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

Optimal. Leaf size=379 \[ -\frac {2 \sqrt {-b} d \sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {\frac {e x}{d}+1} (c d-b e) \left (24 b^2 e^2-71 b c d e+71 c^2 d^2\right ) F\left (\sin ^{-1}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right )|\frac {b e}{c d}\right )}{105 c^{7/2} \sqrt {b x+c x^2} \sqrt {d+e x}}+\frac {16 \sqrt {-b} \sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {d+e x} (2 c d-b e) \left (6 b^2 e^2-11 b c d e+11 c^2 d^2\right ) E\left (\sin ^{-1}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right )|\frac {b e}{c d}\right )}{105 c^{7/2} \sqrt {b x+c x^2} \sqrt {\frac {e x}{d}+1}}+\frac {2 e \sqrt {b x+c x^2} \sqrt {d+e x} \left (24 b^2 e^2-71 b c d e+71 c^2 d^2\right )}{105 c^3}+\frac {12 e \sqrt {b x+c x^2} (d+e x)^{3/2} (2 c d-b e)}{35 c^2}+\frac {2 e \sqrt {b x+c x^2} (d+e x)^{5/2}}{7 c} \]

[Out]

16/105*(-b*e+2*c*d)*(6*b^2*e^2-11*b*c*d*e+11*c^2*d^2)*EllipticE(c^(1/2)*x^(1/2)/(-b)^(1/2),(b*e/c/d)^(1/2))*(-
b)^(1/2)*x^(1/2)*(c*x/b+1)^(1/2)*(e*x+d)^(1/2)/c^(7/2)/(1+e*x/d)^(1/2)/(c*x^2+b*x)^(1/2)-2/105*d*(-b*e+c*d)*(2
4*b^2*e^2-71*b*c*d*e+71*c^2*d^2)*EllipticF(c^(1/2)*x^(1/2)/(-b)^(1/2),(b*e/c/d)^(1/2))*(-b)^(1/2)*x^(1/2)*(c*x
/b+1)^(1/2)*(1+e*x/d)^(1/2)/c^(7/2)/(e*x+d)^(1/2)/(c*x^2+b*x)^(1/2)+12/35*e*(-b*e+2*c*d)*(e*x+d)^(3/2)*(c*x^2+
b*x)^(1/2)/c^2+2/7*e*(e*x+d)^(5/2)*(c*x^2+b*x)^(1/2)/c+2/105*e*(24*b^2*e^2-71*b*c*d*e+71*c^2*d^2)*(e*x+d)^(1/2
)*(c*x^2+b*x)^(1/2)/c^3

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Rubi [A]  time = 0.50, antiderivative size = 379, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 8, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.348, Rules used = {742, 832, 843, 715, 112, 110, 117, 116} \[ \frac {2 e \sqrt {b x+c x^2} \sqrt {d+e x} \left (24 b^2 e^2-71 b c d e+71 c^2 d^2\right )}{105 c^3}-\frac {2 \sqrt {-b} d \sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {\frac {e x}{d}+1} (c d-b e) \left (24 b^2 e^2-71 b c d e+71 c^2 d^2\right ) F\left (\sin ^{-1}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right )|\frac {b e}{c d}\right )}{105 c^{7/2} \sqrt {b x+c x^2} \sqrt {d+e x}}+\frac {16 \sqrt {-b} \sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {d+e x} (2 c d-b e) \left (6 b^2 e^2-11 b c d e+11 c^2 d^2\right ) E\left (\sin ^{-1}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right )|\frac {b e}{c d}\right )}{105 c^{7/2} \sqrt {b x+c x^2} \sqrt {\frac {e x}{d}+1}}+\frac {12 e \sqrt {b x+c x^2} (d+e x)^{3/2} (2 c d-b e)}{35 c^2}+\frac {2 e \sqrt {b x+c x^2} (d+e x)^{5/2}}{7 c} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*e*(71*c^2*d^2 - 71*b*c*d*e + 24*b^2*e^2)*Sqrt[d + e*x]*Sqrt[b*x + c*x^2])/(105*c^3) + (12*e*(2*c*d - b*e)*(
d + e*x)^(3/2)*Sqrt[b*x + c*x^2])/(35*c^2) + (2*e*(d + e*x)^(5/2)*Sqrt[b*x + c*x^2])/(7*c) + (16*Sqrt[-b]*(2*c
*d - b*e)*(11*c^2*d^2 - 11*b*c*d*e + 6*b^2*e^2)*Sqrt[x]*Sqrt[1 + (c*x)/b]*Sqrt[d + e*x]*EllipticE[ArcSin[(Sqrt
[c]*Sqrt[x])/Sqrt[-b]], (b*e)/(c*d)])/(105*c^(7/2)*Sqrt[1 + (e*x)/d]*Sqrt[b*x + c*x^2]) - (2*Sqrt[-b]*d*(c*d -
b*e)*(71*c^2*d^2 - 71*b*c*d*e + 24*b^2*e^2)*Sqrt[x]*Sqrt[1 + (c*x)/b]*Sqrt[1 + (e*x)/d]*EllipticF[ArcSin[(Sqr
t[c]*Sqrt[x])/Sqrt[-b]], (b*e)/(c*d)])/(105*c^(7/2)*Sqrt[d + e*x]*Sqrt[b*x + c*x^2])

