3.14.86 \(\int \frac {(a+b x)^{7/2}}{\sqrt {c+d x}} \, dx\)

Optimal. Leaf size=183 \[ \frac {35 (b c-a d)^4 \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{64 \sqrt {b} d^{9/2}}-\frac {35 \sqrt {a+b x} \sqrt {c+d x} (b c-a d)^3}{64 d^4}+\frac {35 (a+b x)^{3/2} \sqrt {c+d x} (b c-a d)^2}{96 d^3}-\frac {7 (a+b x)^{5/2} \sqrt {c+d x} (b c-a d)}{24 d^2}+\frac {(a+b x)^{7/2} \sqrt {c+d x}}{4 d} \]

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Rubi [A]  time = 0.10, antiderivative size = 183, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 4, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.210, Rules used = {50, 63, 217, 206} \begin {gather*} -\frac {35 \sqrt {a+b x} \sqrt {c+d x} (b c-a d)^3}{64 d^4}+\frac {35 (a+b x)^{3/2} \sqrt {c+d x} (b c-a d)^2}{96 d^3}-\frac {7 (a+b x)^{5/2} \sqrt {c+d x} (b c-a d)}{24 d^2}+\frac {35 (b c-a d)^4 \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{64 \sqrt {b} d^{9/2}}+\frac {(a+b x)^{7/2} \sqrt {c+d x}}{4 d} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*x)^(7/2)/Sqrt[c + d*x],x]

[Out]

(-35*(b*c - a*d)^3*Sqrt[a + b*x]*Sqrt[c + d*x])/(64*d^4) + (35*(b*c - a*d)^2*(a + b*x)^(3/2)*Sqrt[c + d*x])/(9
6*d^3) - (7*(b*c - a*d)*(a + b*x)^(5/2)*Sqrt[c + d*x])/(24*d^2) + ((a + b*x)^(7/2)*Sqrt[c + d*x])/(4*d) + (35*
(b*c - a*d)^4*ArcTanh[(Sqrt[d]*Sqrt[a + b*x])/(Sqrt[b]*Sqrt[c + d*x])])/(64*Sqrt[b]*d^(9/2))

Rule 50

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

Rule 63

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - (a*d)/b + (d*x^p)/b)^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 217

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a,
b}, x] &&  !GtQ[a, 0]

Rubi steps

\begin {align*} \int \frac {(a+b x)^{7/2}}{\sqrt {c+d x}} \, dx &=\frac {(a+b x)^{7/2} \sqrt {c+d x}}{4 d}-\frac {(7 (b c-a d)) \int \frac {(a+b x)^{5/2}}{\sqrt {c+d x}} \, dx}{8 d}\\ &=-\frac {7 (b c-a d) (a+b x)^{5/2} \sqrt {c+d x}}{24 d^2}+\frac {(a+b x)^{7/2} \sqrt {c+d x}}{4 d}+\frac {\left (35 (b c-a d)^2\right ) \int \frac {(a+b x)^{3/2}}{\sqrt {c+d x}} \, dx}{48 d^2}\\ &=\frac {35 (b c-a d)^2 (a+b x)^{3/2} \sqrt {c+d x}}{96 d^3}-\frac {7 (b c-a d) (a+b x)^{5/2} \sqrt {c+d x}}{24 d^2}+\frac {(a+b x)^{7/2} \sqrt {c+d x}}{4 d}-\frac {\left (35 (b c-a d)^3\right ) \int \frac {\sqrt {a+b x}}{\sqrt {c+d x}} \, dx}{64 d^3}\\ &=-\frac {35 (b c-a d)^3 \sqrt {a+b x} \sqrt {c+d x}}{64 d^4}+\frac {35 (b c-a d)^2 (a+b x)^{3/2} \sqrt {c+d x}}{96 d^3}-\frac {7 (b c-a d) (a+b x)^{5/2} \sqrt {c+d x}}{24 d^2}+\frac {(a+b x)^{7/2} \sqrt {c+d x}}{4 d}+\frac {\left (35 (b c-a d)^4\right ) \int \frac {1}{\sqrt {a+b x} \sqrt {c+d x}} \, dx}{128 d^4}\\ &=-\frac {35 (b c-a d)^3 \sqrt {a+b x} \sqrt {c+d x}}{64 d^4}+\frac {35 (b c-a d)^2 (a+b x)^{3/2} \sqrt {c+d x}}{96 d^3}-\frac {7 (b c-a d) (a+b x)^{5/2} \sqrt {c+d x}}{24 d^2}+\frac {(a+b x)^{7/2} \sqrt {c+d x}}{4 d}+\frac {\left (35 (b c-a d)^4\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {c-\frac {a d}{b}+\frac {d x^2}{b}}} \, dx,x,\sqrt {a+b x}\right )}{64 b d^4}\\ &=-\frac {35 (b c-a d)^3 \sqrt {a+b x} \sqrt {c+d x}}{64 d^4}+\frac {35 (b c-a d)^2 (a+b x)^{3/2} \sqrt {c+d x}}{96 d^3}-\frac {7 (b c-a d) (a+b x)^{5/2} \sqrt {c+d x}}{24 d^2}+\frac {(a+b x)^{7/2} \sqrt {c+d x}}{4 d}+\frac {\left (35 (b c-a d)^4\right ) \operatorname {Subst}\left (\int \frac {1}{1-\frac {d x^2}{b}} \, dx,x,\frac {\sqrt {a+b x}}{\sqrt {c+d x}}\right )}{64 b d^4}\\ &=-\frac {35 (b c-a d)^3 \sqrt {a+b x} \sqrt {c+d x}}{64 d^4}+\frac {35 (b c-a d)^2 (a+b x)^{3/2} \sqrt {c+d x}}{96 d^3}-\frac {7 (b c-a d) (a+b x)^{5/2} \sqrt {c+d x}}{24 d^2}+\frac {(a+b x)^{7/2} \sqrt {c+d x}}{4 d}+\frac {35 (b c-a d)^4 \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{64 \sqrt {b} d^{9/2}}\\ \end {align*}

