3.23.51 \(\int \frac {A+B x}{(a+b x)^{3/2} (d+e x)^{7/2}} \, dx\) [2251]

Optimal. Leaf size=187 \[ -\frac {2 (A b-a B)}{b (b d-a e) \sqrt {a+b x} (d+e x)^{5/2}}+\frac {2 (b B d-6 A b e+5 a B e) \sqrt {a+b x}}{5 b (b d-a e)^2 (d+e x)^{5/2}}+\frac {8 (b B d-6 A b e+5 a B e) \sqrt {a+b x}}{15 (b d-a e)^3 (d+e x)^{3/2}}+\frac {16 b (b B d-6 A b e+5 a B e) \sqrt {a+b x}}{15 (b d-a e)^4 \sqrt {d+e x}} \]

[Out]

-2*(A*b-B*a)/b/(-a*e+b*d)/(e*x+d)^(5/2)/(b*x+a)^(1/2)+2/5*(-6*A*b*e+5*B*a*e+B*b*d)*(b*x+a)^(1/2)/b/(-a*e+b*d)^
2/(e*x+d)^(5/2)+8/15*(-6*A*b*e+5*B*a*e+B*b*d)*(b*x+a)^(1/2)/(-a*e+b*d)^3/(e*x+d)^(3/2)+16/15*b*(-6*A*b*e+5*B*a
*e+B*b*d)*(b*x+a)^(1/2)/(-a*e+b*d)^4/(e*x+d)^(1/2)

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Rubi [A]
time = 0.07, antiderivative size = 187, normalized size of antiderivative = 1.00, number of steps used = 4, 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)}{b \sqrt {a+b x} (d+e x)^{5/2} (b d-a e)}+\frac {16 b \sqrt {a+b x} (5 a B e-6 A b e+b B d)}{15 \sqrt {d+e x} (b d-a e)^4}+\frac {8 \sqrt {a+b x} (5 a B e-6 A b e+b B d)}{15 (d+e x)^{3/2} (b d-a e)^3}+\frac {2 \sqrt {a+b x} (5 a B e-6 A b e+b B d)}{5 b (d+e x)^{5/2} (b d-a e)^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*(A*b - a*B))/(b*(b*d - a*e)*Sqrt[a + b*x]*(d + e*x)^(5/2)) + (2*(b*B*d - 6*A*b*e + 5*a*B*e)*Sqrt[a + b*x])
/(5*b*(b*d - a*e)^2*(d + e*x)^(5/2)) + (8*(b*B*d - 6*A*b*e + 5*a*B*e)*Sqrt[a + b*x])/(15*(b*d - a*e)^3*(d + e*
x)^(3/2)) + (16*b*(b*B*d - 6*A*b*e + 5*a*B*e)*Sqrt[a + b*x])/(15*(b*d - a*e)^4*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)^{3/2} (d+e x)^{7/2}} \, dx &=-\frac {2 (A b-a B)}{b (b d-a e) \sqrt {a+b x} (d+e x)^{5/2}}+\frac {(b B d-6 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)}\\ &=-\frac {2 (A b-a B)}{b (b d-a e) \sqrt {a+b x} (d+e x)^{5/2}}+\frac {2 (b B d-6 A b e+5 a B e) \sqrt {a+b x}}{5 b (b d-a e)^2 (d+e x)^{5/2}}+\frac {(4 (b B d-6 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)^2}\\ &=-\frac {2 (A b-a B)}{b (b d-a e) \sqrt {a+b x} (d+e x)^{5/2}}+\frac {2 (b B d-6 A b e+5 a B e) \sqrt {a+b x}}{5 b (b d-a e)^2 (d+e x)^{5/2}}+\frac {8 (b B d-6 A b e+5 a B e) \sqrt {a+b x}}{15 (b d-a e)^3 (d+e x)^{3/2}}+\frac {(8 b (b B d-6 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)^3}\\ &=-\frac {2 (A b-a B)}{b (b d-a e) \sqrt {a+b x} (d+e x)^{5/2}}+\frac {2 (b B d-6 A b e+5 a B e) \sqrt {a+b x}}{5 b (b d-a e)^2 (d+e x)^{5/2}}+\frac {8 (b B d-6 A b e+5 a B e) \sqrt {a+b x}}{15 (b d-a e)^3 (d+e x)^{3/2}}+\frac {16 b (b B d-6 A b e+5 a B e) \sqrt {a+b x}}{15 (b d-a e)^4 \sqrt {d+e x}}\\ \end {align*}