Rule 110

Int[Sqrt[(e_) + (f_.)*(x_)]/(Sqrt[(b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]), x_Symbol] :> Simp[(2*Sqrt[e]*Rt[-(b/d)
, 2]*EllipticE[ArcSin[Sqrt[b*x]/(Sqrt[c]*Rt[-(b/d), 2])], (c*f)/(d*e)])/b, x] /; FreeQ[{b, c, d, e, f}, x] &&
NeQ[d*e - c*f, 0] && GtQ[c, 0] && GtQ[e, 0] &&  !LtQ[-(b/d), 0]

Rule 112

Int[Sqrt[(e_) + (f_.)*(x_)]/(Sqrt[(b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]), x_Symbol] :> Dist[(Sqrt[e + f*x]*Sqrt[
1 + (d*x)/c])/(Sqrt[c + d*x]*Sqrt[1 + (f*x)/e]), Int[Sqrt[1 + (f*x)/e]/(Sqrt[b*x]*Sqrt[1 + (d*x)/c]), x], x] /
; FreeQ[{b, c, d, e, f}, x] && NeQ[d*e - c*f, 0] &&  !(GtQ[c, 0] && GtQ[e, 0])

Rule 116

Int[1/(Sqrt[(b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(e_) + (f_.)*(x_)]), x_Symbol] :> Simp[(2*Rt[-(b/d), 2]*E
llipticF[ArcSin[Sqrt[b*x]/(Sqrt[c]*Rt[-(b/d), 2])], (c*f)/(d*e)])/(b*Sqrt[e]), x] /; FreeQ[{b, c, d, e, f}, x]
&& GtQ[c, 0] && GtQ[e, 0] && (PosQ[-(b/d)] || NegQ[-(b/f)])

Rule 117

Int[1/(Sqrt[(b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(e_) + (f_.)*(x_)]), x_Symbol] :> Dist[(Sqrt[1 + (d*x)/c]
*Sqrt[1 + (f*x)/e])/(Sqrt[c + d*x]*Sqrt[e + f*x]), Int[1/(Sqrt[b*x]*Sqrt[1 + (d*x)/c]*Sqrt[1 + (f*x)/e]), x],
x] /; FreeQ[{b, c, d, e, f}, x] &&  !(GtQ[c, 0] && GtQ[e, 0])

Rule 715

Int[((d_.) + (e_.)*(x_))^(m_)/Sqrt[(b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Dist[(Sqrt[x]*Sqrt[b + c*x])/Sqrt[
b*x + c*x^2], Int[(d + e*x)^m/(Sqrt[x]*Sqrt[b + c*x]), x], x] /; FreeQ[{b, c, d, e}, x] && NeQ[c*d - b*e, 0] &
& NeQ[2*c*d - b*e, 0] && EqQ[m^2, 1/4]

