3.2214 \(\int \frac{(a+b x)^{5/2} (A+B x)}{(d+e x)^{11/2}} \, dx\)

Optimal. Leaf size=95 \[ \frac{2 (a+b x)^{7/2} (-9 a B e+2 A b e+7 b B d)}{63 e (d+e x)^{7/2} (b d-a e)^2}-\frac{2 (a+b x)^{7/2} (B d-A e)}{9 e (d+e x)^{9/2} (b d-a e)} \]

[Out]

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

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Rubi [A]  time = 0.172819, antiderivative size = 95, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083 \[ \frac{2 (a+b x)^{7/2} (-9 a B e+2 A b e+7 b B d)}{63 e (d+e x)^{7/2} (b d-a e)^2}-\frac{2 (a+b x)^{7/2} (B d-A e)}{9 e (d+e x)^{9/2} (b d-a e)} \]

Antiderivative was successfully verified.

[In]  Int[((a + b*x)^(5/2)*(A + B*x))/(d + e*x)^(11/2),x]

[Out]

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

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Rubi in Sympy [A]  time = 12.3986, size = 85, normalized size = 0.89 \[ - \frac{4 \left (a + b x\right )^{\frac{7}{2}} \left (- A b e + \frac{B \left (9 a e - 7 b d\right )}{2}\right )}{63 e \left (d + e x\right )^{\frac{7}{2}} \left (a e - b d\right )^{2}} - \frac{2 \left (a + b x\right )^{\frac{7}{2}} \left (A e - B d\right )}{9 e \left (d + e x\right )^{\frac{9}{2}} \left (a e - b d\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)**(5/2)*(B*x+A)/(e*x+d)**(11/2),x)

[Out]

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

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Mathematica [A]  time = 0.34326, size = 66, normalized size = 0.69 \[ \frac{2 (a+b x)^{7/2} (A (-7 a e+9 b d+2 b e x)+B (-2 a d-9 a e x+7 b d x))}{63 (d+e x)^{9/2} (b d-a e)^2} \]

Antiderivative was successfully verified.

[In]  Integrate[((a + b*x)^(5/2)*(A + B*x))/(d + e*x)^(11/2),x]

[Out]

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

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Maple [A]  time = 0.01, size = 74, normalized size = 0.8 \[ -{\frac{-4\,Abex+18\,Baex-14\,Bbdx+14\,Aae-18\,Abd+4\,Bad}{63\,{a}^{2}{e}^{2}-126\,bead+63\,{b}^{2}{d}^{2}} \left ( bx+a \right ) ^{{\frac{7}{2}}} \left ( ex+d \right ) ^{-{\frac{9}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^(5/2)*(B*x+A)/(e*x+d)^(11/2),x)

[Out]

-2/63*(b*x+a)^(7/2)*(-2*A*b*e*x+9*B*a*e*x-7*B*b*d*x+7*A*a*e-9*A*b*d+2*B*a*d)/(e*
x+d)^(9/2)/(a^2*e^2-2*a*b*d*e+b^2*d^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)*(b*x + a)^(5/2)/(e*x + d)^(11/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 2.4393, size = 528, normalized size = 5.56 \[ -\frac{2 \,{\left (7 \, A a^{4} e -{\left (7 \, B b^{4} d -{\left (9 \, B a b^{3} - 2 \, A b^{4}\right )} e\right )} x^{4} -{\left ({\left (19 \, B a b^{3} + 9 \, A b^{4}\right )} d -{\left (27 \, B a^{2} b^{2} + A a b^{3}\right )} e\right )} x^{3} - 3 \,{\left ({\left (5 \, B a^{2} b^{2} + 9 \, A a b^{3}\right )} d -{\left (9 \, B a^{3} b + 5 \, A a^{2} b^{2}\right )} e\right )} x^{2} +{\left (2 \, B a^{4} - 9 \, A a^{3} b\right )} d -{\left ({\left (B a^{3} b + 27 \, A a^{2} b^{2}\right )} d -{\left (9 \, B a^{4} + 19 \, A a^{3} b\right )} e\right )} x\right )} \sqrt{b x + a} \sqrt{e x + d}}{63 \,{\left (b^{2} d^{7} - 2 \, a b d^{6} e + a^{2} d^{5} e^{2} +{\left (b^{2} d^{2} e^{5} - 2 \, a b d e^{6} + a^{2} e^{7}\right )} x^{5} + 5 \,{\left (b^{2} d^{3} e^{4} - 2 \, a b d^{2} e^{5} + a^{2} d e^{6}\right )} x^{4} + 10 \,{\left (b^{2} d^{4} e^{3} - 2 \, a b d^{3} e^{4} + a^{2} d^{2} e^{5}\right )} x^{3} + 10 \,{\left (b^{2} d^{5} e^{2} - 2 \, a b d^{4} e^{3} + a^{2} d^{3} e^{4}\right )} x^{2} + 5 \,{\left (b^{2} d^{6} e - 2 \, a b d^{5} e^{2} + a^{2} d^{4} e^{3}\right )} x\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*(b*x + a)^(5/2)/(e*x + d)^(11/2),x, algorithm="fricas")

