3.23.60 \(\int \frac {A+B x}{(a+b x)^{5/2} (d+e x)^{7/2}} \, dx\) [2260]

Optimal. Leaf size=246 \[ -\frac {2 (A b-a B)}{3 b (b d-a e) (a+b x)^{3/2} (d+e x)^{5/2}}-\frac {2 (3 b B d-8 A b e+5 a B e)}{3 b (b d-a e)^2 \sqrt {a+b x} (d+e x)^{5/2}}-\frac {4 e (3 b B d-8 A b e+5 a B e) \sqrt {a+b x}}{5 b (b d-a e)^3 (d+e x)^{5/2}}-\frac {16 e (3 b B d-8 A b e+5 a B e) \sqrt {a+b x}}{15 (b d-a e)^4 (d+e x)^{3/2}}-\frac {32 b e (3 b B d-8 A b e+5 a B e) \sqrt {a+b x}}{15 (b d-a e)^5 \sqrt {d+e x}} \]

[Out]

-2/3*(A*b-B*a)/b/(-a*e+b*d)/(b*x+a)^(3/2)/(e*x+d)^(5/2)-2/3*(-8*A*b*e+5*B*a*e+3*B*b*d)/b/(-a*e+b*d)^2/(e*x+d)^
(5/2)/(b*x+a)^(1/2)-4/5*e*(-8*A*b*e+5*B*a*e+3*B*b*d)*(b*x+a)^(1/2)/b/(-a*e+b*d)^3/(e*x+d)^(5/2)-16/15*e*(-8*A*
b*e+5*B*a*e+3*B*b*d)*(b*x+a)^(1/2)/(-a*e+b*d)^4/(e*x+d)^(3/2)-32/15*b*e*(-8*A*b*e+5*B*a*e+3*B*b*d)*(b*x+a)^(1/
2)/(-a*e+b*d)^5/(e*x+d)^(1/2)

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Rubi [A]
time = 0.10, antiderivative size = 246, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {79, 47, 37} \begin {gather*} -\frac {2 (A b-a B)}{3 b (a+b x)^{3/2} (d+e x)^{5/2} (b d-a e)}-\frac {32 b e \sqrt {a+b x} (5 a B e-8 A b e+3 b B d)}{15 \sqrt {d+e x} (b d-a e)^5}-\frac {16 e \sqrt {a+b x} (5 a B e-8 A b e+3 b B d)}{15 (d+e x)^{3/2} (b d-a e)^4}-\frac {4 e \sqrt {a+b x} (5 a B e-8 A b e+3 b B d)}{5 b (d+e x)^{5/2} (b d-a e)^3}-\frac {2 (5 a B e-8 A b e+3 b B d)}{3 b \sqrt {a+b x} (d+e x)^{5/2} (b d-a e)^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(A + B*x)/((a + b*x)^(5/2)*(d + e*x)^(7/2)),x]

[Out]

(-2*(A*b - a*B))/(3*b*(b*d - a*e)*(a + b*x)^(3/2)*(d + e*x)^(5/2)) - (2*(3*b*B*d - 8*A*b*e + 5*a*B*e))/(3*b*(b
*d - a*e)^2*Sqrt[a + b*x]*(d + e*x)^(5/2)) - (4*e*(3*b*B*d - 8*A*b*e + 5*a*B*e)*Sqrt[a + b*x])/(5*b*(b*d - a*e
)^3*(d + e*x)^(5/2)) - (16*e*(3*b*B*d - 8*A*b*e + 5*a*B*e)*Sqrt[a + b*x])/(15*(b*d - a*e)^4*(d + e*x)^(3/2)) -
(32*b*e*(3*b*B*d - 8*A*b*e + 5*a*B*e)*Sqrt[a + b*x])/(15*(b*d - a*e)^5*Sqrt[d + e*x])

Rule 37

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(a + b*x)^(m + 1)*((c + d*x)^(n +
1)/((b*c - a*d)*(m + 1))), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[m + n + 2, 0] && NeQ
[m, -1]

Rule 47

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(a + b*x)^(m + 1)*((c + d*x)^(n + 1
)/((b*c - a*d)*(m + 1))), x] - Dist[d*(Simplify[m + n + 2]/((b*c - a*d)*(m + 1))), Int[(a + b*x)^Simplify[m +
1]*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && ILtQ[Simplify[m + n + 2], 0] &&
NeQ[m, -1] &&  !(LtQ[m, -1] && LtQ[n, -1] && (EqQ[a, 0] || (NeQ[c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && (
SumSimplerQ[m, 1] ||  !SumSimplerQ[n, 1])