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Mathematica [A]
time = 0.25, size = 199, normalized size = 1.06 \begin {gather*} -\frac {2 \left (-3 B d e^2 (a+b x)^3+3 A e^3 (a+b x)^3+10 b B d e (a+b x)^2 (d+e x)-15 A b e^2 (a+b x)^2 (d+e x)+5 a B e^2 (a+b x)^2 (d+e x)-15 b^2 B d (a+b x) (d+e x)^2+45 A b^2 e (a+b x) (d+e x)^2-30 a b B e (a+b x) (d+e x)^2+15 A b^3 (d+e x)^3-15 a b^2 B (d+e x)^3\right )}{15 (b d-a e)^4 \sqrt {a+b x} (d+e x)^{5/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*(-3*B*d*e^2*(a + b*x)^3 + 3*A*e^3*(a + b*x)^3 + 10*b*B*d*e*(a + b*x)^2*(d + e*x) - 15*A*b*e^2*(a + b*x)^2*
(d + e*x) + 5*a*B*e^2*(a + b*x)^2*(d + e*x) - 15*b^2*B*d*(a + b*x)*(d + e*x)^2 + 45*A*b^2*e*(a + b*x)*(d + e*x
)^2 - 30*a*b*B*e*(a + b*x)*(d + e*x)^2 + 15*A*b^3*(d + e*x)^3 - 15*a*b^2*B*(d + e*x)^3))/(15*(b*d - a*e)^4*Sqr
t[a + b*x]*(d + e*x)^(5/2))

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Maple [A]
time = 0.10, size = 281, normalized size = 1.50

method	result	size
default	\(-\frac {2 \left (48 A \,b^{3} e^{3} x^{3}-40 B a \,b^{2} e^{3} x^{3}-8 B \,b^{3} d \,e^{2} x^{3}+24 A a \,b^{2} e^{3} x^{2}+120 A \,b^{3} d \,e^{2} x^{2}-20 B \,a^{2} b \,e^{3} x^{2}-104 B a \,b^{2} d \,e^{2} x^{2}-20 B \,b^{3} d^{2} e \,x^{2}-6 A \,a^{2} b \,e^{3} x +60 A a \,b^{2} d \,e^{2} x +90 A \,b^{3} d^{2} e x +5 B \,a^{3} e^{3} x -49 B \,a^{2} b d \,e^{2} x -85 B a \,b^{2} d^{2} e x -15 B \,b^{3} d^{3} x +3 a^{3} A \,e^{3}-15 A \,a^{2} b d \,e^{2}+45 A a \,b^{2} d^{2} e +15 A \,b^{3} d^{3}+2 B \,a^{3} d \,e^{2}-20 B \,a^{2} b \,d^{2} e -30 B a \,b^{2} d^{3}\right )}{15 \left (e x +d \right )^{\frac {5}{2}} \sqrt {b x +a}\, \left (a e -b d \right )^{4}}\)	\(281\)
gosper	\(-\frac {2 \left (48 A \,b^{3} e^{3} x^{3}-40 B a \,b^{2} e^{3} x^{3}-8 B \,b^{3} d \,e^{2} x^{3}+24 A a \,b^{2} e^{3} x^{2}+120 A \,b^{3} d \,e^{2} x^{2}-20 B \,a^{2} b \,e^{3} x^{2}-104 B a \,b^{2} d \,e^{2} x^{2}-20 B \,b^{3} d^{2} e \,x^{2}-6 A \,a^{2} b \,e^{3} x +60 A a \,b^{2} d \,e^{2} x +90 A \,b^{3} d^{2} e x +5 B \,a^{3} e^{3} x -49 B \,a^{2} b d \,e^{2} x -85 B a \,b^{2} d^{2} e x -15 B \,b^{3} d^{3} x +3 a^{3} A \,e^{3}-15 A \,a^{2} b d \,e^{2}+45 A a \,b^{2} d^{2} e +15 A \,b^{3} d^{3}+2 B \,a^{3} d \,e^{2}-20 B \,a^{2} b \,d^{2} e -30 B a \,b^{2} d^{3}\right )}{15 \sqrt {b x +a}\, \left (e x +d \right )^{\frac {5}{2}} \left (e^{4} a^{4}-4 b \,e^{3} d \,a^{3}+6 b^{2} e^{2} d^{2} a^{2}-4 a \,b^{3} d^{3} e +b^{4} d^{4}\right )}\)	\(322\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x+A)/(b*x+a)^(3/2)/(e*x+d)^(7/2),x,method=_RETURNVERBOSE)