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Mathematica [A]  time = 0.66, size = 189, normalized size = 1.03 \begin {gather*} \frac {\sqrt {d} \sqrt {a+b x} (c+d x) \left (279 a^3 d^3+a^2 b d^2 (326 d x-511 c)+a b^2 d \left (385 c^2-252 c d x+200 d^2 x^2\right )+b^3 \left (-105 c^3+70 c^2 d x-56 c d^2 x^2+48 d^3 x^3\right )\right )+\frac {105 (b c-a d)^{9/2} \sqrt {\frac {b (c+d x)}{b c-a d}} \sinh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b c-a d}}\right )}{b}}{192 d^{9/2} \sqrt {c+d x}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)^(7/2)/Sqrt[c + d*x],x]

[Out]

(Sqrt[d]*Sqrt[a + b*x]*(c + d*x)*(279*a^3*d^3 + a^2*b*d^2*(-511*c + 326*d*x) + a*b^2*d*(385*c^2 - 252*c*d*x +
200*d^2*x^2) + b^3*(-105*c^3 + 70*c^2*d*x - 56*c*d^2*x^2 + 48*d^3*x^3)) + (105*(b*c - a*d)^(9/2)*Sqrt[(b*(c +
d*x))/(b*c - a*d)]*ArcSinh[(Sqrt[d]*Sqrt[a + b*x])/Sqrt[b*c - a*d]])/b)/(192*d^(9/2)*Sqrt[c + d*x])

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IntegrateAlgebraic [A]  time = 0.20, size = 172, normalized size = 0.94 \begin {gather*} \frac {\sqrt {c+d x} (a d-b c)^4 \left (-\frac {105 b^3 (c+d x)^3}{(a+b x)^3}+\frac {385 b^2 d (c+d x)^2}{(a+b x)^2}-\frac {511 b d^2 (c+d x)}{a+b x}+279 d^3\right )}{192 d^4 \sqrt {a+b x} \left (d-\frac {b (c+d x)}{a+b x}\right )^4}+\frac {35 (b c-a d)^4 \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {d} \sqrt {a+b x}}\right )}{64 \sqrt {b} d^{9/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(a + b*x)^(7/2)/Sqrt[c + d*x],x]

[Out]

((-(b*c) + a*d)^4*Sqrt[c + d*x]*(279*d^3 - (511*b*d^2*(c + d*x))/(a + b*x) + (385*b^2*d*(c + d*x)^2)/(a + b*x)
^2 - (105*b^3*(c + d*x)^3)/(a + b*x)^3))/(192*d^4*Sqrt[a + b*x]*(d - (b*(c + d*x))/(a + b*x))^4) + (35*(b*c -
a*d)^4*ArcTanh[(Sqrt[b]*Sqrt[c + d*x])/(Sqrt[d]*Sqrt[a + b*x])])/(64*Sqrt[b]*d^(9/2))