Rule 742

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(e*(d + e*x)^(m - 1)
*(a + b*x + c*x^2)^(p + 1))/(c*(m + 2*p + 1)), x] + Dist[1/(c*(m + 2*p + 1)), Int[(d + e*x)^(m - 2)*Simp[c*d^2
*(m + 2*p + 1) - e*(a*e*(m - 1) + b*d*(p + 1)) + e*(2*c*d - b*e)*(m + p)*x, x]*(a + b*x + c*x^2)^p, x], x] /;
FreeQ[{a, b, c, d, e, m, p}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && NeQ[2*c*d - b*e, 0]
&& If[RationalQ[m], GtQ[m, 1], SumSimplerQ[m, -2]] && NeQ[m + 2*p + 1, 0] && IntQuadraticQ[a, b, c, d, e, m,
p, x]

Rule 832

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Sim
p[(g*(d + e*x)^m*(a + b*x + c*x^2)^(p + 1))/(c*(m + 2*p + 2)), x] + Dist[1/(c*(m + 2*p + 2)), Int[(d + e*x)^(m
- 1)*(a + b*x + c*x^2)^p*Simp[m*(c*d*f - a*e*g) + d*(2*c*f - b*g)*(p + 1) + (m*(c*e*f + c*d*g - b*e*g) + e*(p
+ 1)*(2*c*f - b*g))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, p}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 -
b*d*e + a*e^2, 0] && GtQ[m, 0] && NeQ[m + 2*p + 2, 0] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])
&&  !(IGtQ[m, 0] && EqQ[f, 0])

Rule 843

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Dis
t[g/e, Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p, x], x] + Dist[(e*f - d*g)/e, Int[(d + e*x)^m*(a + b*x + c*x^
2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0]
&&  !IGtQ[m, 0]