Rule 79

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(-(b*e - a*f
))*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(f*(p + 1)*(c*f - d*e))), x] - Dist[(a*d*f*(n + p + 2) - b*(d*e*(n + 1
) + c*f*(p + 1)))/(f*(p + 1)*(c*f - d*e)), Int[(c + d*x)^n*(e + f*x)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e,
f, n}, x] && LtQ[p, -1] && ( !LtQ[n, -1] || IntegerQ[p] ||  !(IntegerQ[n] ||  !(EqQ[e, 0] ||  !(EqQ[c, 0] || L
tQ[p, n]))))

Rubi steps

\begin {align*} \int \frac {A+B x}{(a+b x)^{5/2} (d+e x)^{7/2}} \, dx &=-\frac {2 (A b-a B)}{3 b (b d-a e) (a+b x)^{3/2} (d+e x)^{5/2}}+\frac {(3 b B d-8 A b e+5 a B e) \int \frac {1}{(a+b x)^{3/2} (d+e x)^{7/2}} \, dx}{3 b (b d-a e)}\\ &=-\frac {2 (A b-a B)}{3 b (b d-a e) (a+b x)^{3/2} (d+e x)^{5/2}}-\frac {2 (3 b B d-8 A b e+5 a B e)}{3 b (b d-a e)^2 \sqrt {a+b x} (d+e x)^{5/2}}-\frac {(2 e (3 b B d-8 A b e+5 a B e)) \int \frac {1}{\sqrt {a+b x} (d+e x)^{7/2}} \, dx}{b (b d-a e)^2}\\ &=-\frac {2 (A b-a B)}{3 b (b d-a e) (a+b x)^{3/2} (d+e x)^{5/2}}-\frac {2 (3 b B d-8 A b e+5 a B e)}{3 b (b d-a e)^2 \sqrt {a+b x} (d+e x)^{5/2}}-\frac {4 e (3 b B d-8 A b e+5 a B e) \sqrt {a+b x}}{5 b (b d-a e)^3 (d+e x)^{5/2}}-\frac {(8 e (3 b B d-8 A b e+5 a B e)) \int \frac {1}{\sqrt {a+b x} (d+e x)^{5/2}} \, dx}{5 (b d-a e)^3}\\ &=-\frac {2 (A b-a B)}{3 b (b d-a e) (a+b x)^{3/2} (d+e x)^{5/2}}-\frac {2 (3 b B d-8 A b e+5 a B e)}{3 b (b d-a e)^2 \sqrt {a+b x} (d+e x)^{5/2}}-\frac {4 e (3 b B d-8 A b e+5 a B e) \sqrt {a+b x}}{5 b (b d-a e)^3 (d+e x)^{5/2}}-\frac {16 e (3 b B d-8 A b e+5 a B e) \sqrt {a+b x}}{15 (b d-a e)^4 (d+e x)^{3/2}}-\frac {(16 b e (3 b B d-8 A b e+5 a B e)) \int \frac {1}{\sqrt {a+b x} (d+e x)^{3/2}} \, dx}{15 (b d-a e)^4}\\ &=-\frac {2 (A b-a B)}{3 b (b d-a e) (a+b x)^{3/2} (d+e x)^{5/2}}-\frac {2 (3 b B d-8 A b e+5 a B e)}{3 b (b d-a e)^2 \sqrt {a+b x} (d+e x)^{5/2}}-\frac {4 e (3 b B d-8 A b e+5 a B e) \sqrt {a+b x}}{5 b (b d-a e)^3 (d+e x)^{5/2}}-\frac {16 e (3 b B d-8 A b e+5 a B e) \sqrt {a+b x}}{15 (b d-a e)^4 (d+e x)^{3/2}}-\frac {32 b e (3 b B d-8 A b e+5 a B e) \sqrt {a+b x}}{15 (b d-a e)^5 \sqrt {d+e x}}\\ \end {align*}

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Mathematica [A]
time = 0.28, size = 270, normalized size = 1.10 \begin {gather*} -\frac {2 \left (3 B d e^3 (a+b x)^4-3 A e^4 (a+b x)^4-15 b B d e^2 (a+b x)^3 (d+e x)+20 A b e^3 (a+b x)^3 (d+e x)-5 a B e^3 (a+b x)^3 (d+e x)+45 b^2 B d e (a+b x)^2 (d+e x)^2-90 A b^2 e^2 (a+b x)^2 (d+e x)^2+45 a b B e^2 (a+b x)^2 (d+e x)^2+15 b^3 B d (a+b x) (d+e x)^3-60 A b^3 e (a+b x) (d+e x)^3+45 a b^2 B e (a+b x) (d+e x)^3+5 A b^4 (d+e x)^4-5 a b^3 B (d+e x)^4\right )}{15 (b d-a e)^5 (a+b x)^{3/2} (d+e x)^{5/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(A + B*x)/((a + b*x)^(5/2)*(d + e*x)^(7/2)),x]