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fricas [A]  time = 1.10, size = 542, normalized size = 2.96 \begin {gather*} \left [\frac {105 \, {\left (b^{4} c^{4} - 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + a^{4} d^{4}\right )} \sqrt {b d} \log \left (8 \, b^{2} d^{2} x^{2} + b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2} + 4 \, {\left (2 \, b d x + b c + a d\right )} \sqrt {b d} \sqrt {b x + a} \sqrt {d x + c} + 8 \, {\left (b^{2} c d + a b d^{2}\right )} x\right ) + 4 \, {\left (48 \, b^{4} d^{4} x^{3} - 105 \, b^{4} c^{3} d + 385 \, a b^{3} c^{2} d^{2} - 511 \, a^{2} b^{2} c d^{3} + 279 \, a^{3} b d^{4} - 8 \, {\left (7 \, b^{4} c d^{3} - 25 \, a b^{3} d^{4}\right )} x^{2} + 2 \, {\left (35 \, b^{4} c^{2} d^{2} - 126 \, a b^{3} c d^{3} + 163 \, a^{2} b^{2} d^{4}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{768 \, b d^{5}}, -\frac {105 \, {\left (b^{4} c^{4} - 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + a^{4} d^{4}\right )} \sqrt {-b d} \arctan \left (\frac {{\left (2 \, b d x + b c + a d\right )} \sqrt {-b d} \sqrt {b x + a} \sqrt {d x + c}}{2 \, {\left (b^{2} d^{2} x^{2} + a b c d + {\left (b^{2} c d + a b d^{2}\right )} x\right )}}\right ) - 2 \, {\left (48 \, b^{4} d^{4} x^{3} - 105 \, b^{4} c^{3} d + 385 \, a b^{3} c^{2} d^{2} - 511 \, a^{2} b^{2} c d^{3} + 279 \, a^{3} b d^{4} - 8 \, {\left (7 \, b^{4} c d^{3} - 25 \, a b^{3} d^{4}\right )} x^{2} + 2 \, {\left (35 \, b^{4} c^{2} d^{2} - 126 \, a b^{3} c d^{3} + 163 \, a^{2} b^{2} d^{4}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{384 \, b d^{5}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^(7/2)/(d*x+c)^(1/2),x, algorithm="fricas")

[Out]

[1/768*(105*(b^4*c^4 - 4*a*b^3*c^3*d + 6*a^2*b^2*c^2*d^2 - 4*a^3*b*c*d^3 + a^4*d^4)*sqrt(b*d)*log(8*b^2*d^2*x^
2 + b^2*c^2 + 6*a*b*c*d + a^2*d^2 + 4*(2*b*d*x + b*c + a*d)*sqrt(b*d)*sqrt(b*x + a)*sqrt(d*x + c) + 8*(b^2*c*d
+ a*b*d^2)*x) + 4*(48*b^4*d^4*x^3 - 105*b^4*c^3*d + 385*a*b^3*c^2*d^2 - 511*a^2*b^2*c*d^3 + 279*a^3*b*d^4 - 8
*(7*b^4*c*d^3 - 25*a*b^3*d^4)*x^2 + 2*(35*b^4*c^2*d^2 - 126*a*b^3*c*d^3 + 163*a^2*b^2*d^4)*x)*sqrt(b*x + a)*sq
rt(d*x + c))/(b*d^5), -1/384*(105*(b^4*c^4 - 4*a*b^3*c^3*d + 6*a^2*b^2*c^2*d^2 - 4*a^3*b*c*d^3 + a^4*d^4)*sqrt
(-b*d)*arctan(1/2*(2*b*d*x + b*c + a*d)*sqrt(-b*d)*sqrt(b*x + a)*sqrt(d*x + c)/(b^2*d^2*x^2 + a*b*c*d + (b^2*c
*d + a*b*d^2)*x)) - 2*(48*b^4*d^4*x^3 - 105*b^4*c^3*d + 385*a*b^3*c^2*d^2 - 511*a^2*b^2*c*d^3 + 279*a^3*b*d^4
- 8*(7*b^4*c*d^3 - 25*a*b^3*d^4)*x^2 + 2*(35*b^4*c^2*d^2 - 126*a*b^3*c*d^3 + 163*a^2*b^2*d^4)*x)*sqrt(b*x + a)
*sqrt(d*x + c))/(b*d^5)]

