3.2225 \(\int \frac{A+B x}{\sqrt{a+b x} (d+e x)^{7/2}} \, dx\)

Optimal. Leaf size=145 \[ -\frac{2 \sqrt{a+b x} (B d-A e)}{5 e (d+e x)^{5/2} (b d-a e)}+\frac{4 b \sqrt{a+b x} (-5 a B e+4 A b e+b B d)}{15 e \sqrt{d+e x} (b d-a e)^3}+\frac{2 \sqrt{a+b x} (-5 a B e+4 A b e+b B d)}{15 e (d+e x)^{3/2} (b d-a e)^2} \]

[Out]

(-2*(B*d - A*e)*Sqrt[a + b*x])/(5*e*(b*d - a*e)*(d + e*x)^(5/2)) + (2*(b*B*d + 4
*A*b*e - 5*a*B*e)*Sqrt[a + b*x])/(15*e*(b*d - a*e)^2*(d + e*x)^(3/2)) + (4*b*(b*
B*d + 4*A*b*e - 5*a*B*e)*Sqrt[a + b*x])/(15*e*(b*d - a*e)^3*Sqrt[d + e*x])

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Rubi [A]  time = 0.268115, antiderivative size = 145, 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 \[ -\frac{2 \sqrt{a+b x} (B d-A e)}{5 e (d+e x)^{5/2} (b d-a e)}+\frac{4 b \sqrt{a+b x} (-5 a B e+4 A b e+b B d)}{15 e \sqrt{d+e x} (b d-a e)^3}+\frac{2 \sqrt{a+b x} (-5 a B e+4 A b e+b B d)}{15 e (d+e x)^{3/2} (b d-a e)^2} \]

Antiderivative was successfully verified.

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

[Out]

(-2*(B*d - A*e)*Sqrt[a + b*x])/(5*e*(b*d - a*e)*(d + e*x)^(5/2)) + (2*(b*B*d + 4
*A*b*e - 5*a*B*e)*Sqrt[a + b*x])/(15*e*(b*d - a*e)^2*(d + e*x)^(3/2)) + (4*b*(b*
B*d + 4*A*b*e - 5*a*B*e)*Sqrt[a + b*x])/(15*e*(b*d - a*e)^3*Sqrt[d + e*x])

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Rubi in Sympy [A]  time = 24.4646, size = 134, normalized size = 0.92 \[ - \frac{4 b \sqrt{a + b x} \left (4 A b e - 5 B a e + B b d\right )}{15 e \sqrt{d + e x} \left (a e - b d\right )^{3}} + \frac{2 \sqrt{a + b x} \left (4 A b e - 5 B a e + B b d\right )}{15 e \left (d + e x\right )^{\frac{3}{2}} \left (a e - b d\right )^{2}} - \frac{2 \sqrt{a + b x} \left (A e - B d\right )}{5 e \left (d + e x\right )^{\frac{5}{2}} \left (a e - b d\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((B*x+A)/(e*x+d)**(7/2)/(b*x+a)**(1/2),x)

[Out]

-4*b*sqrt(a + b*x)*(4*A*b*e - 5*B*a*e + B*b*d)/(15*e*sqrt(d + e*x)*(a*e - b*d)**
3) + 2*sqrt(a + b*x)*(4*A*b*e - 5*B*a*e + B*b*d)/(15*e*(d + e*x)**(3/2)*(a*e - b
*d)**2) - 2*sqrt(a + b*x)*(A*e - B*d)/(5*e*(d + e*x)**(5/2)*(a*e - b*d))

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Mathematica [A]  time = 0.208749, size = 131, normalized size = 0.9 \[ \sqrt{a+b x} \sqrt{d+e x} \left (-\frac{2 (A e-B d)}{5 e (d+e x)^3 (a e-b d)}-\frac{4 b (-5 a B e+4 A b e+b B d)}{15 e (d+e x) (a e-b d)^3}+\frac{2 (-5 a B e+4 A b e+b B d)}{15 e (d+e x)^2 (a e-b d)^2}\right ) \]

Antiderivative was successfully verified.