[Out]

(-2*(3*B*d*e^3*(a + b*x)^4 - 3*A*e^4*(a + b*x)^4 - 15*b*B*d*e^2*(a + b*x)^3*(d + e*x) + 20*A*b*e^3*(a + b*x)^3
*(d + e*x) - 5*a*B*e^3*(a + b*x)^3*(d + e*x) + 45*b^2*B*d*e*(a + b*x)^2*(d + e*x)^2 - 90*A*b^2*e^2*(a + b*x)^2
*(d + e*x)^2 + 45*a*b*B*e^2*(a + b*x)^2*(d + e*x)^2 + 15*b^3*B*d*(a + b*x)*(d + e*x)^3 - 60*A*b^3*e*(a + b*x)*
(d + e*x)^3 + 45*a*b^2*B*e*(a + b*x)*(d + e*x)^3 + 5*A*b^4*(d + e*x)^4 - 5*a*b^3*B*(d + e*x)^4))/(15*(b*d - a*
e)^5*(a + b*x)^(3/2)*(d + e*x)^(5/2))

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(448\) vs. \(2(216)=432\).
time = 0.13, size = 449, normalized size = 1.83

method	result	size
default	\(-\frac {2 \left (128 A \,b^{4} e^{4} x^{4}-80 B a \,b^{3} e^{4} x^{4}-48 B \,b^{4} d \,e^{3} x^{4}+192 A a \,b^{3} e^{4} x^{3}+320 A \,b^{4} d \,e^{3} x^{3}-120 B \,a^{2} b^{2} e^{4} x^{3}-272 B a \,b^{3} d \,e^{3} x^{3}-120 B \,b^{4} d^{2} e^{2} x^{3}+48 A \,a^{2} b^{2} e^{4} x^{2}+480 A a \,b^{3} d \,e^{3} x^{2}+240 A \,b^{4} d^{2} e^{2} x^{2}-30 B \,a^{3} b \,e^{4} x^{2}-318 B \,a^{2} b^{2} d \,e^{3} x^{2}-330 B a \,b^{3} d^{2} e^{2} x^{2}-90 B \,b^{4} d^{3} e \,x^{2}-8 A \,a^{3} b \,e^{4} x +120 A \,a^{2} b^{2} d \,e^{3} x +360 A a \,b^{3} d^{2} e^{2} x +40 A \,b^{4} d^{3} e x +5 B \,a^{4} e^{4} x -72 B \,a^{3} b d \,e^{3} x -270 B \,a^{2} b^{2} d^{2} e^{2} x -160 B a \,b^{3} d^{3} e x -15 B \,b^{4} d^{4} x +3 A \,a^{4} e^{4}-20 A \,a^{3} b d \,e^{3}+90 A \,a^{2} b^{2} d^{2} e^{2}+60 A a \,b^{3} d^{3} e -5 A \,b^{4} d^{4}+2 B \,a^{4} d \,e^{3}-30 B \,a^{3} b \,d^{2} e^{2}-90 B \,a^{2} b^{2} d^{3} e -10 B a \,b^{3} d^{4}\right )}{15 \left (e x +d \right )^{\frac {5}{2}} \left (b x +a \right )^{\frac {3}{2}} \left (a e -b d \right )^{5}}\)	\(449\)
gosper	\(-\frac {2 \left (128 A \,b^{4} e^{4} x^{4}-80 B a \,b^{3} e^{4} x^{4}-48 B \,b^{4} d \,e^{3} x^{4}+192 A a \,b^{3} e^{4} x^{3}+320 A \,b^{4} d \,e^{3} x^{3}-120 B \,a^{2} b^{2} e^{4} x^{3}-272 B a \,b^{3} d \,e^{3} x^{3}-120 B \,b^{4} d^{2} e^{2} x^{3}+48 A \,a^{2} b^{2} e^{4} x^{2}+480 A a \,b^{3} d \,e^{3} x^{2}+240 A \,b^{4} d^{2} e^{2} x^{2}-30 B \,a^{3} b \,e^{4} x^{2}-318 B \,a^{2} b^{2} d \,e^{3} x^{2}-330 B a \,b^{3} d^{2} e^{2} x^{2}-90 B \,b^{4} d^{3} e \,x^{2}-8 A \,a^{3} b \,e^{4} x +120 A \,a^{2} b^{2} d \,e^{3} x +360 A a \,b^{3} d^{2} e^{2} x +40 A \,b^{4} d^{3} e x +5 B \,a^{4} e^{4} x -72 B \,a^{3} b d \,e^{3} x -270 B \,a^{2} b^{2} d^{2} e^{2} x -160 B a \,b^{3} d^{3} e x -15 B \,b^{4} d^{4} x +3 A \,a^{4} e^{4}-20 A \,a^{3} b d \,e^{3}+90 A \,a^{2} b^{2} d^{2} e^{2}+60 A a \,b^{3} d^{3} e -5 A \,b^{4} d^{4}+2 B \,a^{4} d \,e^{3}-30 B \,a^{3} b \,d^{2} e^{2}-90 B \,a^{2} b^{2} d^{3} e -10 B a \,b^{3} d^{4}\right )}{15 \left (b x +a \right )^{\frac {3}{2}} \left (e x +d \right )^{\frac {5}{2}} \left (a^{5} e^{5}-5 a^{4} b d \,e^{4}+10 a^{3} b^{2} d^{2} e^{3}-10 a^{2} b^{3} d^{3} e^{2}+5 a \,b^{4} d^{4} e -b^{5} d^{5}\right )}\)	\(505\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x+A)/(b*x+a)^(5/2)/(e*x+d)^(7/2),x,method=_RETURNVERBOSE)