[Out]

-2/15*(48*A*b^3*e^3*x^3-40*B*a*b^2*e^3*x^3-8*B*b^3*d*e^2*x^3+24*A*a*b^2*e^3*x^2+120*A*b^3*d*e^2*x^2-20*B*a^2*b
*e^3*x^2-104*B*a*b^2*d*e^2*x^2-20*B*b^3*d^2*e*x^2-6*A*a^2*b*e^3*x+60*A*a*b^2*d*e^2*x+90*A*b^3*d^2*e*x+5*B*a^3*
e^3*x-49*B*a^2*b*d*e^2*x-85*B*a*b^2*d^2*e*x-15*B*b^3*d^3*x+3*A*a^3*e^3-15*A*a^2*b*d*e^2+45*A*a*b^2*d^2*e+15*A*
b^3*d^3+2*B*a^3*d*e^2-20*B*a^2*b*d^2*e-30*B*a*b^2*d^3)/(e*x+d)^(5/2)/(b*x+a)^(1/2)/(a*e-b*d)^4

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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)^(3/2)/(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. 573 vs. \(2 (179) = 358\).
time = 5.94, size = 573, normalized size = 3.06 \begin {gather*} \frac {2 \, {\left (15 \, B b^{3} d^{3} x + 15 \, {\left (2 \, B a b^{2} - A b^{3}\right )} d^{3} - {\left (3 \, A a^{3} - 8 \, {\left (5 \, B a b^{2} - 6 \, A b^{3}\right )} x^{3} - 4 \, {\left (5 \, B a^{2} b - 6 \, A a b^{2}\right )} x^{2} + {\left (5 \, B a^{3} - 6 \, A a^{2} b\right )} x\right )} e^{3} + {\left (8 \, B b^{3} d x^{3} + 8 \, {\left (13 \, B a b^{2} - 15 \, A b^{3}\right )} d x^{2} + {\left (49 \, B a^{2} b - 60 \, A a b^{2}\right )} d x - {\left (2 \, B a^{3} - 15 \, A a^{2} b\right )} d\right )} e^{2} + 5 \, {\left (4 \, B b^{3} d^{2} x^{2} + {\left (17 \, B a b^{2} - 18 \, A b^{3}\right )} d^{2} x + {\left (4 \, B a^{2} b - 9 \, A a b^{2}\right )} d^{2}\right )} e\right )} \sqrt {b x + a} \sqrt {x e + d}}{15 \, {\left (b^{5} d^{7} x + a b^{4} d^{7} + {\left (a^{4} b x^{4} + a^{5} x^{3}\right )} e^{7} - {\left (4 \, a^{3} b^{2} d x^{4} + a^{4} b d x^{3} - 3 \, a^{5} d x^{2}\right )} e^{6} + 3 \, {\left (2 \, a^{2} b^{3} d^{2} x^{4} - 2 \, a^{3} b^{2} d^{2} x^{3} - 3 \, a^{4} b d^{2} x^{2} + a^{5} d^{2} x\right )} e^{5} - {\left (4 \, a b^{4} d^{3} x^{4} - 14 \, a^{2} b^{3} d^{3} x^{3} - 6 \, a^{3} b^{2} d^{3} x^{2} + 11 \, a^{4} b d^{3} x - a^{5} d^{3}\right )} e^{4} + {\left (b^{5} d^{4} x^{4} - 11 \, a b^{4} d^{4} x^{3} + 6 \, a^{2} b^{3} d^{4} x^{2} + 14 \, a^{3} b^{2} d^{4} x - 4 \, a^{4} b d^{4}\right )} e^{3} + 3 \, {\left (b^{5} d^{5} x^{3} - 3 \, a b^{4} d^{5} x^{2} - 2 \, a^{2} b^{3} d^{5} x + 2 \, a^{3} b^{2} d^{5}\right )} e^{2} + {\left (3 \, b^{5} d^{6} x^{2} - a b^{4} d^{6} x - 4 \, a^{2} b^{3} d^{6}\right )} e\right )}} \end {gather*}