[Out]

-2/63*(7*A*a^4*e - (7*B*b^4*d - (9*B*a*b^3 - 2*A*b^4)*e)*x^4 - ((19*B*a*b^3 + 9*
A*b^4)*d - (27*B*a^2*b^2 + A*a*b^3)*e)*x^3 - 3*((5*B*a^2*b^2 + 9*A*a*b^3)*d - (9
*B*a^3*b + 5*A*a^2*b^2)*e)*x^2 + (2*B*a^4 - 9*A*a^3*b)*d - ((B*a^3*b + 27*A*a^2*
b^2)*d - (9*B*a^4 + 19*A*a^3*b)*e)*x)*sqrt(b*x + a)*sqrt(e*x + d)/(b^2*d^7 - 2*a
*b*d^6*e + a^2*d^5*e^2 + (b^2*d^2*e^5 - 2*a*b*d*e^6 + a^2*e^7)*x^5 + 5*(b^2*d^3*
e^4 - 2*a*b*d^2*e^5 + a^2*d*e^6)*x^4 + 10*(b^2*d^4*e^3 - 2*a*b*d^3*e^4 + a^2*d^2
*e^5)*x^3 + 10*(b^2*d^5*e^2 - 2*a*b*d^4*e^3 + a^2*d^3*e^4)*x^2 + 5*(b^2*d^6*e -
2*a*b*d^5*e^2 + a^2*d^4*e^3)*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)**(5/2)*(B*x+A)/(e*x+d)**(11/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.358746, size = 518, normalized size = 5.45 \[ -\frac{{\left (b x + a\right )}^{\frac{7}{2}}{\left (\frac{{\left (7 \, B b^{12} d^{3}{\left | b \right |} e^{4} - 23 \, B a b^{11} d^{2}{\left | b \right |} e^{5} + 2 \, A b^{12} d^{2}{\left | b \right |} e^{5} + 25 \, B a^{2} b^{10} d{\left | b \right |} e^{6} - 4 \, A a b^{11} d{\left | b \right |} e^{6} - 9 \, B a^{3} b^{9}{\left | b \right |} e^{7} + 2 \, A a^{2} b^{10}{\left | b \right |} e^{7}\right )}{\left (b x + a\right )}}{b^{20} d^{5} e^{10} - 5 \, a b^{19} d^{4} e^{11} + 10 \, a^{2} b^{18} d^{3} e^{12} - 10 \, a^{3} b^{17} d^{2} e^{13} + 5 \, a^{4} b^{16} d e^{14} - a^{5} b^{15} e^{15}} - \frac{9 \,{\left (B a b^{12} d^{3}{\left | b \right |} e^{4} - A b^{13} d^{3}{\left | b \right |} e^{4} - 3 \, B a^{2} b^{11} d^{2}{\left | b \right |} e^{5} + 3 \, A a b^{12} d^{2}{\left | b \right |} e^{5} + 3 \, B a^{3} b^{10} d{\left | b \right |} e^{6} - 3 \, A a^{2} b^{11} d{\left | b \right |} e^{6} - B a^{4} b^{9}{\left | b \right |} e^{7} + A a^{3} b^{10}{\left | b \right |} e^{7}\right )}}{b^{20} d^{5} e^{10} - 5 \, a b^{19} d^{4} e^{11} + 10 \, a^{2} b^{18} d^{3} e^{12} - 10 \, a^{3} b^{17} d^{2} e^{13} + 5 \, a^{4} b^{16} d e^{14} - a^{5} b^{15} e^{15}}\right )}}{64512 \,{\left (b^{2} d +{\left (b x + a\right )} b e - a b e\right )}^{\frac{9}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*(b*x + a)^(5/2)/(e*x + d)^(11/2),x, algorithm="giac")

[Out]

-1/64512*(b*x + a)^(7/2)*((7*B*b^12*d^3*abs(b)*e^4 - 23*B*a*b^11*d^2*abs(b)*e^5
+ 2*A*b^12*d^2*abs(b)*e^5 + 25*B*a^2*b^10*d*abs(b)*e^6 - 4*A*a*b^11*d*abs(b)*e^6
- 9*B*a^3*b^9*abs(b)*e^7 + 2*A*a^2*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*(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^2
0*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^2*d + (b*x + a)*b*e - a*b*e)^(9/2)