[Out]

-2/15*(128*A*b^4*e^4*x^4-80*B*a*b^3*e^4*x^4-48*B*b^4*d*e^3*x^4+192*A*a*b^3*e^4*x^3+320*A*b^4*d*e^3*x^3-120*B*a
^2*b^2*e^4*x^3-272*B*a*b^3*d*e^3*x^3-120*B*b^4*d^2*e^2*x^3+48*A*a^2*b^2*e^4*x^2+480*A*a*b^3*d*e^3*x^2+240*A*b^
4*d^2*e^2*x^2-30*B*a^3*b*e^4*x^2-318*B*a^2*b^2*d*e^3*x^2-330*B*a*b^3*d^2*e^2*x^2-90*B*b^4*d^3*e*x^2-8*A*a^3*b*
e^4*x+120*A*a^2*b^2*d*e^3*x+360*A*a*b^3*d^2*e^2*x+40*A*b^4*d^3*e*x+5*B*a^4*e^4*x-72*B*a^3*b*d*e^3*x-270*B*a^2*
b^2*d^2*e^2*x-160*B*a*b^3*d^3*e*x-15*B*b^4*d^4*x+3*A*a^4*e^4-20*A*a^3*b*d*e^3+90*A*a^2*b^2*d^2*e^2+60*A*a*b^3*
d^3*e-5*A*b^4*d^4+2*B*a^4*d*e^3-30*B*a^3*b*d^2*e^2-90*B*a^2*b^2*d^3*e-10*B*a*b^3*d^4)/(e*x+d)^(5/2)/(b*x+a)^(3
/2)/(a*e-b*d)^5

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Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(b*d-%e*a>0)', see `assume?` fo
r more detai