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]

2/15*(15*B*b^3*d^3*x + 15*(2*B*a*b^2 - A*b^3)*d^3 - (3*A*a^3 - 8*(5*B*a*b^2 - 6*A*b^3)*x^3 - 4*(5*B*a^2*b - 6*
A*a*b^2)*x^2 + (5*B*a^3 - 6*A*a^2*b)*x)*e^3 + (8*B*b^3*d*x^3 + 8*(13*B*a*b^2 - 15*A*b^3)*d*x^2 + (49*B*a^2*b -
60*A*a*b^2)*d*x - (2*B*a^3 - 15*A*a^2*b)*d)*e^2 + 5*(4*B*b^3*d^2*x^2 + (17*B*a*b^2 - 18*A*b^3)*d^2*x + (4*B*a
^2*b - 9*A*a*b^2)*d^2)*e)*sqrt(b*x + a)*sqrt(x*e + d)/(b^5*d^7*x + a*b^4*d^7 + (a^4*b*x^4 + a^5*x^3)*e^7 - (4*
a^3*b^2*d*x^4 + a^4*b*d*x^3 - 3*a^5*d*x^2)*e^6 + 3*(2*a^2*b^3*d^2*x^4 - 2*a^3*b^2*d^2*x^3 - 3*a^4*b*d^2*x^2 +
a^5*d^2*x)*e^5 - (4*a*b^4*d^3*x^4 - 14*a^2*b^3*d^3*x^3 - 6*a^3*b^2*d^3*x^2 + 11*a^4*b*d^3*x - a^5*d^3)*e^4 + (
b^5*d^4*x^4 - 11*a*b^4*d^4*x^3 + 6*a^2*b^3*d^4*x^2 + 14*a^3*b^2*d^4*x - 4*a^4*b*d^4)*e^3 + 3*(b^5*d^5*x^3 - 3*
a*b^4*d^5*x^2 - 2*a^2*b^3*d^5*x + 2*a^3*b^2*d^5)*e^2 + (3*b^5*d^6*x^2 - a*b^4*d^6*x - 4*a^2*b^3*d^6)*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 {3}{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)**(3/2)/(e*x+d)**(7/2),x)

[Out]

