))/(9*e*(b*d - a*e)*(d + e*x)^(9/2)) + (2*(5*b*B*d
+ 4*A*b*e - 9*a*B*e)*(a + b*x)^(5/2))/(63*e*(b*d - a*e)^2*(d + e*x)^(7/2)) + (4
*b*(5*b*B*d + 4*A*b*e - 9*a*B*e)*(a + b*x)^(5/2))/(315*e*(b*d - a*e)^3*(d + e*x)
^(5/2))

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Rubi in Sympy [A]  time = 24.6885, size = 138, normalized size = 0.94 \[ - \frac{4 b \left (a + b x\right )^{\frac{5}{2}} \left (4 A b e - 9 B a e + 5 B b d\right )}{315 e \left (d + e x\right )^{\frac{5}{2}} \left (a e - b d\right )^{3}} + \frac{2 \left (a + b x\right )^{\frac{5}{2}} \left (4 A b e - 9 B a e + 5 B b d\right )}{63 e \left (d + e x\right )^{\frac{7}{2}} \left (a e - b d\right )^{2}} - \frac{2 \left (a + b x\right )^{\frac{5}{2}} \left (A e - B d\right )}{9 e \left (d + e x\right )^{\frac{9}{2}} \left (a e - b d\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)**(3/2)*(B*x+A)/(e*x+d)**(11/2),x)

[Out]

-4*b*(a + b*x)**(5/2)*(4*A*b*e - 9*B*a*e + 5*B*b*d)/(315*e*(d + e*x)**(5/2)*(a*e
- b*d)**3) + 2*(a + b*x)**(5/2)*(4*A*b*e - 9*B*a*e + 5*B*b*d)/(63*e*(d + e*x)**
(7/2)*(a*e - b*d)**2) - 2*(a + b*x)**(5/2)*(A*e - B*d)/(9*e*(d + e*x)**(9/2)*(a*
e - b*d))

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Mathematica [A]  time = 0.313032, size = 135, normalized size = 0.92 \[ \frac{2 (a+b x)^{5/2} \left (A \left (35 a^2 e^2-10 a b e (9 d+2 e x)+b^2 \left (63 d^2+36 d e x+8 e^2 x^2\right )\right )+B \left (5 a^2 e (2 d+9 e x)-2 a b \left (9 d^2+53 d e x+9 e^2 x^2\right )+5 b^2 d x (9 d+2 e x)\right )\right )}{315 (d+e x)^{9/2} (b d-a e)^3} \]

Antiderivative was successfully verified.

[In]  Integrate[((a + b*x)^(3/2)*(A + B*x))/(d + e*x)^(11/2),x]

[Out]

(2*(a + b*x)^(5/2)*(A*(35*a^2*e^2 - 10*a*b*e*(9*d + 2*e*x) + b^2*(63*d^2 + 36*d*
e*x + 8*e^2*x^2)) + B*(5*b^2*d*x*(9*d + 2*e*x) + 5*a^2*e*(2*d + 9*e*x) - 2*a*b*(
9*d^2 + 53*d*e*x + 9*e^2*x^2))))/(315*(b*d - a*e)^3*(d + e*x)^(9/2))

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Maple [A]  time = 0.011, size = 177, normalized size = 1.2 \[ -{\frac{16\,A{b}^{2}{e}^{2}{x}^{2}-36\,Bab{e}^{2}{x}^{2}+20\,B{b}^{2}de{x}^{2}-40\,Aab{e}^{2}x+72\,A{b}^{2}dex+90\,B{a}^{2}{e}^{2}x-212\,Babdex+90\,B{b}^{2}{d}^{2}x+70\,A{a}^{2}{e}^{2}-180\,Aabde+126\,A{b}^{2}{d}^{2}+20\,B{a}^{2}de-36\,Bab{d}^{2}}{315\,{a}^{3}{e}^{3}-945\,{a}^{2}bd{e}^{2}+945\,a{b}^{2}{d}^{2}e-315\,{b}^{3}{d}^{3}} \left ( bx+a \right ) ^{{\frac{5}{2}}} \left ( ex+d \right ) ^{-{\frac{9}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^(3/2)*(B*x+A)/(e*x+d)^(11/2),x)

[Out]