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 896 vs. \(2 (237) = 474\).
time = 30.71, size = 896, normalized size = 3.64 \begin {gather*} -\frac {2 \, {\left (15 \, B b^{4} d^{4} x + 5 \, {\left (2 \, B a b^{3} + A b^{4}\right )} d^{4} - {\left (3 \, A a^{4} - 16 \, {\left (5 \, B a b^{3} - 8 \, A b^{4}\right )} x^{4} - 24 \, {\left (5 \, B a^{2} b^{2} - 8 \, A a b^{3}\right )} x^{3} - 6 \, {\left (5 \, B a^{3} b - 8 \, A a^{2} b^{2}\right )} x^{2} + {\left (5 \, B a^{4} - 8 \, A a^{3} b\right )} x\right )} e^{4} + 2 \, {\left (24 \, B b^{4} d x^{4} + 8 \, {\left (17 \, B a b^{3} - 20 \, A b^{4}\right )} d x^{3} + 3 \, {\left (53 \, B a^{2} b^{2} - 80 \, A a b^{3}\right )} d x^{2} + 12 \, {\left (3 \, B a^{3} b - 5 \, A a^{2} b^{2}\right )} d x - {\left (B a^{4} - 10 \, A a^{3} b\right )} d\right )} e^{3} + 30 \, {\left (4 \, B b^{4} d^{2} x^{3} + {\left (11 \, B a b^{3} - 8 \, A b^{4}\right )} d^{2} x^{2} + 3 \, {\left (3 \, B a^{2} b^{2} - 4 \, A a b^{3}\right )} d^{2} x + {\left (B a^{3} b - 3 \, A a^{2} b^{2}\right )} d^{2}\right )} e^{2} + 10 \, {\left (9 \, B b^{4} d^{3} x^{2} + 4 \, {\left (4 \, B a b^{3} - A b^{4}\right )} d^{3} x + 3 \, {\left (3 \, B a^{2} b^{2} - 2 \, A a b^{3}\right )} d^{3}\right )} e\right )} \sqrt {b x + a} \sqrt {x e + d}}{15 \, {\left (b^{7} d^{8} x^{2} + 2 \, a b^{6} d^{8} x + a^{2} b^{5} d^{8} - {\left (a^{5} b^{2} x^{5} + 2 \, a^{6} b x^{4} + a^{7} x^{3}\right )} e^{8} + {\left (5 \, a^{4} b^{3} d x^{5} + 7 \, a^{5} b^{2} d x^{4} - a^{6} b d x^{3} - 3 \, a^{7} d x^{2}\right )} e^{7} - {\left (10 \, a^{3} b^{4} d^{2} x^{5} + 5 \, a^{4} b^{3} d^{2} x^{4} - 17 \, a^{5} b^{2} d^{2} x^{3} - 9 \, a^{6} b d^{2} x^{2} + 3 \, a^{7} d^{2} x\right )} e^{6} + {\left (10 \, a^{2} b^{5} d^{3} x^{5} - 10 \, a^{3} b^{4} d^{3} x^{4} - 35 \, a^{4} b^{3} d^{3} x^{3} - a^{5} b^{2} d^{3} x^{2} + 13 \, a^{6} b d^{3} x - a^{7} d^{3}\right )} e^{5} - 5 \, {\left (a b^{6} d^{4} x^{5} - 4 \, a^{2} b^{5} d^{4} x^{4} - 5 \, a^{3} b^{4} d^{4} x^{3} + 5 \, a^{4} b^{3} d^{4} x^{2} + 4 \, a^{5} b^{2} d^{4} x - a^{6} b d^{4}\right )} e^{4} + {\left (b^{7} d^{5} x^{5} - 13 \, a b^{6} d^{5} x^{4} + a^{2} b^{5} d^{5} x^{3} + 35 \, a^{3} b^{4} d^{5} x^{2} + 10 \, a^{4} b^{3} d^{5} x - 10 \, a^{5} b^{2} d^{5}\right )} e^{3} + {\left (3 \, b^{7} d^{6} x^{4} - 9 \, a b^{6} d^{6} x^{3} - 17 \, a^{2} b^{5} d^{6} x^{2} + 5 \, a^{3} b^{4} d^{6} x + 10 \, a^{4} b^{3} d^{6}\right )} e^{2} + {\left (3 \, b^{7} d^{7} x^{3} + a b^{6} d^{7} x^{2} - 7 \, a^{2} b^{5} d^{7} x - 5 \, a^{3} b^{4} d^{7}\right )} e\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)/(b*x+a)^(5/2)/(e*x+d)^(7/2),x, algorithm="fricas")

[Out]

-2/15*(15*B*b^4*d^4*x + 5*(2*B*a*b^3 + A*b^4)*d^4 - (3*A*a^4 - 16*(5*B*a*b^3 - 8*A*b^4)*x^4 - 24*(5*B*a^2*b^2
- 8*A*a*b^3)*x^3 - 6*(5*B*a^3*b - 8*A*a^2*b^2)*x^2 + (5*B*a^4 - 8*A*a^3*b)*x)*e^4 + 2*(24*B*b^4*d*x^4 + 8*(17*
B*a*b^3 - 20*A*b^4)*d*x^3 + 3*(53*B*a^2*b^2 - 80*A*a*b^3)*d*x^2 + 12*(3*B*a^3*b - 5*A*a^2*b^2)*d*x - (B*a^4 -
10*A*a^3*b)*d)*e^3 + 30*(4*B*b^4*d^2*x^3 + (11*B*a*b^3 - 8*A*b^4)*d^2*x^2 + 3*(3*B*a^2*b^2 - 4*A*a*b^3)*d^2*x
+ (B*a^3*b - 3*A*a^2*b^2)*d^2)*e^2 + 10*(9*B*b^4*d^3*x^2 + 4*(4*B*a*b^3 - A*b^4)*d^3*x + 3*(3*B*a^2*b^2 - 2*A*
a*b^3)*d^3)*e)*sqrt(b*x + a)*sqrt(x*e + d)/(b^7*d^8*x^2 + 2*a*b^6*d^8*x + a^2*b^5*d^8 - (a^5*b^2*x^5 + 2*a^6*b
*x^4 + a^7*x^3)*e^8 + (5*a^4*b^3*d*x^5 + 7*a^5*b^2*d*x^4 - a^6*b*d*x^3 - 3*a^7*d*x^2)*e^7 - (10*a^3*b^4*d^2*x^
5 + 5*a^4*b^3*d^2*x^4 - 17*a^5*b^2*d^2*x^3 - 9*a^6*b*d^2*x^2 + 3*a^7*d^2*x)*e^6 + (10*a^2*b^5*d^3*x^5 - 10*a^3
*b^4*d^3*x^4 - 35*a^4*b^3*d^3*x^3 - a^5*b^2*d^3*x^2 + 13*a^6*b*d^3*x - a^7*d^3)*e^5 - 5*(a*b^6*d^4*x^5 - 4*a^2
*b^5*d^4*x^4 - 5*a^3*b^4*d^4*x^3 + 5*a^4*b^3*d^4*x^2 + 4*a^5*b^2*d^4*x - a^6*b*d^4)*e^4 + (b^7*d^5*x^5 - 13*a*
b^6*d^5*x^4 + a^2*b^5*d^5*x^3 + 35*a^3*b^4*d^5*x^2 + 10*a^4*b^3*d^5*x - 10*a^5*b^2*d^5)*e^3 + (3*b^7*d^6*x^4 -
9*a*b^6*d^6*x^3 - 17*a^2*b^5*d^6*x^2 + 5*a^3*b^4*d^6*x + 10*a^4*b^3*d^6)*e^2 + (3*b^7*d^7*x^3 + a*b^6*d^7*x^2
- 7*a^2*b^5*d^7*x - 5*a^3*b^4*d^7)*e)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {A + B x}{\left (a + b x\right )^{\frac {5}{2}} \left (d + e x\right )^{\frac {7}{2}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)/(b*x+a)**(5/2)/(e*x+d)**(7/2),x)