Rubi steps

\begin {align*} \int \frac {(d+e x)^{7/2}}{\sqrt {b x+c x^2}} \, dx &=\frac {2 e (d+e x)^{5/2} \sqrt {b x+c x^2}}{7 c}+\frac {2 \int \frac {(d+e x)^{3/2} \left (\frac {1}{2} d (7 c d-b e)+3 e (2 c d-b e) x\right )}{\sqrt {b x+c x^2}} \, dx}{7 c}\\ &=\frac {12 e (2 c d-b e) (d+e x)^{3/2} \sqrt {b x+c x^2}}{35 c^2}+\frac {2 e (d+e x)^{5/2} \sqrt {b x+c x^2}}{7 c}+\frac {4 \int \frac {\sqrt {d+e x} \left (\frac {1}{4} d \left (35 c^2 d^2-17 b c d e+6 b^2 e^2\right )+\frac {1}{4} e \left (71 c^2 d^2-71 b c d e+24 b^2 e^2\right ) x\right )}{\sqrt {b x+c x^2}} \, dx}{35 c^2}\\ &=\frac {2 e \left (71 c^2 d^2-71 b c d e+24 b^2 e^2\right ) \sqrt {d+e x} \sqrt {b x+c x^2}}{105 c^3}+\frac {12 e (2 c d-b e) (d+e x)^{3/2} \sqrt {b x+c x^2}}{35 c^2}+\frac {2 e (d+e x)^{5/2} \sqrt {b x+c x^2}}{7 c}+\frac {8 \int \frac {\frac {1}{8} d (7 c d-3 b e) \left (15 c^2 d^2-11 b c d e+8 b^2 e^2\right )+e (2 c d-b e) \left (11 c^2 d^2-11 b c d e+6 b^2 e^2\right ) x}{\sqrt {d+e x} \sqrt {b x+c x^2}} \, dx}{105 c^3}\\ &=\frac {2 e \left (71 c^2 d^2-71 b c d e+24 b^2 e^2\right ) \sqrt {d+e x} \sqrt {b x+c x^2}}{105 c^3}+\frac {12 e (2 c d-b e) (d+e x)^{3/2} \sqrt {b x+c x^2}}{35 c^2}+\frac {2 e (d+e x)^{5/2} \sqrt {b x+c x^2}}{7 c}+\frac {\left (8 (2 c d-b e) \left (11 c^2 d^2-11 b c d e+6 b^2 e^2\right )\right ) \int \frac {\sqrt {d+e x}}{\sqrt {b x+c x^2}} \, dx}{105 c^3}-\frac {\left (d (c d-b e) \left (71 c^2 d^2-71 b c d e+24 b^2 e^2\right )\right ) \int \frac {1}{\sqrt {d+e x} \sqrt {b x+c x^2}} \, dx}{105 c^3}\\ &=\frac {2 e \left (71 c^2 d^2-71 b c d e+24 b^2 e^2\right ) \sqrt {d+e x} \sqrt {b x+c x^2}}{105 c^3}+\frac {12 e (2 c d-b e) (d+e x)^{3/2} \sqrt {b x+c x^2}}{35 c^2}+\frac {2 e (d+e x)^{5/2} \sqrt {b x+c x^2}}{7 c}+\frac {\left (8 (2 c d-b e) \left (11 c^2 d^2-11 b c d e+6 b^2 e^2\right ) \sqrt {x} \sqrt {b+c x}\right ) \int \frac {\sqrt {d+e x}}{\sqrt {x} \sqrt {b+c x}} \, dx}{105 c^3 \sqrt {b x+c x^2}}-\frac {\left (d (c d-b e) \left (71 c^2 d^2-71 b c d e+24 b^2 e^2\right ) \sqrt {x} \sqrt {b+c x}\right ) \int \frac {1}{\sqrt {x} \sqrt {b+c x} \sqrt {d+e x}} \, dx}{105 c^3 \sqrt {b x+c x^2}}\\ &=\frac {2 e \left (71 c^2 d^2-71 b c d e+24 b^2 e^2\right ) \sqrt {d+e x} \sqrt {b x+c x^2}}{105 c^3}+\frac {12 e (2 c d-b e) (d+e x)^{3/2} \sqrt {b x+c x^2}}{35 c^2}+\frac {2 e (d+e x)^{5/2} \sqrt {b x+c x^2}}{7 c}+\frac {\left (8 (2 c d-b e) \left (11 c^2 d^2-11 b c d e+6 b^2 e^2\right ) \sqrt {x} \sqrt {1+\frac {c x}{b}} \sqrt {d+e x}\right ) \int \frac {\sqrt {1+\frac {e x}{d}}}{\sqrt {x} \sqrt {1+\frac {c x}{b}}} \, dx}{105 c^3 \sqrt {1+\frac {e x}{d}} \sqrt {b x+c x^2}}-\frac {\left (d (c d-b e) \left (71 c^2 d^2-71 b c d e+24 b^2 e^2\right ) \sqrt {x} \sqrt {1+\frac {c x}{b}} \sqrt {1+\frac {e x}{d}}\right ) \int \frac {1}{\sqrt {x} \sqrt {1+\frac {c x}{b}} \sqrt {1+\frac {e x}{d}}} \, dx}{105 c^3 \sqrt {d+e x} \sqrt {b x+c x^2}}\\ &=\frac {2 e \left (71 c^2 d^2-71 b c d e+24 b^2 e^2\right ) \sqrt {d+e x} \sqrt {b x+c x^2}}{105 c^3}+\frac {12 e (2 c d-b e) (d+e x)^{3/2} \sqrt {b x+c x^2}}{35 c^2}+\frac {2 e (d+e x)^{5/2} \sqrt {b x+c x^2}}{7 c}+\frac {16 \sqrt {-b} (2 c d-b e) \left (11 c^2 d^2-11 b c d e+6 b^2 e^2\right ) \sqrt {x} \sqrt {1+\frac {c x}{b}} \sqrt {d+e x} E\left (\sin ^{-1}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right )|\frac {b e}{c d}\right )}{105 c^{7/2} \sqrt {1+\frac {e x}{d}} \sqrt {b x+c x^2}}-\frac {2 \sqrt {-b} d (c d-b e) \left (71 c^2 d^2-71 b c d e+24 b^2 e^2\right ) \sqrt {x} \sqrt {1+\frac {c x}{b}} \sqrt {1+\frac {e x}{d}} F\left (\sin ^{-1}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right )|\frac {b e}{c d}\right )}{105 c^{7/2} \sqrt {d+e x} \sqrt {b x+c x^2}}\\ \end {align*}