-2/315*(b*x+a)^(5/2)*(8*A*b^2*e^2*x^2-18*B*a*b*e^2*x^2+10*B*b^2*d*e*x^2-20*A*a*b
*e^2*x+36*A*b^2*d*e*x+45*B*a^2*e^2*x-106*B*a*b*d*e*x+45*B*b^2*d^2*x+35*A*a^2*e^2
-90*A*a*b*d*e+63*A*b^2*d^2+10*B*a^2*d*e-18*B*a*b*d^2)/(e*x+d)^(9/2)/(a^3*e^3-3*a
^2*b*d*e^2+3*a*b^2*d^2*e-b^3*d^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)^(11/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 2.46947, size = 765, normalized size = 5.2 \[ \frac{2 \,{\left (35 \, A a^{4} e^{2} + 2 \,{\left (5 \, B b^{4} d e -{\left (9 \, B a b^{3} - 4 \, A b^{4}\right )} e^{2}\right )} x^{4} +{\left (45 \, B b^{4} d^{2} - 2 \,{\left (43 \, B a b^{3} - 18 \, A b^{4}\right )} d e +{\left (9 \, B a^{2} b^{2} - 4 \, A a b^{3}\right )} e^{2}\right )} x^{3} - 9 \,{\left (2 \, B a^{3} b - 7 \, A a^{2} b^{2}\right )} d^{2} + 10 \,{\left (B a^{4} - 9 \, A a^{3} b\right )} d e + 3 \,{\left (3 \,{\left (8 \, B a b^{3} + 7 \, A b^{4}\right )} d^{2} - 2 \,{\left (32 \, B a^{2} b^{2} + 3 \, A a b^{3}\right )} d e +{\left (24 \, B a^{3} b + A a^{2} b^{2}\right )} e^{2}\right )} x^{2} +{\left (9 \,{\left (B a^{2} b^{2} + 14 \, A a b^{3}\right )} d^{2} - 2 \,{\left (43 \, B a^{3} b + 72 \, A a^{2} b^{2}\right )} d e + 5 \,{\left (9 \, B a^{4} + 10 \, A a^{3} b\right )} e^{2}\right )} x\right )} \sqrt{b x + a} \sqrt{e x + d}}{315 \,{\left (b^{3} d^{8} - 3 \, a b^{2} d^{7} e + 3 \, a^{2} b d^{6} e^{2} - a^{3} d^{5} e^{3} +{\left (b^{3} d^{3} e^{5} - 3 \, a b^{2} d^{2} e^{6} + 3 \, a^{2} b d e^{7} - a^{3} e^{8}\right )} x^{5} + 5 \,{\left (b^{3} d^{4} e^{4} - 3 \, a b^{2} d^{3} e^{5} + 3 \, a^{2} b d^{2} e^{6} - a^{3} d e^{7}\right )} x^{4} + 10 \,{\left (b^{3} d^{5} e^{3} - 3 \, a b^{2} d^{4} e^{4} + 3 \, a^{2} b d^{3} e^{5} - a^{3} d^{2} e^{6}\right )} x^{3} + 10 \,{\left (b^{3} d^{6} e^{2} - 3 \, a b^{2} d^{5} e^{3} + 3 \, a^{2} b d^{4} e^{4} - a^{3} d^{3} e^{5}\right )} x^{2} + 5 \,{\left (b^{3} d^{7} e - 3 \, a b^{2} d^{6} e^{2} + 3 \, a^{2} b d^{5} e^{3} - a^{3} d^{4} e^{4}\right )} x\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*(b*x + a)^(3/2)/(e*x + d)^(11/2),x, algorithm="fricas")

[Out]