[Out]

Integral((A + B*x)/((a + b*x)**(5/2)*(d + e*x)**(7/2)), x)

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 1936 vs. \(2 (237) = 474\).
time = 1.58, size = 1936, normalized size = 7.87 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)/(b*x+a)^(5/2)/(e*x+d)^(7/2),x, algorithm="giac")

[Out]

-2/15*((b*x + a)*((33*B*b^15*d^8*e^5 - 191*B*a*b^14*d^7*e^6 - 73*A*b^15*d^7*e^6 + 413*B*a^2*b^13*d^6*e^7 + 511
*A*a*b^14*d^6*e^7 - 315*B*a^3*b^12*d^5*e^8 - 1533*A*a^2*b^13*d^5*e^8 - 245*B*a^4*b^11*d^4*e^9 + 2555*A*a^3*b^1
2*d^4*e^9 + 707*B*a^5*b^10*d^3*e^10 - 2555*A*a^4*b^11*d^3*e^10 - 609*B*a^6*b^9*d^2*e^11 + 1533*A*a^5*b^10*d^2*
e^11 + 247*B*a^7*b^8*d*e^12 - 511*A*a^6*b^9*d*e^12 - 40*B*a^8*b^7*e^13 + 73*A*a^7*b^8*e^13)*(b*x + a)/(b^14*d^
12*abs(b)*e^2 - 12*a*b^13*d^11*abs(b)*e^3 + 66*a^2*b^12*d^10*abs(b)*e^4 - 220*a^3*b^11*d^9*abs(b)*e^5 + 495*a^
4*b^10*d^8*abs(b)*e^6 - 792*a^5*b^9*d^7*abs(b)*e^7 + 924*a^6*b^8*d^6*abs(b)*e^8 - 792*a^7*b^7*d^5*abs(b)*e^9 +
495*a^8*b^6*d^4*abs(b)*e^10 - 220*a^9*b^5*d^3*abs(b)*e^11 + 66*a^10*b^4*d^2*abs(b)*e^12 - 12*a^11*b^3*d*abs(b
)*e^13 + a^12*b^2*abs(b)*e^14) + 5*(15*B*b^16*d^9*e^4 - 103*B*a*b^15*d^8*e^5 - 32*A*b^16*d^8*e^5 + 284*B*a^2*b
^14*d^7*e^6 + 256*A*a*b^15*d^7*e^6 - 364*B*a^3*b^13*d^6*e^7 - 896*A*a^2*b^14*d^6*e^7 + 98*B*a^4*b^12*d^5*e^8 +
1792*A*a^3*b^13*d^5*e^8 + 350*B*a^5*b^11*d^4*e^9 - 2240*A*a^4*b^12*d^4*e^9 - 532*B*a^6*b^10*d^3*e^10 + 1792*A
*a^5*b^11*d^3*e^10 + 356*B*a^7*b^9*d^2*e^11 - 896*A*a^6*b^10*d^2*e^11 - 121*B*a^8*b^8*d*e^12 + 256*A*a^7*b^9*d
*e^12 + 17*B*a^9*b^7*e^13 - 32*A*a^8*b^8*e^13)/(b^14*d^12*abs(b)*e^2 - 12*a*b^13*d^11*abs(b)*e^3 + 66*a^2*b^12
*d^10*abs(b)*e^4 - 220*a^3*b^11*d^9*abs(b)*e^5 + 495*a^4*b^10*d^8*abs(b)*e^6 - 792*a^5*b^9*d^7*abs(b)*e^7 + 92
4*a^6*b^8*d^6*abs(b)*e^8 - 792*a^7*b^7*d^5*abs(b)*e^9 + 495*a^8*b^6*d^4*abs(b)*e^10 - 220*a^9*b^5*d^3*abs(b)*e
^11 + 66*a^10*b^4*d^2*abs(b)*e^12 - 12*a^11*b^3*d*abs(b)*e^13 + a^12*b^2*abs(b)*e^14)) + 45*(B*b^17*d^10*e^3 -