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Mathematica [C]  time = 2.26, size = 388, normalized size = 1.02 \[ \frac {2 \sqrt {x} \left (e \sqrt {x} (b+c x) (d+e x) \left (24 b^2 e^2-b c e (89 d+18 e x)+c^2 \left (122 d^2+66 d e x+15 e^2 x^2\right )\right )+\frac {8 (b+c x) (d+e x) \left (-6 b^3 e^3+23 b^2 c d e^2-33 b c^2 d^2 e+22 c^3 d^3\right )}{c \sqrt {x}}+8 i e x \sqrt {\frac {b}{c}} \sqrt {\frac {b}{c x}+1} \sqrt {\frac {d}{e x}+1} \left (-6 b^3 e^3+23 b^2 c d e^2-33 b c^2 d^2 e+22 c^3 d^3\right ) E\left (i \sinh ^{-1}\left (\frac {\sqrt {\frac {b}{c}}}{\sqrt {x}}\right )|\frac {c d}{b e}\right )+\frac {i x \sqrt {\frac {b}{c}} \sqrt {\frac {b}{c x}+1} \sqrt {\frac {d}{e x}+1} \left (48 b^4 e^4-208 b^3 c d e^3+353 b^2 c^2 d^2 e^2-298 b c^3 d^3 e+105 c^4 d^4\right ) F\left (i \sinh ^{-1}\left (\frac {\sqrt {\frac {b}{c}}}{\sqrt {x}}\right )|\frac {c d}{b e}\right )}{b}\right )}{105 c^3 \sqrt {x (b+c x)} \sqrt {d+e x}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*Sqrt[x]*((8*(22*c^3*d^3 - 33*b*c^2*d^2*e + 23*b^2*c*d*e^2 - 6*b^3*e^3)*(b + c*x)*(d + e*x))/(c*Sqrt[x]) + e
*Sqrt[x]*(b + c*x)*(d + e*x)*(24*b^2*e^2 - b*c*e*(89*d + 18*e*x) + c^2*(122*d^2 + 66*d*e*x + 15*e^2*x^2)) + (8
*I)*Sqrt[b/c]*e*(22*c^3*d^3 - 33*b*c^2*d^2*e + 23*b^2*c*d*e^2 - 6*b^3*e^3)*Sqrt[1 + b/(c*x)]*Sqrt[1 + d/(e*x)]
*x*EllipticE[I*ArcSinh[Sqrt[b/c]/Sqrt[x]], (c*d)/(b*e)] + (I*Sqrt[b/c]*(105*c^4*d^4 - 298*b*c^3*d^3*e + 353*b^
2*c^2*d^2*e^2 - 208*b^3*c*d*e^3 + 48*b^4*e^4)*Sqrt[1 + b/(c*x)]*Sqrt[1 + d/(e*x)]*x*EllipticF[I*ArcSinh[Sqrt[b
/c]/Sqrt[x]], (c*d)/(b*e)])/b))/(105*c^3*Sqrt[x*(b + c*x)]*Sqrt[d + e*x])

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fricas [F]  time = 1.80, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (e^{3} x^{3} + 3 \, d e^{2} x^{2} + 3 \, d^{2} e x + d^{3}\right )} \sqrt {e x + d}}{\sqrt {c x^{2} + b x}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral((e^3*x^3 + 3*d*e^2*x^2 + 3*d^2*e*x + d^3)*sqrt(e*x + d)/sqrt(c*x^2 + b*x), x)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (e x + d\right )}^{\frac {7}{2}}}{\sqrt {c x^{2} + b x}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((e*x + d)^(7/2)/sqrt(c*x^2 + b*x), x)