2/315*(35*A*a^4*e^2 + 2*(5*B*b^4*d*e - (9*B*a*b^3 - 4*A*b^4)*e^2)*x^4 + (45*B*b^
4*d^2 - 2*(43*B*a*b^3 - 18*A*b^4)*d*e + (9*B*a^2*b^2 - 4*A*a*b^3)*e^2)*x^3 - 9*(
2*B*a^3*b - 7*A*a^2*b^2)*d^2 + 10*(B*a^4 - 9*A*a^3*b)*d*e + 3*(3*(8*B*a*b^3 + 7*
A*b^4)*d^2 - 2*(32*B*a^2*b^2 + 3*A*a*b^3)*d*e + (24*B*a^3*b + A*a^2*b^2)*e^2)*x^
2 + (9*(B*a^2*b^2 + 14*A*a*b^3)*d^2 - 2*(43*B*a^3*b + 72*A*a^2*b^2)*d*e + 5*(9*B
*a^4 + 10*A*a^3*b)*e^2)*x)*sqrt(b*x + a)*sqrt(e*x + d)/(b^3*d^8 - 3*a*b^2*d^7*e
+ 3*a^2*b*d^6*e^2 - a^3*d^5*e^3 + (b^3*d^3*e^5 - 3*a*b^2*d^2*e^6 + 3*a^2*b*d*e^7
- a^3*e^8)*x^5 + 5*(b^3*d^4*e^4 - 3*a*b^2*d^3*e^5 + 3*a^2*b*d^2*e^6 - a^3*d*e^7
)*x^4 + 10*(b^3*d^5*e^3 - 3*a*b^2*d^4*e^4 + 3*a^2*b*d^3*e^5 - a^3*d^2*e^6)*x^3 +
10*(b^3*d^6*e^2 - 3*a*b^2*d^5*e^3 + 3*a^2*b*d^4*e^4 - a^3*d^3*e^5)*x^2 + 5*(b^3
*d^7*e - 3*a*b^2*d^6*e^2 + 3*a^2*b*d^5*e^3 - a^3*d^4*e^4)*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)**(11/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.342676, size = 709, normalized size = 4.82 \[ -\frac{{\left ({\left (b x + a\right )}{\left (\frac{2 \,{\left (5 \, B b^{11} d^{2}{\left | b \right |} e^{5} - 14 \, B a b^{10} d{\left | b \right |} e^{6} + 4 \, A b^{11} d{\left | b \right |} e^{6} + 9 \, B a^{2} b^{9}{\left | b \right |} e^{7} - 4 \, A a b^{10}{\left | b \right |} e^{7}\right )}{\left (b x + a\right )}}{b^{20} d^{5} e^{10} - 5 \, a b^{19} d^{4} e^{11} + 10 \, a^{2} b^{18} d^{3} e^{12} - 10 \, a^{3} b^{17} d^{2} e^{13} + 5 \, a^{4} b^{16} d e^{14} - a^{5} b^{15} e^{15}} + \frac{9 \,{\left (5 \, B b^{12} d^{3}{\left | b \right |} e^{4} - 19 \, B a b^{11} d^{2}{\left | b \right |} e^{5} + 4 \, A b^{12} d^{2}{\left | b \right |} e^{5} + 23 \, B a^{2} b^{10} d{\left | b \right |} e^{6} - 8 \, A a b^{11} d{\left | b \right |} e^{6} - 9 \, B a^{3} b^{9}{\left | b \right |} e^{7} + 4 \, A a^{2} b^{10}{\left | b \right |} e^{7}\right )}}{b^{20} d^{5} e^{10} - 5 \, a b^{19} d^{4} e^{11} + 10 \, a^{2} b^{18} d^{3} e^{12} - 10 \, a^{3} b^{17} d^{2} e^{13} + 5 \, a^{4} b^{16} d e^{14} - a^{5} b^{15} e^{15}}\right )} - \frac{63 \,{\left (B a b^{12} d^{3}{\left | b \right |} e^{4} - A b^{13} d^{3}{\left | b \right |} e^{4} - 3 \, B a^{2} b^{11} d^{2}{\left | b \right |} e^{5} + 3 \, A a b^{12} d^{2}{\left | b \right |} e^{5} + 3 \, B a^{3} b^{10} d{\left | b \right |} e^{6} - 3 \, A a^{2} b^{11} d{\left | b \right |} e^{6} - B a^{4} b^{9}{\left | b \right |} e^{7} + A a^{3} b^{10}{\left | b \right |} e^{7}\right )}}{b^{20} d^{5} e^{10} - 5 \, a b^{19} d^{4} e^{11} + 10 \, a^{2} b^{18} d^{3} e^{12} - 10 \, a^{3} b^{17} d^{2} e^{13} + 5 \, a^{4} b^{16} d e^{14} - a^{5} b^{15} e^{15}}\right )}{\left (b x + a\right )}^{\frac{5}{2}}}{322560 \,{\left (b^{2} d +{\left (b x + a\right )} b e - a b e\right )}^{\frac{9}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*(b*x + a)^(3/2)/(e*x + d)^(11/2),x, algorithm="giac")

[Out]

-1/322560*((b*x + a)*(2*(5*B*b^11*d^2*abs(b)*e^5 - 14*B*a*b^10*d*abs(b)*e^6 + 4*
A*b^11*d*abs(b)*e^6 + 9*B*a^2*b^9*abs(b)*e^7 - 4*A*a*b^10*abs(b)*e^7)*(b*x + a)/
(b^20*d^5*e^10 - 5*a*b^19*d^4*e^11 + 10*a^2*b^18*d^3*e^12 - 10*a^3*b^17*d^2*e^13
+ 5*a^4*b^16*d*e^14 - a^5*b^15*e^15) + 9*(5*B*b^12*d^3*abs(b)*e^4 - 19*B*a*b^11
*d^2*abs(b)*e^5 + 4*A*b^12*d^2*abs(b)*e^5 + 23*B*a^2*b^10*d*abs(b)*e^6 - 8*A*a*b
^11*d*abs(b)*e^6 - 9*B*a^3*b^9*abs(b)*e^7 + 4*A*a^2*b^10*abs(b)*e^7)/(b^20*d^5*e
^10 - 5*a*b^19*d^4*e^11 + 10*a^2*b^18*d^3*e^12 - 10*a^3*b^17*d^2*e^13 + 5*a^4*b^
16*d*e^14 - a^5*b^15*e^15)) - 63*(B*a*b^12*d^3*abs(b)*e^4 - A*b^13*d^3*abs(b)*e^
4 - 3*B*a^2*b^11*d^2*abs(b)*e^5 + 3*A*a*b^12*d^2*abs(b)*e^5 + 3*B*a^3*b^10*d*abs
(b)*e^6 - 3*A*a^2*b^11*d*abs(b)*e^6 - B*a^4*b^9*abs(b)*e^7 + A*a^3*b^10*abs(b)*e
^7)/(b^20*d^5*e^10 - 5*a*b^19*d^4*e^11 + 10*a^2*b^18*d^3*e^12 - 10*a^3*b^17*d^2*
e^13 + 5*a^4*b^16*d*e^14 - a^5*b^15*e^15))*(b*x + a)^(5/2)/(b^2*d + (b*x + a)*b*
e - a*b*e)^(9/2)