Integral((A + B*x)/((a + b*x)**(3/2)*(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. 1266 vs. \(2 (179) = 358\).
time = 0.74, size = 1266, normalized size = 6.77 \begin {gather*} \frac {4 \, {\left (B^{2} a^{2} b^{7} e - 2 \, A B a b^{8} e + A^{2} b^{9} e\right )}}{{\left (B a b^{\frac {11}{2}} d e^{\frac {1}{2}} - A b^{\frac {13}{2}} d e^{\frac {1}{2}} - B a^{2} b^{\frac {9}{2}} e^{\frac {3}{2}} + A a b^{\frac {11}{2}} e^{\frac {3}{2}} - {\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 )}^{2} B a b^{\frac {7}{2}} e^{\frac {1}{2}} + {\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 )}^{2} A b^{\frac {9}{2}} e^{\frac {1}{2}}\right )} {\left (b^{3} d^{3} {\left | b \right |} - 3 \, a b^{2} d^{2} {\left | b \right |} e + 3 \, a^{2} b d {\left | b \right |} e^{2} - a^{3} {\left | b \right |} e^{3}\right )}} + \frac {2 \, {\left ({\left (b x + a\right )} {\left (\frac {{\left (8 \, B b^{13} d^{6} e^{4} - 15 \, B a b^{12} d^{5} e^{5} - 33 \, A b^{13} d^{5} e^{5} - 45 \, B a^{2} b^{11} d^{4} e^{6} + 165 \, A a b^{12} d^{4} e^{6} + 170 \, B a^{3} b^{10} d^{3} e^{7} - 330 \, A a^{2} b^{11} d^{3} e^{7} - 210 \, B a^{4} b^{9} d^{2} e^{8} + 330 \, A a^{3} b^{10} d^{2} e^{8} + 117 \, B a^{5} b^{8} d e^{9} - 165 \, A a^{4} b^{9} d e^{9} - 25 \, B a^{6} b^{7} e^{10} + 33 \, A a^{5} b^{8} e^{10}\right )} {\left (b x + a\right )}}{b^{11} d^{9} {\left | b \right |} e^{2} - 9 \, a b^{10} d^{8} {\left | b \right |} e^{3} + 36 \, a^{2} b^{9} d^{7} {\left | b \right |} e^{4} - 84 \, a^{3} b^{8} d^{6} {\left | b \right |} e^{5} + 126 \, a^{4} b^{7} d^{5} {\left | b \right |} e^{6} - 126 \, a^{5} b^{6} d^{4} {\left | b \right |} e^{7} + 84 \, a^{6} b^{5} d^{3} {\left | b \right |} e^{8} - 36 \, a^{7} b^{4} d^{2} {\left | b \right |} e^{9} + 9 \, a^{8} b^{3} d {\left | b \right |} e^{10} - a^{9} b^{2} {\left | b \right |} e^{11}} + \frac {5 \, {\left (4 \, B b^{14} d^{7} e^{3} - 13 \, B a b^{13} d^{6} e^{4} - 15 \, A b^{14} d^{6} e^{4} - 6 \, B a^{2} b^{12} d^{5} e^{5} + 90 \, A a b^{13} d^{5} e^{5} + 85 \, B a^{3} b^{11} d^{4} e^{6} - 225 \, A a^{2} b^{12} d^{4} e^{6} - 160 \, B a^{4} b^{10} d^{3} e^{7} + 300 \, A a^{3} b^{11} d^{3} e^{7} + 141 \, B a^{5} b^{9} d^{2} e^{8} - 225 \, A a^{4} b^{10} d^{2} e^{8} - 62 \, B a^{6} b^{8} d e^{9} + 90 \, A a^{5} b^{9} d e^{9} + 11 \, B a^{7} b^{7} e^{10} - 15 \, A a^{6} b^{8} e^{10}\right )}}{b^{11} d^{9} {\left | b \right |} e^{2} - 9 \, a b^{10} d^{8} {\left | b \right |} e^{3} + 36 \, a^{2} b^{9} d^{7} {\left | b \right |} e^{4} - 84 \, a^{3} b^{8} d^{6} {\left | b \right |} e^{5} + 126 \, a^{4} b^{7} d^{5} {\left | b \right |} e^{6} - 126 \, a^{5} b^{6} d^{4} {\left | b \right |} e^{7} + 84 \, a^{6} b^{5} d^{3} {\left | b \right |} e^{8} - 36 \, a^{7} b^{4} d^{2} {\left | b \right |} e^{9} + 9 \, a^{8} b^{3} d {\left | b \right |} e^{10} - a^{9} b^{2} {\left | b \right |} e^{11}}\right )} + \frac {15 \, {\left (B b^{15} d^{8} e^{2} - 5 \, B a b^{14} d^{7} e^{3} - 3 \, A b^{15} d^{7} e^{3} + 7 \, B a^{2} b^{13} d^{6} e^{4} + 21 \, A a b^{14} d^{6} e^{4} + 7 \, B a^{3} b^{12} d^{5} e^{5} - 63 \, A a^{2} b^{13} d^{5} e^{5} - 35 \, B a^{4} b^{11} d^{4} e^{6} + 105 \, A a^{3} b^{12} d^{4} e^{6} + 49 \, B a^{5} b^{10} d^{3} e^{7} - 105 \, A a^{4} b^{11} d^{3} e^{7} - 35 \, B a^{6} b^{9} d^{2} e^{8} + 63 \, A a^{5} b^{10} d^{2} e^{8} + 13 \, B a^{7} b^{8} d e^{9} - 21 \, A a^{6} b^{9} d e^{9} - 2 \, B a^{8} b^{7} e^{10} + 3 \, A a^{7} b^{8} e^{10}\right )}}{b^{11} d^{9} {\left | b \right |} e^{2} - 9 \, a b^{10} d^{8} {\left | b \right |} e^{3} + 36 \, a^{2} b^{9} d^{7} {\left | b \right |} e^{4} - 84 \, a^{3} b^{8} d^{6} {\left | b \right |} e^{5} + 126 \, a^{4} b^{7} d^{5} {\left | b \right |} e^{6} - 126 \, a^{5} b^{6} d^{4} {\left | b \right |} e^{7} + 84 \, a^{6} b^{5} d^{3} {\left | b \right |} e^{8} - 36 \, a^{7} b^{4} d^{2} {\left | b \right |} e^{9} + 9 \, a^{8} b^{3} d {\left | b \right |} e^{10} - a^{9} b^{2} {\left | b \right |} e^{11}}\right )} \sqrt {b x + a}}{15 \, {\left (b^{2} d + {\left (b x + a\right )} b e - a b e\right )}^{\frac {5}{2}}} \end {gather*}