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giac [A]  time = 1.23, size = 268, normalized size = 1.46 \begin {gather*} \frac {{\left (\sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d} {\left (2 \, {\left (b x + a\right )} {\left (4 \, {\left (b x + a\right )} {\left (\frac {6 \, {\left (b x + a\right )}}{b d} - \frac {7 \, {\left (b c d^{5} - a d^{6}\right )}}{b d^{7}}\right )} + \frac {35 \, {\left (b^{2} c^{2} d^{4} - 2 \, a b c d^{5} + a^{2} d^{6}\right )}}{b d^{7}}\right )} - \frac {105 \, {\left (b^{3} c^{3} d^{3} - 3 \, a b^{2} c^{2} d^{4} + 3 \, a^{2} b c d^{5} - a^{3} d^{6}\right )}}{b d^{7}}\right )} \sqrt {b x + a} - \frac {105 \, {\left (b^{4} c^{4} - 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + a^{4} d^{4}\right )} \log \left ({\left | -\sqrt {b d} \sqrt {b x + a} + \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d} \right |}\right )}{\sqrt {b d} d^{4}}\right )} b}{192 \, {\left | b \right |}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^(7/2)/(d*x+c)^(1/2),x, algorithm="giac")

[Out]

1/192*(sqrt(b^2*c + (b*x + a)*b*d - a*b*d)*(2*(b*x + a)*(4*(b*x + a)*(6*(b*x + a)/(b*d) - 7*(b*c*d^5 - a*d^6)/
(b*d^7)) + 35*(b^2*c^2*d^4 - 2*a*b*c*d^5 + a^2*d^6)/(b*d^7)) - 105*(b^3*c^3*d^3 - 3*a*b^2*c^2*d^4 + 3*a^2*b*c*
d^5 - a^3*d^6)/(b*d^7))*sqrt(b*x + a) - 105*(b^4*c^4 - 4*a*b^3*c^3*d + 6*a^2*b^2*c^2*d^2 - 4*a^3*b*c*d^3 + a^4
*d^4)*log(abs(-sqrt(b*d)*sqrt(b*x + a) + sqrt(b^2*c + (b*x + a)*b*d - a*b*d)))/(sqrt(b*d)*d^4))*b/abs(b)

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maple [B]  time = 0.01, size = 650, normalized size = 3.55 \begin {gather*} \frac {35 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, a^{4} \ln \left (\frac {b d x +\frac {1}{2} a d +\frac {1}{2} b c}{\sqrt {b d}}+\sqrt {b d \,x^{2}+a c +\left (a d +b c \right ) x}\right )}{128 \sqrt {b x +a}\, \sqrt {d x +c}\, \sqrt {b d}}-\frac {35 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, a^{3} b c \ln \left (\frac {b d x +\frac {1}{2} a d +\frac {1}{2} b c}{\sqrt {b d}}+\sqrt {b d \,x^{2}+a c +\left (a d +b c \right ) x}\right )}{32 \sqrt {b x +a}\, \sqrt {d x +c}\, \sqrt {b d}\, d}+\frac {105 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, a^{2} b^{2} c^{2} \ln \left (\frac {b d x +\frac {1}{2} a d +\frac {1}{2} b c}{\sqrt {b d}}+\sqrt {b d \,x^{2}+a c +\left (a d +b c \right ) x}\right )}{64 \sqrt {b x +a}\, \sqrt {d x +c}\, \sqrt {b d}\, d^{2}}-\frac {35 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, a \,b^{3} c^{3} \ln \left (\frac {b d x +\frac {1}{2} a d +\frac {1}{2} b c}{\sqrt {b d}}+\sqrt {b d \,x^{2}+a c +\left (a d +b c \right ) x}\right )}{32 \sqrt {b x +a}\, \sqrt {d x +c}\, \sqrt {b d}\, d^{3}}+\frac {35 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, b^{4} c^{4} \ln \left (\frac {b d x +\frac {1}{2} a d +\frac {1}{2} b c}{\sqrt {b d}}+\sqrt {b d \,x^{2}+a c +\left (a d +b c \right ) x}\right )}{128 \sqrt {b x +a}\, \sqrt {d x +c}\, \sqrt {b d}\, d^{4}}+\frac {35 \sqrt {b x +a}\, \sqrt {d x +c}\, a^{3}}{64 d}-\frac {105 \sqrt {b x +a}\, \sqrt {d x +c}\, a^{2} b c}{64 d^{2}}+\frac {105 \sqrt {b x +a}\, \sqrt {d x +c}\, a \,b^{2} c^{2}}{64 d^{3}}-\frac {35 \sqrt {b x +a}\, \sqrt {d x +c}\, b^{3} c^{3}}{64 d^{4}}+\frac {35 \left (b x +a \right )^{\frac {3}{2}} \sqrt {d x +c}\, a^{2}}{96 d}-\frac {35 \left (b x +a \right )^{\frac {3}{2}} \sqrt {d x +c}\, a b c}{48 d^{2}}+\frac {35 \left (b x +a \right )^{\frac {3}{2}} \sqrt {d x +c}\, b^{2} c^{2}}{96 d^{3}}+\frac {7 \left (b x +a \right )^{\frac {5}{2}} \sqrt {d x +c}\, a}{24 d}-\frac {7 \left (b x +a \right )^{\frac {5}{2}} \sqrt {d x +c}\, b c}{24 d^{2}}+\frac {\left (b x +a \right )^{\frac {7}{2}} \sqrt {d x +c}}{4 d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)^(7/2)/(d*x+c)^(1/2),x)