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

[Out]

Sqrt[a + b*x]*Sqrt[d + e*x]*((-2*(-(B*d) + A*e))/(5*e*(-(b*d) + a*e)*(d + e*x)^3
) + (2*(b*B*d + 4*A*b*e - 5*a*B*e))/(15*e*(-(b*d) + a*e)^2*(d + e*x)^2) - (4*b*(
b*B*d + 4*A*b*e - 5*a*B*e))/(15*e*(-(b*d) + a*e)^3*(d + e*x)))

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Maple [A]  time = 0.013, size = 177, normalized size = 1.2 \[ -{\frac{16\,A{b}^{2}{e}^{2}{x}^{2}-20\,Bab{e}^{2}{x}^{2}+4\,B{b}^{2}de{x}^{2}-8\,Aab{e}^{2}x+40\,A{b}^{2}dex+10\,B{a}^{2}{e}^{2}x-52\,Babdex+10\,B{b}^{2}{d}^{2}x+6\,A{a}^{2}{e}^{2}-20\,Aabde+30\,A{b}^{2}{d}^{2}+4\,B{a}^{2}de-20\,Bab{d}^{2}}{15\,{a}^{3}{e}^{3}-45\,{a}^{2}bd{e}^{2}+45\,a{b}^{2}{d}^{2}e-15\,{b}^{3}{d}^{3}}\sqrt{bx+a} \left ( ex+d \right ) ^{-{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x+A)/(e*x+d)^(7/2)/(b*x+a)^(1/2),x)

[Out]