8*B*a*b^16*d^9*e^4 - 2*A*b^17*d^9*e^4 + 27*B*a^2*b^15*d^8*e^5 + 18*A*a*b^16*d^8*e^5 - 48*B*a^3*b^14*d^7*e^6 -
72*A*a^2*b^15*d^7*e^6 + 42*B*a^4*b^13*d^6*e^7 + 168*A*a^3*b^14*d^6*e^7 - 252*A*a^4*b^13*d^5*e^8 - 42*B*a^6*b^
11*d^4*e^9 + 252*A*a^5*b^12*d^4*e^9 + 48*B*a^7*b^10*d^3*e^10 - 168*A*a^6*b^11*d^3*e^10 - 27*B*a^8*b^9*d^2*e^11
+ 72*A*a^7*b^10*d^2*e^11 + 8*B*a^9*b^8*d*e^12 - 18*A*a^8*b^9*d*e^12 - B*a^10*b^7*e^13 + 2*A*a^9*b^8*e^13)/(b^
14*d^12*abs(b)*e^2 - 12*a*b^13*d^11*abs(b)*e^3 + 66*a^2*b^12*d^10*abs(b)*e^4 - 220*a^3*b^11*d^9*abs(b)*e^5 + 4
95*a^4*b^10*d^8*abs(b)*e^6 - 792*a^5*b^9*d^7*abs(b)*e^7 + 924*a^6*b^8*d^6*abs(b)*e^8 - 792*a^7*b^7*d^5*abs(b)*
e^9 + 495*a^8*b^6*d^4*abs(b)*e^10 - 220*a^9*b^5*d^3*abs(b)*e^11 + 66*a^10*b^4*d^2*abs(b)*e^12 - 12*a^11*b^3*d*
abs(b)*e^13 + a^12*b^2*abs(b)*e^14))*sqrt(b*x + a)/(b^2*d + (b*x + a)*b*e - a*b*e)^(5/2) - 4/3*(3*B*b^(17/2)*d
^3*e^(1/2) + 2*B*a*b^(15/2)*d^2*e^(3/2) - 11*A*b^(17/2)*d^2*e^(3/2) - 6*(sqrt(b*x + a)*sqrt(b)*e^(1/2) - sqrt(
b^2*d + (b*x + a)*b*e - a*b*e))^2*B*b^(13/2)*d^2*e^(1/2) - 13*B*a^2*b^(13/2)*d*e^(5/2) + 22*A*a*b^(15/2)*d*e^(
5/2) - 12*(sqrt(b*x + a)*sqrt(b)*e^(1/2) - sqrt(b^2*d + (b*x + a)*b*e - a*b*e))^2*B*a*b^(11/2)*d*e^(3/2) + 24*
(sqrt(b*x + a)*sqrt(b)*e^(1/2) - sqrt(b^2*d + (b*x + a)*b*e - a*b*e))^2*A*b^(13/2)*d*e^(3/2) + 3*(sqrt(b*x + a
)*sqrt(b)*e^(1/2) - sqrt(b^2*d + (b*x + a)*b*e - a*b*e))^4*B*b^(9/2)*d*e^(1/2) + 8*B*a^3*b^(11/2)*e^(7/2) - 11
*A*a^2*b^(13/2)*e^(7/2) + 18*(sqrt(b*x + a)*sqrt(b)*e^(1/2) - sqrt(b^2*d + (b*x + a)*b*e - a*b*e))^2*B*a^2*b^(
9/2)*e^(5/2) - 24*(sqrt(b*x + a)*sqrt(b)*e^(1/2) - sqrt(b^2*d + (b*x + a)*b*e - a*b*e))^2*A*a*b^(11/2)*e^(5/2)
+ 6*(sqrt(b*x + a)*sqrt(b)*e^(1/2) - sqrt(b^2*d + (b*x + a)*b*e - a*b*e))^4*B*a*b^(7/2)*e^(3/2) - 9*(sqrt(b*x
+ a)*sqrt(b)*e^(1/2) - sqrt(b^2*d + (b*x + a)*b*e - a*b*e))^4*A*b^(9/2)*e^(3/2))/((b^4*d^4*abs(b) - 4*a*b^3*d
^3*abs(b)*e + 6*a^2*b^2*d^2*abs(b)*e^2 - 4*a^3*b*d*abs(b)*e^3 + a^4*abs(b)*e^4)*(b^2*d - a*b*e - (sqrt(b*x + a
)*sqrt(b)*e^(1/2) - sqrt(b^2*d + (b*x + a)*b*e - a*b*e))^2)^3)