[Out]

1/4*(b*x+a)^(7/2)*(d*x+c)^(1/2)/d+7/24/d*(b*x+a)^(5/2)*(d*x+c)^(1/2)*a-7/24/d^2*(b*x+a)^(5/2)*(d*x+c)^(1/2)*b*
c+35/96/d*(b*x+a)^(3/2)*(d*x+c)^(1/2)*a^2-35/48/d^2*(b*x+a)^(3/2)*(d*x+c)^(1/2)*a*b*c+35/96/d^3*(b*x+a)^(3/2)*
(d*x+c)^(1/2)*b^2*c^2+35/64/d*(b*x+a)^(1/2)*(d*x+c)^(1/2)*a^3-105/64/d^2*(b*x+a)^(1/2)*(d*x+c)^(1/2)*a^2*b*c+1
05/64/d^3*(b*x+a)^(1/2)*(d*x+c)^(1/2)*a*b^2*c^2-35/64/d^4*(b*x+a)^(1/2)*(d*x+c)^(1/2)*b^3*c^3+35/128*((b*x+a)*
(d*x+c))^(1/2)/(b*x+a)^(1/2)/(d*x+c)^(1/2)*ln((b*d*x+1/2*a*d+1/2*b*c)/(b*d)^(1/2)+(b*d*x^2+a*c+(a*d+b*c)*x)^(1
/2))/(b*d)^(1/2)*a^4-35/32/d*((b*x+a)*(d*x+c))^(1/2)/(b*x+a)^(1/2)/(d*x+c)^(1/2)*ln((b*d*x+1/2*a*d+1/2*b*c)/(b
*d)^(1/2)+(b*d*x^2+a*c+(a*d+b*c)*x)^(1/2))/(b*d)^(1/2)*a^3*b*c+105/64/d^2*((b*x+a)*(d*x+c))^(1/2)/(b*x+a)^(1/2
)/(d*x+c)^(1/2)*ln((b*d*x+1/2*a*d+1/2*b*c)/(b*d)^(1/2)+(b*d*x^2+a*c+(a*d+b*c)*x)^(1/2))/(b*d)^(1/2)*a^2*b^2*c^
2-35/32/d^3*((b*x+a)*(d*x+c))^(1/2)/(b*x+a)^(1/2)/(d*x+c)^(1/2)*ln((b*d*x+1/2*a*d+1/2*b*c)/(b*d)^(1/2)+(b*d*x^
2+a*c+(a*d+b*c)*x)^(1/2))/(b*d)^(1/2)*a*b^3*c^3+35/128/d^4*((b*x+a)*(d*x+c))^(1/2)/(b*x+a)^(1/2)/(d*x+c)^(1/2)
*ln((b*d*x+1/2*a*d+1/2*b*c)/(b*d)^(1/2)+(b*d*x^2+a*c+(a*d+b*c)*x)^(1/2))/(b*d)^(1/2)*b^4*c^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)^(7/2)/(d*x+c)^(1/2),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(a*d-b*c>0)', see `assume?` for
more details)Is a*d-b*c zero or nonzero?

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\left (a+b\,x\right )}^{7/2}}{\sqrt {c+d\,x}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x)^(7/2)/(c + d*x)^(1/2),x)

[Out]

int((a + b*x)^(7/2)/(c + d*x)^(1/2), x)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)**(7/2)/(d*x+c)**(1/2),x)

[Out]

Timed out

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