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maple [B]  time = 0.10, size = 918, normalized size = 2.42 \[ \frac {2 \sqrt {e x +d}\, \sqrt {\left (c x +b \right ) x}\, \left (15 c^{5} e^{4} x^{5}-3 b \,c^{4} e^{4} x^{4}+81 c^{5} d \,e^{3} x^{4}+6 b^{2} c^{3} e^{4} x^{3}-26 b \,c^{4} d \,e^{3} x^{3}+188 c^{5} d^{2} e^{2} x^{3}+24 b^{3} c^{2} e^{4} x^{2}-83 b^{2} c^{3} d \,e^{3} x^{2}+99 b \,c^{4} d^{2} e^{2} x^{2}+122 c^{5} d^{3} e \,x^{2}+48 \sqrt {\frac {c x +b}{b}}\, \sqrt {-\frac {\left (e x +d \right ) c}{b e -c d}}\, \sqrt {-\frac {c x}{b}}\, b^{5} e^{4} \EllipticE \left (\sqrt {\frac {c x +b}{b}}, \sqrt {\frac {b e}{b e -c d}}\right )-232 \sqrt {\frac {c x +b}{b}}\, \sqrt {-\frac {\left (e x +d \right ) c}{b e -c d}}\, \sqrt {-\frac {c x}{b}}\, b^{4} c d \,e^{3} \EllipticE \left (\sqrt {\frac {c x +b}{b}}, \sqrt {\frac {b e}{b e -c d}}\right )+24 \sqrt {\frac {c x +b}{b}}\, \sqrt {-\frac {\left (e x +d \right ) c}{b e -c d}}\, \sqrt {-\frac {c x}{b}}\, b^{4} c d \,e^{3} \EllipticF \left (\sqrt {\frac {c x +b}{b}}, \sqrt {\frac {b e}{b e -c d}}\right )+448 \sqrt {\frac {c x +b}{b}}\, \sqrt {-\frac {\left (e x +d \right ) c}{b e -c d}}\, \sqrt {-\frac {c x}{b}}\, b^{3} c^{2} d^{2} e^{2} \EllipticE \left (\sqrt {\frac {c x +b}{b}}, \sqrt {\frac {b e}{b e -c d}}\right )-95 \sqrt {\frac {c x +b}{b}}\, \sqrt {-\frac {\left (e x +d \right ) c}{b e -c d}}\, \sqrt {-\frac {c x}{b}}\, b^{3} c^{2} d^{2} e^{2} \EllipticF \left (\sqrt {\frac {c x +b}{b}}, \sqrt {\frac {b e}{b e -c d}}\right )+24 b^{3} c^{2} d \,e^{3} x -440 \sqrt {\frac {c x +b}{b}}\, \sqrt {-\frac {\left (e x +d \right ) c}{b e -c d}}\, \sqrt {-\frac {c x}{b}}\, b^{2} c^{3} d^{3} e \EllipticE \left (\sqrt {\frac {c x +b}{b}}, \sqrt {\frac {b e}{b e -c d}}\right )+142 \sqrt {\frac {c x +b}{b}}\, \sqrt {-\frac {\left (e x +d \right ) c}{b e -c d}}\, \sqrt {-\frac {c x}{b}}\, b^{2} c^{3} d^{3} e \EllipticF \left (\sqrt {\frac {c x +b}{b}}, \sqrt {\frac {b e}{b e -c d}}\right )-89 b^{2} c^{3} d^{2} e^{2} x +176 \sqrt {\frac {c x +b}{b}}\, \sqrt {-\frac {\left (e x +d \right ) c}{b e -c d}}\, \sqrt {-\frac {c x}{b}}\, b \,c^{4} d^{4} \EllipticE \left (\sqrt {\frac {c x +b}{b}}, \sqrt {\frac {b e}{b e -c d}}\right )-71 \sqrt {\frac {c x +b}{b}}\, \sqrt {-\frac {\left (e x +d \right ) c}{b e -c d}}\, \sqrt {-\frac {c x}{b}}\, b \,c^{4} d^{4} \EllipticF \left (\sqrt {\frac {c x +b}{b}}, \sqrt {\frac {b e}{b e -c d}}\right )+122 b \,c^{4} d^{3} e x \right )}{105 \left (c e \,x^{2}+b e x +c d x +b d \right ) c^{5} x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/105*(e*x+d)^(1/2)*((c*x+b)*x)^(1/2)*(15*c^5*e^4*x^5+24*((c*x+b)/b)^(1/2)*(-(e*x+d)/(b*e-c*d)*c)^(1/2)*(-1/b*
c*x)^(1/2)*EllipticF(((c*x+b)/b)^(1/2),(1/(b*e-c*d)*b*e)^(1/2))*b^4*c*d*e^3-95*((c*x+b)/b)^(1/2)*(-(e*x+d)/(b*
e-c*d)*c)^(1/2)*(-1/b*c*x)^(1/2)*EllipticF(((c*x+b)/b)^(1/2),(1/(b*e-c*d)*b*e)^(1/2))*b^3*c^2*d^2*e^2+142*((c*
x+b)/b)^(1/2)*(-(e*x+d)/(b*e-c*d)*c)^(1/2)*(-1/b*c*x)^(1/2)*EllipticF(((c*x+b)/b)^(1/2),(1/(b*e-c*d)*b*e)^(1/2
))*b^2*c^3*d^3*e-71*((c*x+b)/b)^(1/2)*(-(e*x+d)/(b*e-c*d)*c)^(1/2)*(-1/b*c*x)^(1/2)*EllipticF(((c*x+b)/b)^(1/2
),(1/(b*e-c*d)*b*e)^(1/2))*b*c^4*d^4+48*((c*x+b)/b)^(1/2)*(-(e*x+d)/(b*e-c*d)*c)^(1/2)*(-1/b*c*x)^(1/2)*Ellipt
icE(((c*x+b)/b)^(1/2),(1/(b*e-c*d)*b*e)^(1/2))*b^5*e^4-232*((c*x+b)/b)^(1/2)*(-(e*x+d)/(b*e-c*d)*c)^(1/2)*(-1/
b*c*x)^(1/2)*EllipticE(((c*x+b)/b)^(1/2),(1/(b*e-c*d)*b*e)^(1/2))*b^4*c*d*e^3+448*((c*x+b)/b)^(1/2)*(-(e*x+d)/
(b*e-c*d)*c)^(1/2)*(-1/b*c*x)^(1/2)*EllipticE(((c*x+b)/b)^(1/2),(1/(b*e-c*d)*b*e)^(1/2))*b^3*c^2*d^2*e^2-440*(
(c*x+b)/b)^(1/2)*(-(e*x+d)/(b*e-c*d)*c)^(1/2)*(-1/b*c*x)^(1/2)*EllipticE(((c*x+b)/b)^(1/2),(1/(b*e-c*d)*b*e)^(
1/2))*b^2*c^3*d^3*e+176*((c*x+b)/b)^(1/2)*(-(e*x+d)/(b*e-c*d)*c)^(1/2)*(-1/b*c*x)^(1/2)*EllipticE(((c*x+b)/b)^
(1/2),(1/(b*e-c*d)*b*e)^(1/2))*b*c^4*d^4-3*b*c^4*e^4*x^4+81*c^5*d*e^3*x^4+6*b^2*c^3*e^4*x^3-26*b*c^4*d*e^3*x^3
+188*c^5*d^2*e^2*x^3+24*b^3*c^2*e^4*x^2-83*b^2*c^3*d*e^3*x^2+99*b*c^4*d^2*e^2*x^2+122*c^5*d^3*e*x^2+24*b^3*c^2
*d*e^3*x-89*b^2*c^3*d^2*e^2*x+122*b*c^4*d^3*e*x)/c^5/x/(c*e*x^2+b*e*x+c*d*x+b*d)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (e x + d\right )}^{\frac {7}{2}}}{\sqrt {c x^{2} + b x}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((e*x + d)^(7/2)/sqrt(c*x^2 + b*x), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (d + e x\right )^{\frac {7}{2}}}{\sqrt {x \left (b + c x\right )}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral((d + e*x)**(7/2)/sqrt(x*(b + c*x)), x)

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