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]

4*(B^2*a^2*b^7*e - 2*A*B*a*b^8*e + A^2*b^9*e)/((B*a*b^(11/2)*d*e^(1/2) - A*b^(13/2)*d*e^(1/2) - B*a^2*b^(9/2)*
e^(3/2) + A*a*b^(11/2)*e^(3/2) - (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
^(7/2)*e^(1/2) + (sqrt(b*x + a)*sqrt(b)*e^(1/2) - sqrt(b^2*d + (b*x + a)*b*e - a*b*e))^2*A*b^(9/2)*e^(1/2))*(b
^3*d^3*abs(b) - 3*a*b^2*d^2*abs(b)*e + 3*a^2*b*d*abs(b)*e^2 - a^3*abs(b)*e^3)) + 2/15*((b*x + a)*((8*B*b^13*d^
6*e^4 - 15*B*a*b^12*d^5*e^5 - 33*A*b^13*d^5*e^5 - 45*B*a^2*b^11*d^4*e^6 + 165*A*a*b^12*d^4*e^6 + 170*B*a^3*b^1
0*d^3*e^7 - 330*A*a^2*b^11*d^3*e^7 - 210*B*a^4*b^9*d^2*e^8 + 330*A*a^3*b^10*d^2*e^8 + 117*B*a^5*b^8*d*e^9 - 16
5*A*a^4*b^9*d*e^9 - 25*B*a^6*b^7*e^10 + 33*A*a^5*b^8*e^10)*(b*x + a)/(b^11*d^9*abs(b)*e^2 - 9*a*b^10*d^8*abs(b
)*e^3 + 36*a^2*b^9*d^7*abs(b)*e^4 - 84*a^3*b^8*d^6*abs(b)*e^5 + 126*a^4*b^7*d^5*abs(b)*e^6 - 126*a^5*b^6*d^4*a
bs(b)*e^7 + 84*a^6*b^5*d^3*abs(b)*e^8 - 36*a^7*b^4*d^2*abs(b)*e^9 + 9*a^8*b^3*d*abs(b)*e^10 - a^9*b^2*abs(b)*e
^11) + 5*(4*B*b^14*d^7*e^3 - 13*B*a*b^13*d^6*e^4 - 15*A*b^14*d^6*e^4 - 6*B*a^2*b^12*d^5*e^5 + 90*A*a*b^13*d^5*
e^5 + 85*B*a^3*b^11*d^4*e^6 - 225*A*a^2*b^12*d^4*e^6 - 160*B*a^4*b^10*d^3*e^7 + 300*A*a^3*b^11*d^3*e^7 + 141*B
*a^5*b^9*d^2*e^8 - 225*A*a^4*b^10*d^2*e^8 - 62*B*a^6*b^8*d*e^9 + 90*A*a^5*b^9*d*e^9 + 11*B*a^7*b^7*e^10 - 15*A
*a^6*b^8*e^10)/(b^11*d^9*abs(b)*e^2 - 9*a*b^10*d^8*abs(b)*e^3 + 36*a^2*b^9*d^7*abs(b)*e^4 - 84*a^3*b^8*d^6*abs
(b)*e^5 + 126*a^4*b^7*d^5*abs(b)*e^6 - 126*a^5*b^6*d^4*abs(b)*e^7 + 84*a^6*b^5*d^3*abs(b)*e^8 - 36*a^7*b^4*d^2
*abs(b)*e^9 + 9*a^8*b^3*d*abs(b)*e^10 - a^9*b^2*abs(b)*e^11)) + 15*(B*b^15*d^8*e^2 - 5*B*a*b^14*d^7*e^3 - 3*A*
b^15*d^7*e^3 + 7*B*a^2*b^13*d^6*e^4 + 21*A*a*b^14*d^6*e^4 + 7*B*a^3*b^12*d^5*e^5 - 63*A*a^2*b^13*d^5*e^5 - 35*
B*a^4*b^11*d^4*e^6 + 105*A*a^3*b^12*d^4*e^6 + 49*B*a^5*b^10*d^3*e^7 - 105*A*a^4*b^11*d^3*e^7 - 35*B*a^6*b^9*d^
2*e^8 + 63*A*a^5*b^10*d^2*e^8 + 13*B*a^7*b^8*d*e^9 - 21*A*a^6*b^9*d*e^9 - 2*B*a^8*b^7*e^10 + 3*A*a^7*b^8*e^10)
/(b^11*d^9*abs(b)*e^2 - 9*a*b^10*d^8*abs(b)*e^3 + 36*a^2*b^9*d^7*abs(b)*e^4 - 84*a^3*b^8*d^6*abs(b)*e^5 + 126*
a^4*b^7*d^5*abs(b)*e^6 - 126*a^5*b^6*d^4*abs(b)*e^7 + 84*a^6*b^5*d^3*abs(b)*e^8 - 36*a^7*b^4*d^2*abs(b)*e^9 +
9*a^8*b^3*d*abs(b)*e^10 - a^9*b^2*abs(b)*e^11))*sqrt(b*x + a)/(b^2*d + (b*x + a)*b*e - a*b*e)^(5/2)