-2/15*(b*x+a)^(1/2)*(8*A*b^2*e^2*x^2-10*B*a*b*e^2*x^2+2*B*b^2*d*e*x^2-4*A*a*b*e^
2*x+20*A*b^2*d*e*x+5*B*a^2*e^2*x-26*B*a*b*d*e*x+5*B*b^2*d^2*x+3*A*a^2*e^2-10*A*a
*b*d*e+15*A*b^2*d^2+2*B*a^2*d*e-10*B*a*b*d^2)/(e*x+d)^(5/2)/(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)/(sqrt(b*x + a)*(e*x + d)^(7/2)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.646446, size = 427, normalized size = 2.94 \[ \frac{2 \,{\left (3 \, A a^{2} e^{2} - 5 \,{\left (2 \, B a b - 3 \, A b^{2}\right )} d^{2} + 2 \,{\left (B a^{2} - 5 \, A a b\right )} d e + 2 \,{\left (B b^{2} d e -{\left (5 \, B a b - 4 \, A b^{2}\right )} e^{2}\right )} x^{2} +{\left (5 \, B b^{2} d^{2} - 2 \,{\left (13 \, B a b - 10 \, A b^{2}\right )} d e +{\left (5 \, B a^{2} - 4 \, A a b\right )} e^{2}\right )} x\right )} \sqrt{b x + a} \sqrt{e x + d}}{15 \,{\left (b^{3} d^{6} - 3 \, a b^{2} d^{5} e + 3 \, a^{2} b d^{4} e^{2} - a^{3} d^{3} e^{3} +{\left (b^{3} d^{3} e^{3} - 3 \, a b^{2} d^{2} e^{4} + 3 \, a^{2} b d e^{5} - a^{3} e^{6}\right )} x^{3} + 3 \,{\left (b^{3} d^{4} e^{2} - 3 \, a b^{2} d^{3} e^{3} + 3 \, a^{2} b d^{2} e^{4} - a^{3} d e^{5}\right )} x^{2} + 3 \,{\left (b^{3} d^{5} e - 3 \, a b^{2} d^{4} e^{2} + 3 \, a^{2} b d^{3} e^{3} - a^{3} d^{2} e^{4}\right )} x\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/15*(3*A*a^2*e^2 - 5*(2*B*a*b - 3*A*b^2)*d^2 + 2*(B*a^2 - 5*A*a*b)*d*e + 2*(B*b
^2*d*e - (5*B*a*b - 4*A*b^2)*e^2)*x^2 + (5*B*b^2*d^2 - 2*(13*B*a*b - 10*A*b^2)*d
*e + (5*B*a^2 - 4*A*a*b)*e^2)*x)*sqrt(b*x + a)*sqrt(e*x + d)/(b^3*d^6 - 3*a*b^2*
d^5*e + 3*a^2*b*d^4*e^2 - a^3*d^3*e^3 + (b^3*d^3*e^3 - 3*a*b^2*d^2*e^4 + 3*a^2*b
*d*e^5 - a^3*e^6)*x^3 + 3*(b^3*d^4*e^2 - 3*a*b^2*d^3*e^3 + 3*a^2*b*d^2*e^4 - a^3
*d*e^5)*x^2 + 3*(b^3*d^5*e - 3*a*b^2*d^4*e^2 + 3*a^2*b*d^3*e^3 - a^3*d^2*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)/(e*x+d)**(7/2)/(b*x+a)**(1/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.263141, size = 477, normalized size = 3.29 \[ -\frac{{\left ({\left (b x + a\right )}{\left (\frac{2 \,{\left (B b^{6} d{\left | b \right |} e^{3} - 5 \, B a b^{5}{\left | b \right |} e^{4} + 4 \, A b^{6}{\left | b \right |} e^{4}\right )}{\left (b x + a\right )}}{b^{12} d^{3} e^{6} - 3 \, a b^{11} d^{2} e^{7} + 3 \, a^{2} b^{10} d e^{8} - a^{3} b^{9} e^{9}} + \frac{5 \,{\left (B b^{7} d^{2}{\left | b \right |} e^{2} - 6 \, B a b^{6} d{\left | b \right |} e^{3} + 4 \, A b^{7} d{\left | b \right |} e^{3} + 5 \, B a^{2} b^{5}{\left | b \right |} e^{4} - 4 \, A a b^{6}{\left | b \right |} e^{4}\right )}}{b^{12} d^{3} e^{6} - 3 \, a b^{11} d^{2} e^{7} + 3 \, a^{2} b^{10} d e^{8} - a^{3} b^{9} e^{9}}\right )} - \frac{15 \,{\left (B a b^{7} d^{2}{\left | b \right |} e^{2} - A b^{8} d^{2}{\left | b \right |} e^{2} - 2 \, B a^{2} b^{6} d{\left | b \right |} e^{3} + 2 \, A a b^{7} d{\left | b \right |} e^{3} + B a^{3} b^{5}{\left | b \right |} e^{4} - A a^{2} b^{6}{\left | b \right |} e^{4}\right )}}{b^{12} d^{3} e^{6} - 3 \, a b^{11} d^{2} e^{7} + 3 \, a^{2} b^{10} d e^{8} - a^{3} b^{9} e^{9}}\right )} \sqrt{b x + a}}{960 \,{\left (b^{2} d +{\left (b x + a\right )} b e - a b e\right )}^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/960*((b*x + a)*(2*(B*b^6*d*abs(b)*e^3 - 5*B*a*b^5*abs(b)*e^4 + 4*A*b^6*abs(b)
*e^4)*(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) + 5*(B*b^7*d^2*abs(b)*e^2 - 6*B*a*b^6*d*abs(b)*e^3 + 4*A*b^7*d*abs(b)*e^3 + 5
*B*a^2*b^5*abs(b)*e^4 - 4*A*a*b^6*abs(b)*e^4)/(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*a*b^7*d^2*abs(b)*e^2 - A*b^8*d^2*abs(b
)*e^2 - 2*B*a^2*b^6*d*abs(b)*e^3 + 2*A*a*b^7*d*abs(b)*e^3 + B*a^3*b^5*abs(b)*e^4
- A*a^2*b^6*abs(b)*e^4)/(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)