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Mupad [B]
time = 2.81, size = 444, normalized size = 1.80 \begin {gather*} \frac {\sqrt {d+e\,x}\,\left (\frac {32\,b^2\,x^4\,\left (5\,B\,a\,e-8\,A\,b\,e+3\,B\,b\,d\right )}{15\,{\left (a\,e-b\,d\right )}^5}+\frac {-4\,B\,a^4\,d\,e^3-6\,A\,a^4\,e^4+60\,B\,a^3\,b\,d^2\,e^2+40\,A\,a^3\,b\,d\,e^3+180\,B\,a^2\,b^2\,d^3\,e-180\,A\,a^2\,b^2\,d^2\,e^2+20\,B\,a\,b^3\,d^4-120\,A\,a\,b^3\,d^3\,e+10\,A\,b^4\,d^4}{15\,b\,e^3\,{\left (a\,e-b\,d\right )}^5}+\frac {4\,x^2\,\left (a^2\,e^2+10\,a\,b\,d\,e+5\,b^2\,d^2\right )\,\left (5\,B\,a\,e-8\,A\,b\,e+3\,B\,b\,d\right )}{5\,e^2\,{\left (a\,e-b\,d\right )}^5}+\frac {2\,x\,\left (5\,B\,a\,e-8\,A\,b\,e+3\,B\,b\,d\right )\,\left (-a^3\,e^3+15\,a^2\,b\,d\,e^2+45\,a\,b^2\,d^2\,e+5\,b^3\,d^3\right )}{15\,b\,e^3\,{\left (a\,e-b\,d\right )}^5}+\frac {16\,b\,x^3\,\left (3\,a\,e+5\,b\,d\right )\,\left (5\,B\,a\,e-8\,A\,b\,e+3\,B\,b\,d\right )}{15\,e\,{\left (a\,e-b\,d\right )}^5}\right )}{x^4\,\sqrt {a+b\,x}+\frac {a\,d^3\,\sqrt {a+b\,x}}{b\,e^3}+\frac {x^3\,\left (a\,e+3\,b\,d\right )\,\sqrt {a+b\,x}}{b\,e}+\frac {3\,d\,x^2\,\left (a\,e+b\,d\right )\,\sqrt {a+b\,x}}{b\,e^2}+\frac {d^2\,x\,\left (3\,a\,e+b\,d\right )\,\sqrt {a+b\,x}}{b\,e^3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((A + B*x)/((a + b*x)^(5/2)*(d + e*x)^(7/2)),x)

[Out]

((d + e*x)^(1/2)*((32*b^2*x^4*(5*B*a*e - 8*A*b*e + 3*B*b*d))/(15*(a*e - b*d)^5) + (10*A*b^4*d^4 - 6*A*a^4*e^4
+ 20*B*a*b^3*d^4 - 4*B*a^4*d*e^3 + 180*B*a^2*b^2*d^3*e + 60*B*a^3*b*d^2*e^2 - 180*A*a^2*b^2*d^2*e^2 - 120*A*a*
b^3*d^3*e + 40*A*a^3*b*d*e^3)/(15*b*e^3*(a*e - b*d)^5) + (4*x^2*(a^2*e^2 + 5*b^2*d^2 + 10*a*b*d*e)*(5*B*a*e -
8*A*b*e + 3*B*b*d))/(5*e^2*(a*e - b*d)^5) + (2*x*(5*B*a*e - 8*A*b*e + 3*B*b*d)*(5*b^3*d^3 - a^3*e^3 + 45*a*b^2
*d^2*e + 15*a^2*b*d*e^2))/(15*b*e^3*(a*e - b*d)^5) + (16*b*x^3*(3*a*e + 5*b*d)*(5*B*a*e - 8*A*b*e + 3*B*b*d))/
(15*e*(a*e - b*d)^5)))/(x^4*(a + b*x)^(1/2) + (a*d^3*(a + b*x)^(1/2))/(b*e^3) + (x^3*(a*e + 3*b*d)*(a + b*x)^(
1/2))/(b*e) + (3*d*x^2*(a*e + b*d)*(a + b*x)^(1/2))/(b*e^2) + (d^2*x*(3*a*e + b*d)*(a + b*x)^(1/2))/(b*e^3))

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