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Mupad [B]
time = 2.46, size = 288, normalized size = 1.54 \begin {gather*} \frac {\sqrt {d+e\,x}\,\left (\frac {2\,x\,\left (-a^2\,e^2+10\,a\,b\,d\,e+15\,b^2\,d^2\right )\,\left (5\,B\,a\,e-6\,A\,b\,e+B\,b\,d\right )}{15\,e^3\,{\left (a\,e-b\,d\right )}^4}-\frac {4\,B\,a^3\,d\,e^2+6\,A\,a^3\,e^3-40\,B\,a^2\,b\,d^2\,e-30\,A\,a^2\,b\,d\,e^2-60\,B\,a\,b^2\,d^3+90\,A\,a\,b^2\,d^2\,e+30\,A\,b^3\,d^3}{15\,e^3\,{\left (a\,e-b\,d\right )}^4}+\frac {16\,b^2\,x^3\,\left (5\,B\,a\,e-6\,A\,b\,e+B\,b\,d\right )}{15\,e\,{\left (a\,e-b\,d\right )}^4}+\frac {8\,b\,x^2\,\left (a\,e+5\,b\,d\right )\,\left (5\,B\,a\,e-6\,A\,b\,e+B\,b\,d\right )}{15\,e^2\,{\left (a\,e-b\,d\right )}^4}\right )}{x^3\,\sqrt {a+b\,x}+\frac {d^3\,\sqrt {a+b\,x}}{e^3}+\frac {3\,d\,x^2\,\sqrt {a+b\,x}}{e}+\frac {3\,d^2\,x\,\sqrt {a+b\,x}}{e^2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((d + e*x)^(1/2)*((2*x*(15*b^2*d^2 - a^2*e^2 + 10*a*b*d*e)*(5*B*a*e - 6*A*b*e + B*b*d))/(15*e^3*(a*e - b*d)^4)
- (6*A*a^3*e^3 + 30*A*b^3*d^3 - 60*B*a*b^2*d^3 + 4*B*a^3*d*e^2 + 90*A*a*b^2*d^2*e - 30*A*a^2*b*d*e^2 - 40*B*a
^2*b*d^2*e)/(15*e^3*(a*e - b*d)^4) + (16*b^2*x^3*(5*B*a*e - 6*A*b*e + B*b*d))/(15*e*(a*e - b*d)^4) + (8*b*x^2*
(a*e + 5*b*d)*(5*B*a*e - 6*A*b*e + B*b*d))/(15*e^2*(a*e - b*d)^4)))/(x^3*(a + b*x)^(1/2) + (d^3*(a + b*x)^(1/2
))/e^3 + (3*d*x^2*(a + b*x)^(1/2))/e + (3*d^2*x*(a + b*x)^(1/2))/e^2)

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